Modeling Financial Market Volatility: A Component Model ... · GARCH is a state-of-the-art volatility model that uses realized measures of volatility for predicting daily latent volatility

Modeling Financial Market Volatility:

A Component Model Perspective

2018-1

Johan Stax Jakobsen

PhD Thesis

DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS

AARHUS BSS � AARHUS UNIVERSITY � DENMARK

MODELING FINANCIAL MARKET VOLATILITY: A

COMPONENT MODEL PERSPECTIVE

By Johan Stax Jakobsen

A PhD thesis submitted to

School of Business and Social Sciences, Aarhus University,

in partial fulfilment of the requirements of

the PhD degree in

Economics and Business Economics

January 2018

CREATESCenter for Research in Econometric Analysis of Time Series

This version: January 17, 2018 © Johan Stax Jakobsen

PREFACE

This dissertation is the final product of four years of PhD studies at the Department of

Economics and Business Economics, Aarhus School of Business and Social Sciences,

Aarhus University. I am thankful for the financial support and excellent research

facilities provided by the department that have made everything possible. I am also

grateful and proud for being a member of the renowned Center for Research in

Econometric Analysis of Time Series (CREATES) funded by the National Research

Foundation (DNRF78) that has allowed me to attend inspiring courses, taught by

some of the world’s leading scholars, and interesting seminars. Also, the rich group of

researchers has given me the option of valuable feedback for almost any problem.

I will furthermore acknowledge the external support from Augustinus Fonden, Etly

og Jørgen Stjerngrens Fond, Knud Højgaards Fond, Oticon Fonden and Rudolph

Als Fondet that made my two stays at Queensland University of Technology (QUT)

possible.

I would like to extend my gratitude to a number of people that have helped me

and made my life the last four years endlessly more pleasant than it otherwise would

have been without them. First and foremost, I wish to thank my main supervisor Prof.

Timo Teräsvirta who has shown great patience with me and always provided valuable

feedback to paper drafts and research ideas. His kind and insightful help will never

be forgotten.

During the winter months of the last two years, I had the pleasure of visiting

Prof. Stan Hurn and Dr. Annastiina Silvennoinen at the School of Economics and

Finance, QUT in Brisbane, Australia. I appreciate the hospitality and generosity of

the faculty and hope that I someday will return. In addition to Prof. Stan Hurn and Dr.

Annastiina Silvennoinen, I would, in particular, like to thank Prof. Adam Clements

and Prof. Russell Davidson for enlightening discussions. Also, I am grateful to John

Polichronis for allowing me to spend Christmas with him and his family while away

from my own, and to the PhD students at QUT for a lot of memorable weekend trips,

discussions about French politics and much more.

I appreciate the abundance of great colleagues at the Department of Economics

and Business Economics and CREATES. I am grateful to all of them for creating

an outstanding academic and social environment. Special thanks go to Prof. Niels

Haldrup for establishing and directing CREATES and to Solveig Nygaard Sørensen

i

ii

for always being helpful, proof-reading papers and making CREATES run efficiently.

I would also like to thank my fellow PhD students and in particular Alexander, Bo,

Boris, Carsten, Christian, Daniel, Jakob, Jonas, Jorge, Kasper, Mikkel, Simon, Strange,

and Søren for many interesting conversations, social activities and so much more.

It would not have been the same without you. Many of you will be close and dear

friends for the rest of my life.

I would like to put special thanks forward to a group of people that I hold close

to my heart. To Boris, my Bulgarian friend; I promise that I will continue to be a

great friend of you and your country. To Nicolai, my study mate from the first year

at the university and business partner; I am sure that the future will bring plenty of

new projects. To Daniel, my co-author with whom I share many interests; I hope

that the future will bring scientific contributions and plenty of fishing trips. Finally,

I would express my gratitude towards friends and family. Your unconditional love

and support mean everything to me. Special thanks to my twin brother Thomas for

sharing so many great experiences with me. In particular, countless fishing trips have

been peaceful escapes from the hazels and problems encountered during the last

four years.

Johan Stax Jakobsen

Aarhus, November 2017

UPDATED PREFACE

The predefence took place on December 21, 2017. I am grateful to the members

of the assessment committee consisting of Asger Lunde (Aarhus University and

CREATES), Peter Reinhardt Hansen (University of North Carolina and CREATES), and

Esther Ruiz Ortega (Charles III University of Madrid) for their valuable comments

and suggestions. Some of the suggestions have been incorporated into the present

version of the dissertation while other remain for future research.

Johan Stax Jakobsen

Aarhus, January 2018

iii

CONTENTS

Summary vii

Danish summary xi

1 Volatility persistence in the Realized Exponential GARCH model 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Persistence in a multiplicative REGARCH . . . . . . . . . . . . . . . . 4

1.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Realized EGARCH models with time-varying unconditional variance 652.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.2 Four new Realized EGARCH models . . . . . . . . . . . . . . . . . . . 68

2.3 Estimation and inference . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.5 A VaR framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.6 Empirical application to stock market volatility . . . . . . . . . . . . . 78

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3 Introducing macro-finance variables into the Realized EGARCH frame-work 1093.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.2 Realized measures of volatility . . . . . . . . . . . . . . . . . . . . . . 111

3.3 Modeling framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.5 Forecasting methodology . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.6 Forecast evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.7 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

v

vi CONTENTS

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

SUMMARY

This dissertation presents three self-contained papers on the modeling of latent

volatility of financial return series that is one of the most scrutinized research areas

in financial econometrics. This is with good reason since proper understanding

and modeling of volatility are of practical importance in the financial industry in

relation to for example risk management, portfolio allocation and pricing of financial

instruments. The AutoRegressive Conditional Heteroskedasticity (ARCH) model of

Engle (1982) and the Generalized ARCH (GARCH) model of Bollerslev (1986) have

fostered a huge and successful literature able to explain numerous stylized facts

of financial return series such as volatility clustering (Mandelbrot, 1963) and the

leverage effect (Black, 1976). Research has extended the early models in numerous

ways to deal with issues such as structural breaks, long-range dependence, time-

varying parameters, etc.

A successful part of the literature has focused on the development of multi-

component models to adequately capture the complex dynamics of financial volatil-

ity in a parsimonious way. Additive models with a quickly mean-reverting short-run

(high-frequency) and a persistent long-run (low-frequency) component were intro-

duced by Ding and Granger (1996) and Engle and Lee (1999). This notion of a short-

run and a long-run component is a common theme in the component literature and

rationalized using e.g. economic arguments (see e.g. Andersen and Bollerselv (1997)

(heterogeneity in news arrival) or Müller, Dacorogna, Davé, Olsen, Pictet, and von

Weizsäcker (1997) (investor time-horizon)) or structural breaks (see e.g. Lamoureux

and Lastrapes (1990)). More recently, there has been growing focus on multiplicative

component models (Engle and Rangle (2008) and Engle, Ghysels, and Sohn (2013),

among others). This dissertation in particular extends the literature on multiplicative

component models.

All chapters in this dissertation concern themselves with component structures

for modeling latent volatility. The papers extend the Realized Exponential GARCH

model (REGARCH) of Hansen and Huang (2016) in different directions. The RE-

GARCH is a state-of-the-art volatility model that uses realized measures of volatility

for predicting daily latent volatility. The first paper utilizes the idea of the GARCH-

MIDAS model of Engle et al. (2013) to increase the flexibility of the REGARCH in order

to accommodate evident long-range dependence in financial market volatility. The

vii

viii SUMMARY

second paper investigates whether insights from the literature on changing level of

unconditional variance in the GARCH framework are transferable to the Realized

EGARCH framework. The third paper examines the information content in different

macro-finance indicators for predicting latent volatility.

Chapter 1 - Volatility persistence in the Realized Exponential GARCH model (joint

work with Daniel Borup) - introduces parsimonious extensions of the Realized Expo-

nential GARCH model (REGARCH) of Hansen and Huang (2016) to capture evident

high-persistence in the conditional variance process. The extensions decompose the

conditional variance into a short-term and a long-term component. The latter utilizes

mixed-data sampling or a heterogeneous autoregressive structure, avoiding parame-

ter proliferation otherwise incurred by using the classical ARMA structures embedded

in the REGARCH. The proposed models are dynamically complete, which facilitates

multi-period forecasting. A thorough empirical investigation with the exchange-

traded index fund SPY that tracks the S&P 500 Index and 20 individual stocks shows

that our models better capture the autocorrelation structure in volatility. This leads

to substantial improvements in empirical fit and predictive ability (particular beyond

short horizons) relative to the original REGARCH.

Chapter 2 - Realized EGARCH models with time-varying unconditional variance (joint

work with Bo Laursen) - extends the Realized Exponential GARCH model (REGARCH)

of Hansen and Huang (2016) such that the unconditional variance is allowed to

change smoothly as a function of time. The model specification allows the conditional

variance to be multiplicatively decomposed into a stationary and non-stationary part.

The stationary part is specified as a zero mean REGARCH and the non-stationary

part represents the unconditional variance as a flexible function of time. We propose

four parametric alternatives inspired by the existing GARCH-literature: a smooth

transition time-varying structure, a flexible Fourier form, a quadratic spline, and a

cubic spline.

An application using data on the exchange-traded index fund SPY tests the models

empirically with both a forecasting and a Value-at-Risk exercise. The analysis shows

that the introduction of a non-stationary component modeled as a function of time

improves the in-sample fit of the model, but generally fails to provide out-of-sample

improvements.

Chapter 3 - Introducing macro-finance variables into the Realized EGARCH frame-

work - proposes two ways of including macro-finance indicators into the Realized

EGARCH model (REGARCH) of Hansen and Huang (2016). First, an additive speci-

fication, where the exogenous variables are added directly to the GARCH equation.

Secondly, a multiplicative component structure that separates the latent volatility

into a part modeled as a zero mean REGARCH and a part modeling the baseline

ix

volatility as a functions of the exogenous variables.

An empirical investigation with the exchange-traded index fund SPY that tracks

the S&P 500 Index and 20 individual stocks involving three macro-finance variables:

the policy uncertainty index of Baker, Bloom, and Davis (2016), the Arouba-Diebold-

Scotti business condition index of Aruoba, Diebold, and Scotti (2009) and VIX leads

to several interesting results. For the multiplicative decomposition, we realize large

in-sample and modest short horizon out-of-sample gains from including VIX as a

covariate, while the gains are smaller for the additive specification. This stipulates

that a multiplicative specification may be the preferred avenue when incorporating

implied volatility in GARCH type models. For ADS and EPU, we also find more mod-

est evidence of superior in-sample performance, but close to none out-of-sample

gains. Furthermore, our results corroborate that the additional information content

from including exogenous covariates is much smaller when working in a framework

utilizing realized measures of volatility.

x SUMMARY

References

Andersen, T. G., Bollerselv, T., 1997. Heterogeneous information arrivals and return

volatility dynamics: Uncovering the long-run in high frequency returns. The Journal

of Finance 52 (3), 975–1005.

Aruoba, S. B., Diebold, F. X., Scotti, C., 2009. Real-time measurement of business

conditions. Journal of Business & Economic Statistics 27 (4), 417–427.

Baker, S. R., Bloom, N., Davis, S. J., 2016. Measuring economic policy uncertainty. The

Quarterly Journal of Economics.

Black, F., 1976. Studies of stock market volatility changes. Proceedings of the American

Statistical Association, Business and Economics Statistics Section, 177–181.

Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal

of Econometrics 31 (3), 307–327.

Ding, Z., Granger, C. W., 1996. Modeling volatility persistence of speculative returns:

A new approach. Journal of Econometrics 73 (1), 185 – 215.

Engle, R. F., 1982. Autoregressive conditional heteroskedasticity with estimates of the

variance of united kingdom inflation. Econometrica 50 (4), 987–1008.

Engle, R. F., Ghysels, E., Sohn, B., 2013. Stock market volatility and macroeconomic

fundamentals. The Review of Economics and Statistics 95 (3), 776–797.

Engle, R. F., Lee, G., 1999. A long-run and short-run component model of stock

return volatility. In R. F. Engle and H. White (eds.), Cointegration, Causality, and

Forecasting: A Festschrift in Honour of Clive WJ Granger, 475–497.

Engle, R. F., Rangle, J. G., 2008. The Spline-GARCH model for low-frequency volatility

and its global macroeconomic causes. The Review of Financial Studies 21 (3),

1187–1222.

Hansen, P. R., Huang, Z., 2016. Exponential GARCH modeling with realized measures

of volatility. Journal of Business and Economic Statistics 34 (2), 269–287.

Lamoureux, C. G., Lastrapes, W. D., 1990. Persistence in variance, structural change,

and the GARCH model. Journal of Business & Economic Statistics 8, 225–234.

Mandelbrot, B., 1963. The variation of certain speculative prices. The Journal of

Business 36, 394–394.

Müller, U. A., Dacorogna, M. M., Davé, R. D., Olsen, R. B., Pictet, O. V., von Weizsäcker,

J. E., 1997. Volatilities of different time resolutions - analyzing the dynamics of

market components. Journal of Empirical Finance 4 (2), 213 – 239.

DANISH SUMMARY

Denne afhandling præsenterer tre selvstændige artikler omhandlende modellering af

latent volatilitet af finansielle afkastserier, der er et af de vigtigste forskningsområder

inden for finansiel økonometri. Dette er med god grund, da forståelse og modellering

af volatilitet er af praktisk relevans for den finansielle sektor inden for områder såsom

risikostyring, porteføljeallokering og prisfastsættelse af finansielle instrumenter. Den

AutoRegressive Conditional Heteroskedasticity (ARCH) model af Engle (1982) og

den Generalized ARCH (GARCH) model af Bollerslev (1986) har inspireret en stor og

succesfuld litteratur, der er i stand til at forklare en række karakteristika for finansielle

afkastserier såsom volatilitetsklynger (Mandelbrot, 1963) og gearingseffekten (Black,

1976). Forskningen har videreudviklet de tidlige modeller i et utal af retninger for

at tage højde for en række problemer som f.eks. strukturelle brud, long-memory og

tidsvarierende parametre, mv.

En succesfuld del af litteraturen har fokuseret på udviklingen af komponentmo-

deller, der på en simpel måde er i stand til at beskrive de komplekse dynamikker

for finansiel volatilitet. Additive modeller med en kortsigtet (højfrekvent) og persi-

stent langsigtet (lavfrekvent) komponent blev introduceret af Ding og Granger (1996)

og Engle og Lee (1999). Idéen om en kortsigtet og langsigtet komponent er et gen-

nemgående tema i komponentlitteraturen og rationaliseret ved bl.a. økonomiske

argumenter (se f.eks. Andersen og Bollerselv (1997) (heterogenitet i ankomsten af

nyheder) eller Müller et al. (1997) (forskelle i investorers tidshorisont)) eller struk-

turelle brud (se f.eks. Lamoureux og Lastrapes (1990)). Mere nyligt har der været

et voksende fokus på multiplikative komponentmodeller (Engle og Rangle (2008)

og Engle et al. (2013), med flere). Denne afhandling udvider primært litteraturen

vedrørende multiplikative komponentmodeller.

Alle kapitler i denne afhandling omhandler komponentstrukturer for modellering

af latent volatilitet. Artiklerne udvider den Realized Exponential GARCH model (RE-

GARCH) af Hansen og Huang (2016) i forskellige retninger. Denne model er en ’state

of the art’ volatilitetsmodel som bruger realiserede mål for volatilitet til at prediktere

daglig volatilitet. Den første artikel udnytter ideen bag GARCH-MIDAS modellen af

Engle et al. (2013) således, at den foreslåede model er i stand til at approksimere den

evidente long-range afhængighed i finansiel volatilitet. Den anden artikel undersøger,

om resultaterne fra litteraturen omkring niveauskift i den ubetingede forventning

xi

xii DANISH SUMMARY

af varians kan overføres til REGARCH-litteraturen. Den tredje artikel undersøger

informationsindholdet i forskellige makroøkonomiske og finansielle indikatorer for

prediktering af latent volatilitet.

Kapitel 1 - Volatility persistence in the Realized Exponential GARCH model (fælles

med Daniel Borup) - introducerer en simpel udvidelse af den Realized Exponential

GARCH model (REGARCH) af Hansen og Huang (2016), der er i stand til at beskrive

den evidente long-range afhængighed i den betingede varians-proces. Udvidelsen

dekomponerer den betingede varians i en kortsigtet og langsigtet del. Den langsig-

tede komponent udnytter et MIDAS-filter eller en heterogen autoregressiv struktur

således, at det store antal parametre krævet ved den klassiske ARMA struktur inde-

holdt i REGARCH undgås. De foreslåede modeller er dynamisk komplette, hvilket

muliggør forecasting flere perioder frem. En gennemarbejdet empirisk undersøgelse,

der anvender data fra den børshandlede indeksfond SPY, som tracker S&P 500 indek-

set, og 20 individuelle aktier, viser, at de nye modeller bedre er i stand til at matche

autokorrelationsstrukturen for volatilitet. Dette medfører substantiel forbedring af

empirisk fit og prediktiv evne.

Kapitel 2 - Realized EGARCH models with time-varying unconditional variance (fæl-

les med Bo Laursen) - udvider den Realized Exponential GARCH model (REGARCH)

af Hansen og Huang (2016) således, at den ubetingede varians er en funktion af

tid. Modelspecifikationen tillader en multiplikativ dekomponering af den betingede

varians i en stationær og ikke-stationær del. Den stationære del er specificeret som

en REGARCH med forventet værdi lig med nul og den ikke-stationære repræsenterer

den ubetingede varians som en fleksibel funktion af tid. Vi foreslår fire forskellige

parametriske alternativer inspireret af den eksisterende GARCH-litteratur: En smooth

transiton struktur, fleksibel fourier form, en kvadratisk spline og en kubisk spline. En

applikation med data fra den børshandlede fond SPY tester modellerne empirisk ved

hjælp af både en forecasting og en Value-at-Risk øvelse. Analysen viser, at introduk-

tionen af en ikke-stationær komponent specificeret som en funktion af tid forbedrer

in-sample fit, men generelt ikke medfører bedre out-of-sample fit.

Kapitel 3 - Introducing macro-finance variables into the Realized EGARCH framework -

foreslår to måder at inkludere makroøkonomiske og finansielle variable i den Realized

Exponential GARCH model (REGARCH) af Hansen og Huang (2016) og den robuste

udvidelse af Banulescu, Hansen, Huang, og Matei (2014). Den første specifikation er

en additiv version, hvor de eksogene variable er tilføjet GARCH-ligningen. Den anden

specifikation er en multiplikativ komponent struktur, som separerer den latente

volatilitet i en del beskrevet ved en REGARCH med forventet værdi lig med nul og en

del, der specificere baseline volatiliteten som en funktion af de eksogene variable.

En empirisk undersøgelse på SPY, som tracker S&P 500 indekset, og 20 individuel-

xiii

le aktier med tre forskellige makroøkonomiske og finansielle indikatorer: Economic

Policy Uncertainty index (EPU) af Baker et al. (2016), Arouba-Diebold-Scotti business

condition index (ADS) af Aruoba et al. (2009) og VIX fører til flere interessante resulta-

ter. For den multiplikative dekomponering, så finder vi store in-sample og moderat

out-of-sample forbedringer ved at inkludere VIX som en kovariat, mens forbedringer-

ne er mindre for den additive specification. Dette understreger, at den multiplikative

specifikation kan være at foretrække, når man anvender implied volatilitet i GARCH

modeller. For ADS og EPU, så finder vi også moderate evidens for bedre in-sample

performance, men tæt på ingen forbedringer out-of-sample. Derudover, så underbyg-

ger vores resultater, at den yderligere information fra inklusion af eksogene kovariater

er betydelig mindre, når man arbejder i et framework som udnytter realiserede mål

for volatilitet.

xiv DANISH SUMMARY

Litteratur

Andersen, T. G., Bollerselv, T., 1997. Heterogeneous information arrivals and return

volatility dynamics: Uncovering the long-run in high frequency returns. The Journal

of Finance 52 (3), 975–1005.





Banulescu, G. D., Hansen, P. R., Huang, Z., Matei, M., 2014. Volatility during the finan-

cial crisis through the lens of high frequency data: A Realized EGARCH approach.



Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal

of Econometrics 31 (3), 307–327.

Ding, Z., Granger, C. W., 1996. Modeling volatility persistence of speculative returns:

A new approach. Journal of Econometrics 73 (1), 185 – 215.

Engle, R. F., 1982. Autoregressive conditional heteroskedasticity with estimates of the

variance of united kingdom inflation. Econometrica 50 (4), 987–1008.






Engle, R. F., Rangle, J. G., 2008. The Spline-GARCH model for low-frequency volatility


1187–1222.




and the GARCH model. Journal of Business & Economic Statistics 8, 225–234.


Business 36, 394–394.

xv

Müller, U. A., Dacorogna, M. M., Davé, R. D., Olsen, R. B., Pictet, O. V., von Weizsäcker,

J. E., 1997. Volatilities of different time resolutions - analyzing the dynamics of

market components. Journal of Empirical Finance 4 (2), 213 – 239.

C H A P T E R 1VOLATILITY PERSISTENCE IN THE REALIZED

EXPONENTIAL GARCH MODEL

Daniel BorupAarhus University and CREATES

Johan Stax JakobsenAarhus University and CREATES

Abstract

We introduce parsimonious extensions of the Realized Exponential GARCH model

(REGARCH) to capture evident long-range dependence in the conditional variance

process. The extensions decompose conditional variance into a short-term and a

long-term component. The latter utilizes mixed-data sampling or a heterogeneous

autoregressive structure, avoiding parameter proliferation otherwise incurred by us-

ing the classical ARMA structures embedded in the REGARCH. The proposed models

are dynamically complete, facilitating multi-period forecasting. A thorough empirical

investigation with an exchange-traded fund that tracks the S&P 500 Index and 20

individual stocks shows that our models better capture the autocorrelation structure

of volatility. This leads to substantial improvements in empirical fit and predictive

ability (particularly beyond short horizons) relative to the original REGARCH.

1

2 CHAPTER 1.

1.1 Introduction

The Realized GARCH model (RGARCH) and Realized Exponential GARCH model1

(REGARCH) (Hansen, Huang, and Shek, 2012; Hansen and Huang, 2016) provide an

advantageous structure for the joint modeling of stock returns and realized measures

of their volatility. The models facilitate exploitation of granular information in high-

frequency data by including realized measures, which constitute a much stronger

signal of latent volatility than squared returns (Andersen, Bollerslev, Diebold, and

Labys, 2001, 2003). Various models have been proposed to utilize similar information

with notable innovations including the GARCH-X model (Engle, 2002), the multi-

plicative error model (Engle and Gallo, 2006), and the HEAVY model (Shephard and

Sheppard, 2010).

It is, however, generally recognized that volatility is highly persistent. This persistence

is typically documented via a positive and slowly decaying autocorrelation function

(long-range dependence) or a persistence parameter close to unity, known as the

"integrated GARCH effect". Despite the empirical success of the R(E)GARCH mod-

els, these models do not adequately capture this dependency structure in volatility

(both latent and realized) without proliferation in parameters. Indeed, Hansen and

Huang (2016) point out that the REGARCH does a good job modeling the returns,

but falls short in terms of describing the dynamic properties of the realized mea-

sure. In the class of GARCH models without realized measures, several contributions

have been made to account for these two stylized features. A few notable references

include the Integrated GARCH (Engle and Bollerselv, 1986), the Fractionally Inte-

grated (E)GARCH (Baillie, Bollerslev, and Mikkelsen, 1996; Bollerslev and Mikkelsen,

1996), FIAPARCH (Tse, 1998), regime-switching GARCH (Diebold and Inoue, 2001),

HYGARCH (Davidson, 2004), the Spline-GARCH (Engle and Rangel, 2008), and the

time-varying component GJR-GARCH (Amado and Teräsvirta, 2013). In the class

of R(E)GARCH models, Vander Elst (2015) proposed a fractionally integrated RE-

GARCH, whereas Huang, Liu, and Wang (2016) suggested the addition of a weekly

and a monthly averaged realized measure in the GARCH equation of the RGARCH.

In this paper, we introduce parsimonious extensions of the REGARCH to capture this

evident high persistence by means of a decomposition of the conditional variance. We

utilize a multiplicative decomposition into a short-term and long-term component.

This structure is particularly useful since it enables explicit modelling of a "baseline

volatility", whose level arguably shifts over time, and is the basis around which short-

term movements occur. Such as structure is motivated by Mikosch and Starica (2004),

who show that long-range dependence and the integrated GARCH effect may be

1The REGARCH is a generalization of the RGARCH model with a more flexible specification of theleverage function supposed to better capture the asymmetric relationship between stock returns andvolatility.

1.1. INTRODUCTION 3

explained by level shifts in the unconditional variance, and by Amado and Teräsvirta

(2013), who support this finding empirically in a multiplicative component version of

the GJR-GARCH model.2

The idea of decomposing volatility originates from Engle and Lee (1999) and has pri-

marily been used to empirically support countercyclicality in stock market volatility

(see e.g. Engle et al. (2013) and Dominicy and Vander Elst (2015)). The multiplica-

tive component structure (see e.g. Feng (2004), Engle and Rangel (2008), Engle et al.

(2013) and Laursen and Jakobsen (2017)) is appealing since it is intuitive and facili-

tates parsimonious specifications of a slow-moving component in volatility. Moreover,

it allows for great flexibility as opposed to formal long-memory models employing,

e.g., fractional integration. Whether the high persistence arises due to structural

breaks, fractional integration or another source (see e.g. Lamoureux and Lastrapes

(1990), Diebold and Inoue (2001), Hillebrand (2005), McCloskey and Perron (2013),

and Varneskov and Perron (2017)) our proposed models are able to reproduce the

high persistence of volatility observed in stock return data and alleviate the integrated

GARCH effect, without formally belonging to the class of long-memory models. This

plays an important role in stationarity of the short-term component and existence

of the unconditional variance (which requires the persistence parameter |β| < 1),

but also provides a means to obtain improved multi-step forecasts by reducing the

long-lasting impact of the short-term component and its innovations (via faster con-

vergence to the baseline volatility).

When specifying our models, we retain the dynamics of the short-term component

like those from a first-order REGARCH, but model the long-term component either

via mixed-data sampling (MIDAS) or a heterogeneous autoregressive (HAR) structure.

The former specifies the slow-moving component as a weighted average of weekly

or monthly aggregates of the realized measure with the backward-looking window

and weights estimated from the data. The MIDAS concept was originally introduced

in a regression framework (Ghysels, Santa-Clara, and Valkanov, 2004, 2005; Ghysels,

Sinko, and Valkanov, 2007), allowing for the left-hand and right-hand variables to

be sampled at different frequencies. It has recently been incorporated successfully

into the GARCH framework with the GARCH-MIDAS proposal of Engle et al. (2013).

The latter is motivated by the simple, yet empirically successful HAR model by Corsi

(2009), which approximates the dependencies in volatility by a simple additive cas-

cade structure of a daily, weekly and monthly component of realized measures. Both

our extensions introduce only two or three additional parameters, hence avoid pa-

rameter proliferation otherwise incurred by means of the classical ARMA structures

embedded in the original REGARCH. Moreover, they remain dynamically complete.

2Conrad and Kleen (2016) also show formally that the autocorrelation function of squared returns isbetter captured by a multiplicative GARCH specification rather than its nested GARCH(1,1) model, arisingfrom the persistence in the long-term component.

4 CHAPTER 1.

That is, the models fully characterize the dynamic properties of all variables included

in the model. This property is especially relevant for forecasting purposes, since

it allows for multi-period forecasting. This contrasts GARCH-X models, which only

provide forecasts one period into the future, and related extensions including macroe-

conomic factors who typically rely on questionable assumptions about the included

variables’ dynamics.3

We apply our REGARCH-MIDAS and REGARCH-HAR to the exchange-traded in-

dex fund, SPY, which tracks the S&P 500 Index, and 20 individual stocks and compare

their performances to a quadratic REGARCH-Spline and a fractionally integrated

REGARCH, the FloEGARCH, (Vander Elst, 2015). We find that both our proposed

models better capture the autocorrelation structure of latent and realized volatility

relative to the original REGARCH, which is only able to capture the dependency

over the very short term. This leads to substantial improvements in empirical fit

(log-likelihood and information criteria) and predictive ability, particularly beyond

shorter horizons, when benchmarked to the original REGARCH. We document, ad-

ditionally, that the backward-looking horizon of the HAR specification is too short

to sufficiently capture autocorrelation beyond approximately one month. While the

REGARCH-Spline comes short relative to our proposals (with four-five extra param-

eters), the FloEGARCH performs well. It does, however, not perform better than

our best-performing REGARCH-MIDAS specifications in-sample and lack predictive

accuracy in the short-term. This leaves the REGARCH-MIDAS as a very attractive

model for capturing volatility persistence in the REGARCH framework and improving

forecasting performance.

The remainder of the paper is laid out as follows. Section 1.2 introduces our ex-

tensions to the original REGARCH: the REGARCH-MIDAS and the REGARCH-HAR.

Section 3.4 outlines the associated estimation procedure. Section 1.4 summarizes our

data set, examines the empirical fit and predictive ability of our proposed models, and

introduces a procedure for generating multi-period forecasts. Section 1.5 concludes.

Technical details concerning Proposition 1 are presented in the Appendix.

1.2 Persistence in a multiplicative REGARCH

Let {rt } denote a time series of returns, {xt } a (vector) time series of realized measures,

and {Ft } a filtration so that {rt , xt } is adapted to Ft . We define the conditional mean

by µt = E[rt |Ft−1] and the conditional variance by σ2t = Var[rt |Ft−1]. Our aim is to

allow for more flexible dependence structures in the state-of-the-art specification of

3For instance, the assumption of a random walk (Dominicy and Vander Elst, 2015)), use of outside-generated forecasts (usually from a standard autoregressive specification) of the exogenous variables inthe model (Conrad and Loch, 2015) or the assumption that the long-term component is constant for theforecasting horizon (Engle et al., 2013).

1.2. PERSISTENCE IN A MULTIPLICATIVE REGARCH 5

conditional variance provided by the REGARCH of Hansen and Huang (2016). To that

end, we define

rt =µt +σt zt , (1.1)

where {zt } is an i.i.d. innovation process with zero mean and unit variance, and

assume that the conditional variance can be multiplicatively decomposed into two

components

σ2t = ht g t . (1.2)

We refer to ht as the short-term component, supposed to capture day-to-day (high-

frequency) fluctuations in the conditional variance (see e.g. Engle et al. (2013), and

Wang and Ghysels (2015)). On the contrary, g t is supposed to capture secular (low-

frequency) movements in the conditional variance, henceforth referred to as the

long-term component or baseline volatility. With the multiplicative decomposition

in (1.2), we extend a daily REGARCH(1,1) (with a single realized measure) to

rt =µt +σt zt , (1.3)

loght =β loght−1 +τ(zt−1)+αut−1, (1.4)

log xt = ξ+φ logσ2t +δ(zt )+ut , (1.5)

log g t =ω+ f (xt−2, xt−3, . . . ;η), (1.6)

where f (·;η) is a Ft−1-measurable function, which can be linear or non-linear. The

equations are labelled as the "return equation", the "GARCH equation", the "mea-

surement equation", and the "long-term equation", respectively. For identification

purposes, we have omitted an intercept in (1.4). The leverage functions, τ(·) and

δ(·), facilitate modeling of the dependence between return innovations and volatility

innovations shown to be empirically important (see e.g. Christensen, Nielsen, and

Zhu (2010)). In addition, they play an important role in making the assumption of

independence between zt and ut empirically realistic (Hansen and Huang, 2016).

We adopt the quadratic form of the leverage functions based on the second-order

Hermite polynomial,

τ(z) = τ1z +τ2(z2 −1), (1.7)

δ(z) = δ1z +δ2(z2 −1). (1.8)

The leverage functions have a flexible form and imply E[τ(z)

] = E[δ(z)

] = 0 when

E [z] = 0 and Var[z] = 1. Thus, if |β| < 1, our identification restriction implies that

E[loght

] = 0 such that E[

logσ2t

]= E

[log g t

].4 In the (Quasi-)Maximum Likelihood

analysis below, we employ a Gaussian specification like Hansen and Huang (2016)

4The GARCH equation implies that loght = β j loght− j +∑ j−1

i=0 βi [τ(zt−1−i )+αut−1−i

]such that

loght has a stationary representation if |β| < 1.

6 CHAPTER 1.

with zt ∼ N (0,1) and ut ∼ N (0,σ2u), and zt ,ut mutually and serially independent.5

We check the validity of this approach via a parametric bootstrap in Section 3.4 below.

The return and GARCH equation are canonical in the GARCH literature. In the re-

turn equation, the conditional mean, µt , may be modeled in various ways including

a GARCH-in-Mean specification or simply as a constant.6 Following the latter ap-

proach, we estimate the constant µt = µ. In our multiplicative specification, the

GARCH equation drives the dynamics of the high-frequency part of latent volatility.

The dynamics are specified as a slightly modified version of the EGARCH model

of Nelson (1991) (different leverage function) with the addition of the term αut−1

that relates the latent volatility with the innovation to the realized measure. Hence,

α represents how informative the realized measure is about future volatility. The

persistence parameter β can be interpreted as the AR-coefficient in an AR(1) model

for loght with innovations τ(zt−1)+αut−1.

The measurement equation is the true innovation in the R(E)GARCH, which makes

the model dynamically complete. The equation links the ex-post realized measure

with the ex-ante conditional variance. Discrepancies between the two measures are

expected, since the conditional variance (and returns) refers to a close-to-close mar-

ket interval, whereas the realized measure is computed from a shorter, open-to-close

market interval. Hence, the realized measure is expected to be smaller than the con-

ditional variance on average. Additionally, the realized measure may be an imperfect

measure of volatility. Therefore, the equation includes both a proportional, ξ, and an

exponential, φ, correction parameter. The innovation term, ut , can be seen as the

true difference between ex-ante and ex-post volatility.

Given the high persistence of the conditional variance (documented in the empirical

section below), simply including additional lags in the ARMA structure embedded

in the original REGARCH is not a viable solution, keeping parameter proliferation

in mind (cf. Section 1.4). Instead, we utilize the multiplicative component structure,

which is both intuitively appealing and maintain parsimony. This is motivated by

Mikosch and Starica (2004) who showed that the high persistence can be explained

by level shifts in the unconditional variance (see also Diebold (1986) and Lamoureux

and Lastrapes (1990)). On this basis, Amado and Teräsvirta (2013) proposed a multi-

plicative decomposition of the GJR-GARCH model, where the "baseline volatility"

changes deterministically according to the passage of time. We may, therefore, enable

5Watanabe (2012), Louzis, Xanthopoulos-Sisinis, and Refenes (2013) and Louzis, Xanthopoulos-Sisinis,and Refenes (2014) assumed a skewed t-distribution in their Value-at-Risk applications.

6The mean is typically modeled as a constant since stock market returns generally are found to be closeto serially uncorrelated, see e.g. Ding, Granger, and Engle (1993) and references therein. Sometimes theassumption of zero mean, µ= 0, is imposed for simplicity and may in fact generate better out-of-sampleperformance, see e.g. Hansen and Huang (2016). However, in option-pricing applications a GARCH-in-Mean specification is usually employed, see e.g. Huang, Wang, and Hansen (2017).


capturing high persistence via the structure proposed above, when the long-term

component in (1.6) is specified as a slow-moving baseline volatility around which

stationary short-term fluctuations occur via the standard GARCH equation. Natu-

rally, this interpretation (and the existence of the unconditional variance) depends

on whether |β| < 1 holds in practice, which may be questionable on the basis on

former evidence for the original REGARCH (confirmed in Section 1.4). However, this

integrated GARCH effect is alleviated in our proposed models, where β is notably

below unity.

Whether high persistence of the conditional variance process arises due to structural

breaks, fractional integration or any other source, the long-term component, if mod-

eled accurately, facilitates high persistence in the REGARCH framework. That is, we

do not explicitly take a stance on the reason for the presence of high persistence. We

resort to this approach rather than developing a formal long-memory model (see e.g.

Bollerslev and Mikkelsen (1996) and Vander Elst (2015)), since prevailing ambiguity

about the origination of long memory somewhat distorts the judgement on the cor-

rect formal modeling. There exists a long list of explanations for long memory in a

time series of which a few are; (i) cross-sectional aggregation of short-memory time

series (Granger, 1980; Abadir and Talmain, 2002; Zaffaroni, 2004; Haldrup and Valdés,

2017), (ii) temporal aggregation across mixed-frequency series (Chambers, 1998),

(iii) aggregation through networks (Schennach, 2013), (iv) hidden cross-section de-

pendence in large-dimensional vector autoregressive systems (Chevillon, Hecq, and

Laurent, 2015), (v) structural breaks (Granger and Ding, 1996; Parke, 1999; Diebold

and Inoue, 2001; Perron and Qu, 2007), (vi) certain types of nonlinearity (Davidson

and Sibbertsen, 2005; Miller and Park, 2010), and (vii) economic agents’ learning

(Chevillon and Mavroeidis, 2017). The various explanations do, however, not nec-

essarily imply the same type of long memory (see e.g. Haldrup and Valdés (2017)

for several definitions). For instance, Parke (1999) formalizes the relation between

structural changes and fractional integration, whereas the expectation formation of

economic agents in Chevillon and Mavroeidis (2017) do not yield fractional integra-

tion, but rather apparent or spurious long memory (see e.g. Davidson and Sibbertsen

(2005) and Haldrup and Kruse (2014)).

For the remainder of this paper, we assume for clarity of exposition that xt is one-

dimensional, containing a single (potentially robust) realized measure consistently

estimating integrated variance (see e.g. (Andersen et al., 2001, 2003)), such as the

realized variance or the realized kernel (Barndorff-Nielsen, Hansen, Lunde, and Shep-

hard, 2008).7 We facilitate level shifts in the baseline volatility via the function f (·;η),

7This assumption is without loss of generality in the sense that additional realized measures (and theirassociated measurement equations) can be added, though we still approximate the long-range dependenceusing only past information of the realized variance, realized kernel or another related consistent estimatorfor integrated variance.

8 CHAPTER 1.

which takes as input past values of the realized measure. We make the dependence

on η explicit in the function f (·;η), and prefer that it is low-dimensional. If f (·;η) is

constant, we obtain the REGARCH as a special case. If f (·;η) is time-varying, past

information may assist in capturing the dependency structure of conditional variance

better, potentially leading to improved in-sample and out-of-sample properties of the

models. We propose in the following sections two ways to parsimoniously formulate

f (·;η) using non-overlapping weekly and monthly averages of the realized measure

to be consistent with the idea of a slow-moving, low-frequency component.8 We

model low-frequency movements in conditional variance using (aggregates of) past

information of the realized measure rather than tying it to macroeconomic state vari-

ables as in Engle et al. (2013) and Dominicy and Vander Elst (2015). Besides proving

empirically preferable (see e.g. Andersen and Varneskov (2014)), such a procedure

renders the model in (1.3)-(1.6) complete with dynamic specifications of all variables

included in the model. Consequently, forecasting can be conducted on the basis of

the (jointly estimated) empirical dynamics, which stands in contrast to incomplete

specifications using exogenous information (from e.g. macroeconomic variables).

The latter usually relies on unrealistic assumptions on the dynamics of the exogenous

variables (e.g. random walks (Dominicy and Vander Elst, 2015)), outside-generated

forecasts (usually from a standard autoregressive specification) of the exogenous

variables in the model (Conrad and Loch, 2015) or the assumption that the long-term

component is constant for the forecasting horizon (Engle et al., 2013). We do, however,

emphasize that our proposed model accommodates well the inclusion of exogenous

information if deemed appropriate.

In the following, we introduce two ways of modeling the low-frequency component,

g t , via formulations of f (·;η) that parsimoniously enable high persistence in the RE-

GARCH formulation, leading to the REGARCH-MIDAS model and the REGARCH-HAR

model.

1.2.1 The Realized EGARCH-MIDAS model

Inspired by the GARCH-MIDAS model of Engle et al. (2013), we consider the following

MIDAS specification of the long-term component

log g t =ω+λK∑

k=1Γk

(γ)

y (N )t−1,k , (1.9)

where Γk(γ)

is a parametrized (by the vector γ) non-negative weighting function sat-

isfying the restriction∑K

k=1Γk(γ)= 1, and y (N )

t ,k = 1N

∑Ni=1 log xt−N (k−1)−i is an N -day

8Excluding information in the realized measure on day t −1 from the function f (·;η) is consistent withthe formulations in the GARCH-MIDAS framework of Engle et al. (2013). The idea is to separate the effectsof the realized measure into two, such that the day-to-day effects is (mainly) contained in the short-termcomponent ht via ut−1 and the long-term component captures the information contained in the realizedmeasure further back in time.


average of the logarithm of the realized measure. Hence, the value of N determines

the frequency of the data feeding into the low-frequency component. We consider in

the following N ∈ {5,22}, corresponding to weekly and monthly averages.

By estimatingγ, for a given weighting function and choice of K , the term∑K

k=1Γk(γ)

yt−1,k

acts as a filter, which extracts the empirically relevant information from past values

of the realized measure with assigned importance given by the estimated λ. That is,

the lag selection process is allowed to be data driven. In practice, we need to choose

a value for K and a weighting scheme. Conventional weighting schemes are based on

the exponential, exponential Almon lag, or the beta-weight specification. A detailed

discussion can be found in Ghysels et al. (2007), who studied the choice of weighting

function in the context of MIDAS regression models. We employ in the following the

two-parameter beta-weight specification defined by

Γk(γ1,γ2

)= (k/K

)γ1−1 (1−k/K

)γ2−1∑Kj=1

(j /K

)γ1−1 (1− j /K

)γ2−1 (1.10)

due to its flexible form. We restrict γ2 > 1, which ensures a monotonically decreas-

ing weighting scheme and avoid counterintuitive schemes with, e.g., most weight

assigned to the most distant observation (see Engle et al. (2013) and Asgharian,

Christiansen, and Hou (2016) for a similar restriction).9 We then examine a single-

parameter case in which we impose γ1 = 1 and a case where γ1 is a free parameter.

More rich structures for the weighting scheme can obviously be considered by intro-

ducing additional parameters, but we will not explore that route, since one important

aim of the MIDAS models is parsimony.

As long as the weighting function is reasonably flexible, the choice of lag length

of the MIDAS component, K , is of limited importance if chosen reasonably large. The

reason is that the estimated γ assigns the relevant weights to each lag simultaneously

while estimating the entire model. Should one want to determine an ‘optimal’ K , we

simply suggest to estimate the model for a range of values of K and choose that for

which higher values lead to no sizeable gain in the maximized log-likelihood value

(see also the empirical section below).

The REGARCH-MIDAS framework proposed here is easily extendable in several

ways. For instance, a multivariate extension is simply obtained by adding additional

MIDAS components to (1.9). Hence, we may add additional high-frequency based

measures such as the daily range, the realized quarticity (see e.g. Bollerslev, Patton,

and Quaedvlieg (2016)) or additional, different estimators of integrated variance. If

the relationship between macroeconomic variables and volatility is of interest, one

may also include indicators such as GDP and production growth rates, or inflation

9We found in our empirical section below that this restriction was only binding in a few cases.

10 CHAPTER 1.

rates (see e.g. Engle et al. (2013)), despite them being of different frequencies. An-

other direction of interest is the understanding of different aggregation schemes of

higher-frequency variables. For example, by considering a rolling window of non-

overlapping averages, our approach differs slightly from that initially proposed in

Engle et al. (2013) who used overlapping averages in the GARCH-MIDAS context.

1.2.2 The Realized EGARCH-HAR model

Inspired by Corsi (2009), we suggest the following HAR-specification of the long-term

component

log g t =ω+γ11

5

5∑i=1

log xt−i−1 +γ21

22

22∑i=i

log xt−i−1. (1.11)

The argument for this particular lag structure is motived by the heterogeneous market

hypothesis (Müller et al., 1993), which suggests an account of the heterogeneity

in information arrival due to e.g. different trading frequencies of financial market

participants. See Corsi (2009) for a more detailed discussion. This particular choice

of lag structure including the lagged weekly and monthly average of the logarithm

of the realized measure is intuitive and has been empirically successful, but is not

data driven as opposed to the MIDAS lag structure. The lag structure can be seen as a

special case of the step-function MIDAS specification in Forsberg and Ghysels (2007),

which was, indeed, inspired by Corsi (2009).

1.3 Estimation

We estimate the models using (Quasi-)Maximum Likelihood (QML) consistent with

the procedures in Hansen et al. (2012) and Hansen and Huang (2016). The log-

likelihood function can be factorized as

L(r, x;θ) =T∑

t=1`t (rt , xt ;θ) =

T∑t=1

[`t (rt ;θ)+`t (xt |rt ;θ)], (1.12)

where θ = (µ,β,τ1,τ2,α,ξ,φ,δ1,δ2,ω,η,σ2u)′ is the vector of parameters in (1.3)-(1.6),

and `t (rt ;θ) is the partial log-likelihood, measuring the goodness of fit of the return

distribution. Given the distributional assumptions, zt ∼ N (0,1) and ut ∼ N (0,σ2u),

and zt ,ut mutually and serially independent, we have

`t (rt ;θ) =−1

2

[log2π+ logσ2

t + z2t

], (1.13)

`t (xt |rt ;θ) =−1

2

[log2π+ logσ2

u + u2t

σ2u

], (1.14)

where zt = zt (θ) = (rt −µ)/σt . We initialize the conditional variance process to be

equal to its unconditional mean, i.e. logh0 = 0. Alternatively, one can treat logh0 as an

1.3. ESTIMATION 11

unknown parameter and estimate it as in Hansen and Huang (2016), who show that

the initial value is asymptotically negligible. To initialize the long-term component,

log g t , at the beginning of the sample, we simply set past values of log xt equal to

log x1 for the length of the backward-looking horizon in the MIDAS-filter. This is

done to avoid giving our proposed models an unfair advantage by utilizing more

data than the benchmark REGARCH. To avoid inferior local optima in the numerical

optimization, we perturb starting values and re-estimate the parameters for each

perturbation.

1.3.1 Score function

Since the scores define the first order conditions for the maximum-likelihood esti-

mator and facilitate direct computation of standard errors for the coefficients, we

present closed-form expressions for the scores in the following. To simplify nota-

tion, we write τ(z) = τ′a(z) and δ(z) = δ′b(z) with a(z) = b(z) =(z, z2 −1

)′, and

let azt = ∂a(zt )/∂zt and bzt = ∂b(zt )/∂zt . In addition, we define θ1 = (β,τ1,τ2,α)′,θ2 = (ξ,φ,δ1,δ2)′, mt = (loght , a(zt )′,ut )′, and nt = (1, logσ2

t ,b(zt )′)′.

Proposition 1 (Scores). The scores, ∂`∂θ =∑Tt=1

∂`t∂θ , are given from

∂`t

∂θ=

B(zt ,ut )hµ,t −[

zt −δ′ ut

σ2u

bzt

]1σt

B(zt ,ut )hθ1,t

B(zt ,ut )hθ2,t + ut

σ2u

nt

B(zt ,ut )hω,t +D(zt ,ut )gω,t

B(zt ,ut )hη,t +D(zt ,ut )gη,t

12

u2t −σ2

u

σ4u

, (1.15)

where

A(zt ) = ∂ loght+1

∂ loght= (

β−αφ)+ 1

2

(αδ′bzt −τ′azt

)zt , (1.16)

B(zt ,ut ) = ∂`t

∂ loght=−1

2

[(1− z2

t )+ ut

σ2u

(δ′bzt zt −2φ

)], (1.17)

C (zt ) = ∂ loght+1

∂ log g t=−αφ+ 1

2


)zt , (1.18)

D(zt ,ut ) = ∂`t

∂ log g t=−1

2

[(1− z2

t )+ ut

σ2u

(δ′bzt zt −2φ

)]. (1.19)

12 CHAPTER 1.

Furthermore, we have

hµ,t+1 = ∂ loght+1

∂µ= A(zt )hµ,t +


) 1

σt, (1.20)

hθ1,t+1 = ∂ loght+1

∂θ1= A(zt )hθ1,t +mt , (1.21)


∂θ2= A(zt )hθ2,t +αnt , (1.22)

hω,t+1 = ∂ loght+1

∂ω= A(zt )hω,t +C (zt ), (1.23)

hη,t+1 = ∂ loght+1

∂η= A(zt )hη,t +C (zt )gη,t , (1.24)

where gη,t depends on the specification of f (·;η) and is therefore presented in Appendix

A.1.

By corollary, the score function is a Martingale Difference Sequence (MDS), pro-

vided that E[zt |Ft−1

]= 0, E[

z2t |Ft−1

]= 1, E

[ut |zt ,Ft−1

]= 0, and E[

u2t |zt ,Ft−1

]=σ2

u ,

which is useful for future analysis of the asymptotic properties of the QML estima-

tor.10

1.3.2 Asymptotic Properties

It is commonly acknowledged that the asymptotic analysis of even conventional

GARCH models is challenging (see e.g. Francq and Zakoïan (2010)), causing most

models to be introduced without accompanying asymptotic properties of their esti-

mators. Most recently, the asymptotic theory of the EGARCH(1,1) model was devel-

oped by Wintenberger (2013). Han and Kristensen (2014) and Han (2015) conclude

that inference for the QML estimator is quite robust to the level of persistence in

covariates included in GARCH-X models, irrespective of them being stationary or not.

However, no such analysis has, to our knowledge, been developed for the original RE-

GARCH. The MDS properties following Proposition 1 apply to the original REGARCH

as well, leading Hansen and Huang (2016) to conjecture that the limiting distribution

of the estimators is normal. We follow the same route and leave the development of

the asymptotic theory for estimators of the REGARCH-MIDAS and REGARCH-HAR

for future research. Hence, we conjecture that

pT (θ−θ)

d−→ N (0,T J−1I J−1), (1.25)

where I is the limit of the outer-product of the scores and J is the negative limit of

the Hessian matrix for the log-likelihood function. In practice, we rely on estimates

of these two components in the sandwich formula for computing robust standard

10These are the same conditions as in Hansen and Huang (2016) and we refer the reader hereto forfurther details.

1.4. EMPIRICAL RESULTS 13

errors of the coefficients.

To check the validity of this approach, we employ a parametric bootstrapping tech-

nique (Paparoditis and Politis, 2009) with 999 replications and a sample size of 2,500

observations (approximately 10 years, similar to the size of the rolling in-sample

window used in the forecasting exercise below). Figure A.1 depicts the empirical

standardized distribution of a subset of the estimated parameters.

¿ Insert Figure A.1 about here À

It stands out that the in-sample distribution of the estimated parameters for both the

REGARCH, REGARCH-MIDAS and REGRACH-HAR is generally in agreement with a

standard normal distribution. We also compared the bootstrapped standard errors

with the robust QML standard errors computed from the sandwich-formula in (1.25),

which are reported in the empirical section below. The standard errors were quite

similar, which suggests in conjunction with Figure A.1 that the QML approach and

associated inferences are valid. We do, however, note that the QML standard errors

are slightly smaller on average relative to the bootstrapped standard errors, causing

us to be careful in not putting too much weight on the role of standard errors in the

interpretation of the results below.

1.4 Empirical results

In this section, we examine the empirical fit as well as the forecasting performance of

the REGARCH-MIDAS and REGARCH-HAR, including an outline of the forecasting

procedures involved with the proposed models. We mainly comment on the weekly

REGARCH-MIDAS, since its empirical results are qualitatively similar to those from

the monthly version.

1.4.1 Data

The full sample data set consists of daily close-to-close returns and the daily realized

kernels (RK) of the SPY exchange-traded fund that tracks the S&P 500 Index and 20

individual stocks for the 2002/01-2013/12 period. In the computation of the realized

kernel, we use tick-by-tick data, restrict attention to the official trading hours 9:30:00

and 16:00:00 New York time, and employ the Parzen kernel as in Barndorff-Nielsen,

Hansen, Lunde, and Shephard (2011). See also Barndorff-Nielsen et al. (2008) and

Barndorff-Nielsen, Hansen, Lunde, and Shephard (2009) for additional details.11 For

each stock, we remove short trading days where trading occurred in a span of less

than 20,000 seconds (compared to typically 23,400 for a full trading day). We also

remove data on February 27, 2007, which contains an extreme outlier associated

11The data was kindly provided to us by Asger Lunde.

14 CHAPTER 1.

with a computer glitch on the New York Exchange that day. This leaves a sample

size for each stock of about 3,000 observations. Table A.1 reports summary statistics

of the daily returns and the logarithm of daily realized kernels. Figure A.2 depicts

the evolution of returns, squared returns, realized kernel and the autocorrelation

function (ACF) of the logarithm of the realized kernel for SPY.

¿ Insert Table A.1 about here À


We compute outlier-robust estimates of return skewness and kurtosis (Kim and White,

2004; Teräsvirta and Zhao, 2011) along with their conventional estimates. The robust

measures point to negligible skewness and quite mild kurtosis in the return series.

This stands in contrast to the moderately skewed, severely fat-tailed distributions sug-

gested by the conventional measures, corroborating the findings in Kim and White

(2004) that stylized facts of returns series change when using robust estimators.

We estimate the fractional integrated parameter d in the logarithm of the realized

kernel with the two-step exact local Whittle estimator of Shimotsu (2010). Over the

full sample all series have d > 0.5, suggesting that volatility is highly persistent.12

This finding is supported by the slowly decaying ACF of the logarithm of the realized

kernel for SPY. Since the conventional ACF may be biased for the unobserved ACF

of the logarithm conditional variance due to the presence of measurement errors,13

we also compute the instrumented ACF proposed by Hansen and Lunde (2014). We

use the authors’ preferred specification with multiple instruments (four through

ten) and optimal combination. The instrumented ACF show a similar pattern as

the conventional ACF, but points toward an even higher degree of persistence. We

also conducted a (Dickey-Fuller) unit root test across all asset considered using the

instrumented persistence parameter (cf. Table A.2).


The (biased) conventional least square estimates point to moderate persistence and

strong rejection of a unit root. The persistence parameter is, as expected, notably

higher when using the instrumented variables estimator of Hansen and Lunde (2014),

however the null hypothesis of a unit root remains rejected for all assets. Collectively,

these findings motivate a modeling framework that is capable of capturing a high

12We estimated the parameters with m = bT q c for q =∈ {0.5,0.55, . . . ,0.8}, leading to no alterations of theconclusions obtained for q = 0.65. See also Wenger, Leschinski, and Sibbertsen (2017) for a comprehensiveempirical study on long memory in volatility and the choice of estimator of d .

13The element of microstructure noise is, arguably, low, given the construction of the realized kernel,however sampling error may still be present, causing the differences in the conventional and instrumentedACF.


degree of persistence. Given the requirement that |β| < 1, this also motivates a frame-

work that pulls β away from unity. This is where the proposed REGARCH-MIDAS and

REGARCH-HAR prove useful.

1.4.2 In-sample results

In this section, we examine the empirical fit of the proposed REGARCH-HAR and

REGARCH-MIDAS using the full sample of observations for SPY and the 20 individual

stocks. We start out by discussing the choice of lag length for the MIDAS component,

K , in the following subsection.

Choice of lag length, K

As noted above, the REGARCH-HAR utilizes by construction lagged information

equal to four weeks (approximately one month) to describe the dynamics of the

realized measure, whereas the REGARCH-MIDAS allows the researcher to explore

and subsequently choose a suitable lag length, possibly beyond four weeks. For the

original two-parameter setting as well as the single-parameter setting, Figure A.3

depicts the estimated lag weights and associated maximized log-likelihood values of

the weekly REGRACH-MIDAS on SPY for a range of K starting with four lags up to

104 lags (approximately two years).


The figure yields a number of interesting insights. First, the maximized log-likelihood

values and associated patterns are very similar across the single-parameter and

two-parameter case. The maximized log-likelihood values initially increase until lag

25-50, after which the values reach a ceiling. This observation is corroborated by the

estimated lag functions in the lower panel of the figure. Their patterns show that

recent information matters the most with the information content decaying to zero

for lags approximately equal to 20 in the two-parameter setting and 25 in the single-

parameter setting. Hence, based on the figure we may conclude that information

up to half a year in the past is most important for explaining the dynamics of the

conditional variance. This is generally supported by a similar analysis using monthly

averages rather than weekly in the MIDAS component, but the monthly specification

seems to indicate that additional past information is relevant (cf. Figure A.4).

Secondly, a REGARCH-MIDAS with information only up to the past four weeks pro-

vides only a slightly greater log-likelihood value than the REGARCH-HAR (cf. Table

A.3 below). This indicates that the step-function approximation in the REGARCH-

HAR does a reasonable job at capturing the information content up to four weeks

in the past. Collectively, however, these findings also suggest that the information

lag in the REGARCH-HAR is too short. Based on these findings, we proceed in the

16 CHAPTER 1.

following with a value of K = 52 for the weekly MIDAS and K = 12 for the monthly

MIDAS uniformly in all subsequent analyses, including the individual stock results.

Note that we choose K larger than what the initial analysis suggests for the weekly

specification, since we want consistency between the weekly and monthly specifica-

tions and greater flexibility when applying the choice to the individual stocks. We do,

however, emphasize that it is free for the researcher to optimize over the choice of K

for each individual asset to achieve an even better fit.

Benchmark models

For comparative purposes, we estimate (using QML) two direct antecedents of

the REGARCH-MIDAS and REGARCH-HAR proposed in this paper. The first is a

REGARCH-Spline (REGARCH-S), with the only difference stemming from the spec-

ification of the long-term component. That is, we consider the quadratic spline

formulation

log g t =ω+ c0t

T+

K∑k=1

ck

(max

{ t

T− tk−1

T,0

})2

, (1.26)

where {t0 = 0, t1, t2, . . . , tK = T } denotes a partition of the time horizon T in K + 1

equidistant intervals. Consequently, the smooth fluctuations in the long-term com-

ponent arises from the (deterministic) passage of time instead of (stochastic) move-

ments in the realized kernel as prescribed by the REGARCH-HAR and REGARCH-

MIDAS.14 The formulation of the long-term component originates from Engle and

Rangel (2008) and is also examined in Engle et al. (2013) and Laursen and Jakobsen

(2017), to which we refer for further details. The number of knots, K , is selected using

the BIC information criterion.15

The second benchmark is the FloEGRACH of Vander Elst (2015), which incorpo-

rates fractional integration in the GARCH equation of the REGARCH in a similar vein

to the development of the FI(E)GARCH model of Baillie et al. (1996) and Bollerslev

and Mikkelsen (1996). The model, thus, explicitly incorporates long-memory via frac-

tionally integrated polynomials in the ARMA structure defined via the parameter d .

In contrast to our proposals and the REGARCH-S, the FloEGARCH do not formulate a

multiplicative component structure. Following Vander Elst (2015), we implement a

FloEGARCH(1,d ,1), which is defined as

rt =µ+σt zt , (1.27)

logσ2t =ω+ (1−β)L−1(1−L)−d (

τ(zt−1)+αut−1)

, (1.28)

log xt = ξ+φ logσ2t +δ(zt )+ut , (1.29)

14When the long-term component is specified as a deterministic component it follows that E[logσ2t ] =

log gt .15In a similar spirit to the choice of K for the REGARCH-MIDAS, we apply the number of knots

determined in the estimation on SPY uniformly in all subsequent analyses.


where (1−L)d is the fractional differencing operator. The infinite polynomial can be

written as

(1−β)L−1(1−L)−d =∞∑

n=0

(n∑

m=0βmψ−d ,n−m

)Ln , (1.30)

where ψ−d ,k =ψ−d ,k−1k−1+d

k and ψ−d ,0 = 1. In the implementation, we truncate the

infinite sum at 1,000, similar to Bollerslev and Mikkelsen (1996) and Vander Elst

(2015), and initialize the process similarly to Vander Elst (2015).

For completeness, we also estimate a multiplicative version of the EGARCH(1,1)

model (Nelson, 1991) defined by

rt =µ+σt zt , (1.31)

loght =β loght−1 +τ1zt−1 +α(|zt−1|−

p2/π

), (1.32)

log g t =ω. (1.33)

Results for the S&P 500 Index

In Table A.3, we report estimated parameters, their standard errors, and the associated

maximized log-likelihood values for the models under consideration.

¿ Insert Table A.3 about here ÀWe derive a number of notable findings. First, the multiplicative component struc-

tures lead to substantial increases in the maximized log-likelihood value relative

to the original REGARCH. It is worth noting that the null hypothesis of no MIDAS

component, λ= 0 such that f (·;η) = 0, renders γ1 and γ2 unidentified nuisance pa-

rameters. Hence, assessing the statistical significance of the differences in maximized

log-likelihood values via a standard LR test and a limiting χ2 distribution is infeasible.

We follow conventional approaches (see e.g. Hansen et al. (2012); Engle et al. (2013);

Hansen and Huang (2016)) and comment only on log-likelihood differences relative

to the original REGARCH, but note that comparing twice this difference with the

critical value of the χ2 distribution with appropriate degrees of freedom can be indica-

tive of significance.16 For instance, the LR statistic associated with the log-likelihood

gain of the weekly REGARCH-MIDAS is 92.06, compared to a 5% critical value of

5.99, which strongly indicates significance of the log-likelihood improvement. On a

similar data set, Huang et al. (2016) find a log-likelihood gain of approximately 16.5

points (LR statistic of 32.91), when introducing a HAR modification of the RGARCH

of Hansen et al. (2012).17 Addressing this issue, we nuance our interpretation of the

16Most recently, Conrad and Kleen (2016) have developed a misspecification test for comparison of theGARCH-MIDAS model of Engle et al. (2013) and its nested GARCH model.

17The RGARCH by Hansen et al. (2012) is obtained as a special case of the REGARCH (with similarrealized measures) by a proportionality restriction on the leverage function in the GARCH equation, (1.4),via τ(zt ) =αδ(zt ).

18 CHAPTER 1.

log-likelihood gains by information criteria, which hold the number of parameters

up against the maximized log-likelihood.

The substantial increases in log-likelihood value by only a small increase in the

number of parameters in the REGACRH-MIDAS and REGARCH-HAR lead to system-

atic improvements in information criteria. Despite the noticeably greater number

of parameters in the REGARCH-S, the increase in the log-likelihood value is only

comparable to that of the REGARCH-HAR, leading to a modest improvement in the

AIC, only a slight improvement in the BIC, and even a worsening of the HQIC. The

FloEGARCH comes closest to the REGARCH-MIDAS specifications, but is still short

about seven likelihood points. Since it only introduces one additional parameter, the

information criteria are comparable to those of the REGARCH-MIDAS.18 We have also

considered higher-order versions of the original REGARCH(p,q), with p, q ∈ {1, . . . ,5}.

The best fitting version, the REGARCH(5,5), provides a likelihood gain close to, but

still less than the REGARCH-MIDAS models. This gain is, however, obtained with

the inclusion of an additional eight parameters, causing the information criteria to

deteriorate.19

Secondly, we confirm the finding in the former section that the single-parameter

REGARCH-MIDAS performs comparable to the two-parameter version. Additionally,

for the same number of parameters, the single-parameter REGARCH-MIDAS provides

a considerable 16-point likelihood gain relative to the REGARCH-HAR. This suggests

that the HAR formulation is too short-sighted to fully capture the conditional variance

dynamics (despite providing a substantial gain relative to the original REGARCH)

by using only the most recent month’s realized kernels. The differences of the lag

functions, as depicted in Figure A.5, corroborate this point, by attaching a positive

weight on observations further than a month in the past.


The cascade structure as evidenced in Corsi (2009) and Huang et al. (2016) of the

HAR formulation is clear from the figure as well, leading to the conclusion that it con-

stitutes a rather successful, yet suboptimal, approximation of the beta-lag function

used in the MIDAS formulation.

18It is also noteworthy that the FloEGARCH attaches a positive weight to information four years inthe past (1,000 daily lags), whereas the REGARCH-MIDAS only carries information from the last year.This suggests that the outperformance of the REGARCH-MIDAS relative to the FloEGARCH is somewhatconservative.

19It also stands out from Table A.3 that the improvements in maximised value from all models underconsideration arises from a better modeling of the realized measure and not returns, which comes asno surprise given the motivation behind their development and that the original REGARCH is already avery successful model in fitting returns while lacking adequate modelling of the realized measure, as putforward in Hansen and Huang (2016).


In Figure A.6, we depict the fitted conditional variance along with the long-term

components of each multiplicative component model under consideration.


The long-term component of the REGARCH-MIDAS models appear smooth and do,

indeed, resemble a time-varying baseline volatility. The long-term component in the

REGARCH-HAR is less smooth in contrast to that from the REGACRH-Spline, which

is excessively smooth. To elaborate on the pertinence of the long-term component,

we compute for each model the variance ratio given by

VR = Var[log g t ]

Var[loght g t ], (1.34)

which reveals how much of the variation in the fitted conditional variance can be

attributed to the long-term component. The last row in Table A.3 suggests that the

long-term component contribution is important with more than two-thirds of the

variation for the REGARCH-HAR and REGARCH-MIDAS formulations - noticeably

larger than that for the REGARCH-S. Moreover, the monthly aggregation scheme for

the realized kernel leads to a smoother slow-moving component and, by implication,

a smaller VR ratio.

In terms of parameter estimates and associated standard errors, the values are very

similar across the various REGARCH extensions for most of the intersection of pa-

rameters. The leverage effect appears to be supported in all model formulations,

and estimated values of φ are less than unity with relatively small standard errors,

consistent with the realized measure being computed from open-to-close data and

conditional variance referring to the close-to-close period. Moreover, estimated λ is

close to 0.9 and precisely estimated, suggesting that past information in the realized

kernels are highly informative on conditional variance. The fractional integration pa-

rameter, d , is estimated to 0.65 in the FloEGARCH, confirming the high persistence in

the conditional variance process also suggested by the summary statistics presented

above. Note also that the parameters of the beta-weight function are imprecisely

estimated when γ1 = 1 is not imposed. The reason is that two almost identical weight

structures may be obtained for two (possibly very) different combinations of γ1 and

γ2, leaving the pair imprecisely estimated. Importantly, the estimated values of β are

considerably smaller in our proposed models relative to the original REGARCH. A

similar, but less pronounced result, is obtained for the REGARCH-S. This reduction

in estimated β plays an important role in satisfying the condition that |β| < 1 and

alleviating the integrated GARCH effect. This occurs intuitively since we enable a

flexible level of the baseline volatility which the short-term movements fluctuates

around. Lastly, the measurement equations in the REGARCH-MIDAS and REGARCH-

HAR have smaller estimated residual variances, σ2u , than the original REGARCH. This

20 CHAPTER 1.

may indicate that the new models also provide a better empirical fit of the realized

measure via the multiplicative component specifications proposed here.

Autocorrelation function of conditional variance and realized kernel

In this section, we consider the implications of the REGARCH-HAR and REGARCH-

MIDAS on the ACF of the conditional variance and the realized kernel relative to

the original formulation in REGARCH. We depict in Figure A.7 the simulated and

sample ACF of the logarithm of the conditional variance, logσ2t , for the REGARCH,

REGARCH(5,5), REGARCH-HAR, single-parameter and two-parameter REGARCH-

MIDAS, and FloEGARCH on SPY. The simulated ACF is obtained using the estimated

parameters in Table A.3 with a sample size of 3,750 (approximately 15 years) and

10,000 Monte Carlo replications, whereas the sample ACF is based on the fitted

conditional variance.


In general and for a given model, the closer the simulated and sample ACF are to

each other, the larger is the degree of internal consistency in modeling the depen-

dency structure of conditional variance. We note that the original REGARCH is only

able to capture the autocorrelation structure over the very short term. Moreover,

the REGARCH(5,5) does not substantially improve upon the REGARCH. The simu-

lated ACF of the REGARCH-HAR is closer to the sample ACF, but starts diverging at

about lag 30. Only the REGARCH-MIDAS models and the FloEGARCH are capable of

capturing the pattern of the autocorrelation structure over a long horizon. It should

also be noted that the results for the REGARCH-MIDAS is for a particular choice

of K = 52 and K = 12 for the weekly and monthly versions, respectively. Larger val-

ues of K , for a given model, may provide an even greater degree of fit. Indeed, the

monthly REGARCH-MIDAS trades off some fit in the short term for improved ac-

curacy in the long term by using a cruder aggregation scheme of the realized measure.

In Figure A.8, we depict simulated and sample ACFs of the logarithm of the real-

ized kernel for each model to provide an insight into whether the models are able to

capture the autocorrelation structure of the market realized variance.


The picture is, expectedly, similar to the one in Figure A.7. With only two or three

additional parameters, the REGARCH-HAR and especially the REGARCH-MIDAS

specifications provide a noticeable increase in the ability to capture the dynamics of

the realized measure relative to the REGARCH. This suggests that the multiplicative

component structure used in the REGARCH-HAR and REGARCH-MIDAS consti-

tutes a very appealing and parsimonious way of capturing high persistence in the

REGARCH framework.


Results for individual stocks

The conclusions for the SPY above also apply to individual stocks, for which detailed

results are presented in Appendix A.4. In summary, Table A.4 reports the differences

in log-likelihood values for our proposed models and their benchmarks relative to

the original REGARCH.


First, the REGARCH-HAR and REGARCH-MIDAS provide systematically large gains

relative to the original REGARCH for all stocks. The two competing benchmarks,

REGARCH-S and FloEGARCH, also provide sizeable gains. Despite this, the REGARCH-

MIDAS specification is the preferred choice for all but two stocks. It also stands out

that the weekly REGARCH-MIDAS consistently outperforms the REGARCH-HAR.

This is generally the case for the monthly REGARCH-MIDAS as well, albeit with a few

exceptions. These exceptions may relate to its crude aggregation scheme, which sac-

rifices too much fit of the autocorrelation structure in the short term for better fit in

the long-term compared to the relatively short-sighted formulation in the REGARCH-

HAR. On this basis, we may conjecture that a framework which incorporates both

daily, weekly and monthly aggregates (sort of hybrid between a HAR and MIDAS

specification) would fit particularly well. The information criteria in the Appendix

corroborate these findings.

In Table A.5 we report the estimated β for all stocks.


They are all very similar and close to unity in the original REGARCH, but are substan-

tially reduced in the REGARCH-MIDAS and REGARCH-HAR - even more so than for

the S&P 500 Index.

1.4.3 Forecasting with the REGARCH-MIDAS and REGARCH-HAR

In this section, we detail how to generate one- and multi-step forecasts using the

REGARCH-MIDAS and REGARCH-HAR. We note that our models are dynamically

complete. By implication, they are capable of generating multi-period forecasts with-

out imposing (unrealistic) assumptions on the dynamics of the realized measure

(such as the random walk), as usually done in the GARCH-X model that otherwise

are only suitable for one-step ahead forecasting. This feature turns out to be valu-

able below, when we evaluate the predictive ability of the REGARCH-MIDAS and

REGARCH-HAR relative to that of the original REGARCH and the benchmark models.

22 CHAPTER 1.

One-step and multi-step forecasting

Denote by k, k ≥ 1, the forecast horizon measured in days. Our aim is to forecast

the conditional variance k days into the future. To that end, we note that for k = 1

one-step ahead forecasting can be easily achieved directly via the GARCH equation

in (1.4). For multi-period forecasting (k > 1), we note that recursive substitution of

the GARCH equation implies

loght+k =βk loght +k∑

j=1β j−1

(τ(zt+k− j )+αut+k− j

), (1.35)

such that

logσ2t+k = loght+k g t+k =βk loght +

k∑j=1

β j−1(τ(zt+k− j )+αut+k− j

)+ log g t+k .

(1.36)

Multi-period forecasts of logσ2t+k may then be obtained via

logσ2t+k|t ≡ E[logσ2

t+k |Ft ] =βk loght +βk−1 (τ(zt )+αut

)+ log g t+k . (1.37)

Consequently, the contribution of the short-term component to the forecast is easily

computed with known quantities at time t , namely ht ,ut , zt . To obtain g t+k , we gen-

erate recursively, using estimated parameters, the future path of the realized measure

using the measurement equation in (1.5). It is worth noting that for multi-step fore-

cast horizons a lower magnitude of β causes the forecast to converge more rapidly

towards the baseline volatility, determined by (the forecast of) the long-term compo-

nent. Because this baseline volatility is allowed to be time-varying, a lower magnitude

of β is preferable since it generates more flexibility and reduces a long-lasting impact

on the forecast from the most recent ht and its innovation. By implication, the abil-

ity to generate reasonable forecasts of the long-term component is valuable, which

strongly motivates the dynamic completeness of the models.20

Jensen’s inequality stipulates that exp{E[logσ2t+k |Ft ]} 6= E[exp{logσ2

t+k }|Ft ] such that

we need to consider the distributional aspects of logσ2t+k|t to obtain an unbiased

forecast of σ2t+k|t . As a solution, we utilize a simulation procedure with empirical dis-

tributions of zt and ut . Using M simulations and re-sampling the estimated residuals,

the resulting forecast of the conditional variance given by

σ2t+k|t =

1

M

M∑m=1

exp{logσ2t+k|t ,m} (1.38)

20We found, indeed, that setting gt+k = gt leads to notably inferior forecasting performance relative tothe case that exploits the estimated dynamics of the realized kernel.


is unbiased. In the implementation, we estimate model parameters on a rolling

basis with 10 years of data (2,500 observations) and leave the remaining (about 500)

observations for (pseudo) out-of-sample evaluation. The empirical distribution of

zt and ut is similarly obtained using the same historical window of observations.

Forecasting with the REGARCH follows directly from the above with log g t+h =ω.

Forecast evaluation

Given the latent nature of the conditional variance, we require a proxy, σ2t , of σ2

t for

forecast evaluation. To that end, we employ the adjusted realized kernel in line with

e.g. Huang et al. (2016) and Sharma and Vipul (2016) given by σ2t = κRKt , where

κ=∑T

t=1 r 2t∑T

t=1 RKt. (1.39)

The adjustment is needed since the realized measure is a measure of open-to-close

variance, whereas the forecast generated by the REGARCH framework measures close-

to-close variance. We compute κ on the basis of the out-of-sample period. A second

implication of using the realized kernel as proxy is that we implicitly restrict ourselves

to the choice of robust loss functions (Hansen and Lunde, 2006; Patton, 2011) when

quantifying the forecast precisions in order to obtain consistent ranking of forecasts.

Let Li ,t+k (σ2t+k ,σ2

t+k|t ) denote the loss function for the i ’th k-step ahead forecast. Two

such robust functions are the Squared Prediction Error (SPE) and Quasi-Likelihood

(QLIKE) loss function given as

L(SPE)i ,t+k (σ2

t+k ,σ2t+k|t ) = (σ2

t+k −σ2t+k|t )2, (1.40)

L(QLIKE)i ,t+k (σ2

t+k ,σ2t+k|t ) = σ2

t+k

σ2t+k|t

− log

σ2t+k

σ2t+k|t

−1. (1.41)

In both cases, a value of zero is obtained for a perfect forecast. The SPE (QLIKE) loss

function penalizes forecast error symmetrically (asymmetrically), and the QLIKE

often gives rise to more power in statistical forecast evaluation procedures, espe-

cially when comparing losses across different regimes (see e.g. Borup and Thyrsgaard

(2017)).

Given the objective of evaluating whether the REGARCH-MIDAS and REGARCH-HAR

provide an improvement in forecasts relative to the REGARCH, we use the Diebold-

Mariano test (Diebold and Mariano, 1995).21 Let the loss differentials from the i ’th

21We acknowledge that the Diebold-Mariano test is technically not appropriate for comparing forecastsof nested models since the limiting distribution is non-standard under the null hypothesis (see e.g. Clarkand McCracken (2001) and Clark and West (2007)). The adjusted mean squared errors of Clark andWest (2007) or the bootstrapping procedure of Clark and McCracken (2015) are appropriate alterationsto standard inferences. However, since we estimate our models on a rolling basis with a finite, fixed

24 CHAPTER 1.

model relative to the REGARCH (abbreviated REG) be given by di ,t = Li ,t+k (σ2t ,σ2

t+k|t )−LREG,t+k (σ2

t ,σ2t+k|t ). The Diebold-Mariano test of equal predictive ability can be con-

ducted using the conventional t-statistic

S = T 1/2 d√V

, (1.42)

where d = T −1 ∑Tt=1 di ,t and V is an estimate of the long-run variance of the loss

differentials. We employ in the following a HAC estimator and follow state-of-the art

good practice by using the data-dependent bandwidth selection by Andrews (1991)

based on an AR(1) approximation and a Bartlett kernel.22 We perform the test against

the alternative that the i ’th forecast losses are smaller than the ones arising from the

original REGARCH and evaluate S in the standard normal distribution.

We also do a Model Confidence Set (MCS) procedure (Hansen, Lunde, and Nason,

2011) to compare the predictive accuracy of all our proposed models to that of the

REGACRH-Spline and the FloEGARCH. For a fixed significance level, α, the proce-

dure identifies the MCS, M∗α, from the set of competing models, M0, which contains

the best models with 1−α probability (asymptotically as the length of the out-of-

sample window approaches infinity). The procedure is conducted recursively based

on an equivalence test for any M ⊆ M0 and an elimination rule, which identifies

and removes a given model from M in case of rejection of the equivalence test. The

equivalence test is based on pairwise comparisons using the statistic Si j in (1.42)

for all i , j ∈ M and the range statistic TM = maxi , j∈M {|Si j |}, where the eliminated

model is identified by argmaxi∈M sup j∈M {Si j }. Following Hansen et al. (2011), we

implement the procedure using a block bootstrap and 105 replications.

Forecasting results

Figure A.9 depicts Theil’s U statistic in terms of the ratio of forecast losses on the

SPY arising from forecasts generated by the original REGARCH to those from the

REGARCH-HAR and the weekly REGARCH-MIDAS (single-parameter) on horizons

k = 1, . . . ,22. It depicts their associated statistical significance, too. Quantitatively and

qualitatively similar results for the remaining MIDAS specifications are left out, but

are available upon request.

window size, the asymptotic framework of Giacomini and White (2006) provides a rigorous justification forproceeding with the Diebold-Mariano test statistic evaluated in a standard normal distribution. See alsoDiebold (2015) for a discussion.

22Admittedly, the high persistence in both the realized kernels and the forecasts generated by themodels under consideration may transmit to the loss differentials, leading to a potential need for a long-memory robust variance estimator in (1.42). In fact, Kruse, Leschinski, and Will (2016) show that thestandard Diebold-Mariano test statistic is most likely oversized in these cases. However, this transmissioncritically depends on the unbiasedness and (loading on) a common long memory between the forecasts(see their Propositions 2-4), leaving a further examination out of the scope of this paper.



The figure convincingly concludes that both the REGARCH-HAR and REGARCH-

MIDAS improve upon the forecasting performance of the original REGARCH for all

forecast horizons. These improvements tend to grow as the forecast horizon increases

from a few percentages to roughly 30-40% depending on the loss function. This

indicates the usefulness of modeling a slow-moving component, particularly for

forecasting beyond short horizons. In general, the improvements are statistically

significant for all horizons, except for the shorter horizons in the REGARCH-MIDAS

case.23 Table A.6 reports results from a similar analysis on the 20 individual stocks.


Also on the individual stock basis, both the REGARCH-HAR and REGARCH-MIDAS

provide substantial improvements on the original REGARCH, in particular at longer

horizons. The REGARCH-MIDAS outperforms the REGARCH-HAR with a system-

atically larger improvement for all horizons and based on statistical significance.

Moreover, only a few stocks are not significantly favoring the REGARCH-MIDAS over

the original REGARCH.

Having established the improvement upon the original REGARCH, we turn to a

complete comparison of all our proposed models, the REGARCH-Spline and the

FloEGARCH. Table A.7 reports the percentage of stocks (including SPY) for which a

given model is included in the MCS at an α= 10% significance level.


The inclusion frequency of our proposed REGARCH-MIDAS models are high and

indicate superiority over all competing models in both the short-term and beyond.

Interestingly, the cruder, monthly aggregation scheme dominates for longer horizons,

whereas the finer, weekly scheme is preferred for short horizons. The REGARCH-

Spline shows moderate improvement over the original REGARCH, but is less fre-

quently included in the MCS compared to our proposed REGARCH-MIDAS and

REGARCH-HAR. The FloEGARCH performs relatively bad for horizons 2,3,4 and 5,

but is increasingly included in the MCS as the forecast horizon increases, reaching

similar performance as the REGARCH-MIDAS models at monthly predictions. These

findings indicate the usefulness of the flexibility obtained via the multiplicative com-

ponent structure as opposed to, e.g., incorporating fractional integration as in the

FloEGARCH.

23We have also examined the models’ predictive ability of cumulative forecasts for a 5,10, and 22horizon. Consistent with the findings for the point forecasts, both the REGARCH-HAR and REGARCH-MIDAS provide substantial and statistically significant improvements relative to the original REGARCH.

26 CHAPTER 1.

1.5 Conclusion

We introduce two extensions of the otherwise successful REGARCH model to capture

the evident high persistence observed in stock return volatility series. Both exten-

sions exploit a multiplicative decomposition of the conditional variance process

into a short-term and a long-term component. The latter is modeled either using

mixed-data sampling or a heterogeneous autoregressive structure, giving rise to the

REGARCH-MIDAS and REGARCH-HAR models, respectively. Both models lead to

substantial in-sample improvements of the REGARCH with the REGARCH-MIDAS

dominating the REGARCH-HAR. Evidently, the backward-looking horizon of the

HAR specification is too short to adequately capture the autocorrelation structure of

volatility for horizons longer than a month.

Our suggested models are dynamically complete, facilitating multi-period forecasting

in contrast to e.g. the GARCH-X or models tying the slow-moving behavior of volatil-

ity to e.g. macroeconomic state variables. Coupled with a lower estimated β and

time-varying baseline volatility, we show in a forecasting exercise that the REGARCH-

MIDAS and REGARCH-HAR leads to significant improvements in predictive ability of

the REGARCH, particularly beyond short horizons.

Similarly to the original REGARCH, our proposed models involve an easy multi-

variate extension, enabling the inclusion of for instance additional realized measures,

macroeconomic variables or event-related dummies (e.g. from policy announce-

ments). Some additional questions remain for future research. On the empirical side,

applications to other asset classes exhibiting high persistence such as commodities,24

bonds or exchange rates, or the use of our proposed models in estimating the (term

structure of) variance risk premia, or investigating the risk-return relationship via

the return equation (see e.g. Christensen et al. (2010)) are of potential interest. On

the theoretical side, development of a misspecification tests for comparison of our

models with the nested REGARCH and asymptotic properties of the QML estimator

would prove very useful.

Acknowledgement

We thank Timo Teräsvirta, Asger Lunde, Peter Reinhard Hansen, Esther Ruiz Ortega,

Bent Jesper Christensen, Jorge Wolfgang Hansen and participants at research semi-

nars at Aarhus University for useful comments and suggestions. We also thank Asger

Lunde for providing cleaned high-frequency tick data. The authors acknowledge

support from CREATES - Center for Research in Econometric Analysis of Time Se-

ries (DNRF78), funded by the Danish National Research Foundation. Some of this

24See e.g. Lunde and Olesen (2013) for an application of the REGARCH to commodities.

1.5. CONCLUSION 27

research was carried out while D. Borup was visiting the Department of Economics,

University of Pennsylvania, and the generosity and hospitality of the department is

gratefully acknowledged. An earlier version of this paper was circulated under the

title "Long-range dependence in the Realized (Exponential) GARCH framework".

28 CHAPTER 1.

1.6 References

Abadir, K., Talmain, G., 2002. Aggregation, persistence and volatility in a macro model.

The Review of Economic Studies 69, 749–779.

Amado, C., Teräsvirta, T., 2013. Modelling volatility by variance decomposition. Jour-

nal of Econometrics 175, 142–153.

Andersen, T. G., Bollerslev, T., Diebold, F. X., Labys, P., 2001. The distribution of realized

exchange rate volatility. Journal of the American Statistical Association 96 (453),

42–55.

Andersen, T. G., Bollerslev, T., Diebold, F. X., Labys, P., 2003. Modeling and forecasting

realized volatility. Econometrica 71 (2), 579–625.

Andersen, T. G., Varneskov, R., 2014. On the informational efficiency of option-implied

and time series forecasts of realized volatility, CREATES Research Paper.

Andrews, D. W. K., 1991. Heteroskedasticity and autocorrelation consistent covariance

matrix estimation. Econometrica 59 (3), 817–858.

Asgharian, H., Christiansen, C., Hou, A. J., 2016. Macro-finance determinants of

the long-run stock-bond correlation: The DCC-MIDAS specification. Journal of

Financial Econometrics 14 (3), 617.

Baillie, R. T., Bollerslev, T., Mikkelsen, H. O., 1996. Fractionally integrated generalized

autoregressive conditional heteroskedasticity. Journal of Econometrics 74 (1), 3–30.

Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2008. Designing

realized kernels to measure the ex post variation of equity prices in the presence of

noise. Econometrica 76 (6), 1481–1536.

Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2009. Realised kernels

in practice: Trades and quotes. Econometrics Journal 12 (3), 1–32.

Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2011. Multivariate

realised kernels: Consistent positive semi-definite estimators of the covariation of

equity prices with noise and non-synchronous trading. Journal of Econometrics

162 (2), 149–169.

Bollerslev, T., Mikkelsen, H. O., 1996. Modeling and pricing long memory in stock

market volatility. Journal of Econometrics 73 (1), 542–547.

Bollerslev, T., Patton, A. J., Quaedvlieg, R., 2016. Exploiting the errors: A simple ap-

proach for improved volatility forecasting. Journal of Econometrics 192 (1), 1–18.

Borup, D., Thyrsgaard, M., 2017. Statistical tests for equal predictive ability across

multiple forecasting methods, CREATES Research Paper.

1.6. REFERENCES 29

Chambers, M. J., 1998. Long memory and aggregation in macroeconomic time series.

International Economic Review 39, 1053–1072.

Chevillon, G., Hecq, A., Laurent, S., 2015. Long memory through marginalization of

large systems and hidden cross-section dependence, working paper.

Chevillon, G., Mavroeidis, S., 2017. Learning can generate long memory. Journal of

Econometrics 198, 1–9.

Christensen, B. J., Nielsen, M. O., Zhu, J., 2010. Long memory in stock market volatility

and the volatility-in-mean effect: The FIEGARCH-M model. Journal of Empirical

Finance 17, 460–470.

Clark, T. E., McCracken, M., 2001. Tests of equal forecast accuracy and encompassing

for nested models. Journal of Econometrics 105 (1), 85–110.

Clark, T. E., McCracken, M., 2015. Nested forecast model comparisons: A new ap-

proach to testing equal accuracy. Journal of Econometrics 186, 160–177.

Clark, T. E., West, K. D., 2007. Approximately normal tests for equal predictive accuracy

in nested models. Journal of Econometrics 138 (1), 291–311.

Conrad, C., Kleen, O., 2016. On the statistical properties of multiplicative GARCH

models, working paper.

Conrad, C., Loch, K., 2015. Anticipating long-term stock market volatility. Journal of

Applied Econometrics 30, 1090–1114.

Corsi, F., 2009. A simple approximate long-memory model of realised volatility. Jour-

nal of Financial Econometrics 7 (2), 174–196.

Davidson, J., 2004. Moment and memory properties of linear conditional het-

eroscedasticity models, and a new model. Journal of Business and Economic

Statistics 22 (1), 16–29.

Davidson, J., Sibbertsen, P., 2005. Generating schemes for long memory processes:

Regimes, aggregation and linearity. Journal of Econometrics 128, 253–282.

Diebold, F. X., 1986. Modeling the persistence of conditional variance: A comment.

Econometric Reviews 5 (1), 51–56.

Diebold, F. X., 2015. Comparing predictive accuracy, twenty years later: A personal

perspective on the use and abuse of the diebold-mariano tests. Journal of Business

and Economic Statistics 33 (1), 1–9.

Diebold, F. X., Inoue, A., 2001. Long memory and regime switching. Journal of Econo-

metrics 105, 131–159.

30 CHAPTER 1.

Diebold, F. X., Mariano, R. S., 1995. Comparing predictive accuracy. Journal of Busi-

ness and Economic Statistics 13 (3), 253–263.

Ding, Z., Granger, C. W., Engle, R. F., 1993. A long memory property of stock market

returns and a new model. Journal of Empirical Finance 1 (1), 83 – 106.

Dominicy, Y., Vander Elst, H., 2015. Macro-driven VaR forecasts: From very high to

very low-frequency data, working paper.

Engle, R., 2002. New frontiers for ARCH models. Journal of Applied Econometrics 17,

425–446.

Engle, R., Bollerselv, T., 1986. Modeling the persistence in conditional variances.

Econometric Reviews 5 (1), 1–50.

Engle, R. F., Gallo, G. M., 2006. A multiple indicators model for volatility using intra-

daily data. Journal of Econometrics 131, 3–27.






Engle, R. F., Rangel, J. G., 2008. The Spline-GARCH model for low-frequency volatility


1187–1222.

Feng, Y., 2004. Simultaneously modeling conditional heteroskedasticity and scale

change. Econometric Theory 20 (3), 563–596.

Forsberg, L., Ghysels, E., 2007. Why do absolute returns predict volatility so well?

Journal of Financial Econometrics 5 (1), 31.

Francq, C., Zakoïan, J.-M., 2010. GARCH Models: Structure, statistical inference and

financial applications,. New York: Wiley.

Fuller, W. A., 1996. Introduction to statistical time series, 2nd Edition. Wiley.

Ghysels, E., Santa-Clara, P., Valkanov, R., 2004. The MIDAS touch: Mixed data sampling

regression models, working paper.

Ghysels, E., Santa-Clara, P., Valkanov, R., 2005. There is a risk-return trade-off after all.

Journal of Financial Economics 76 (3), 509 – 548.

Ghysels, E., Sinko, A., Valkanov, R., 2007. MIDAS regressions: Further results and new

directions. Econometric Reviews 26 (1), 53–90.

1.6. REFERENCES 31

Giacomini, R., White, H., 2006. Tests of conditional predictive ability. Econometrica

74 (6), 1545–1578.

Granger, C. W., 1980. Long memory relationships and the aggregation of dynamic

models. Journal of Econometrics 14, 227–238.

Granger, C. W., Ding, Z., 1996. Varieties of long memory models. Journal of Econo-

metrics 73, 61–77.

Haldrup, N., Kruse, R., 2014. Discriminating between fractional integration and spuri-

ous long memory, working Paper.

Haldrup, N., Valdés, J. E. V., 2017. Long memory, fractional integration, and cross-

sectional aggregation. Journal of Econometrics 199, 1–11.

Han, H., 2015. Asymptotic properties of GARCH-X processes. Journal of Financial

Econometrics 13 (1), 188–221.

Han, H., Kristensen, D., 2014. Asymptotic theory for the QMLE in GARCH-X models

with stationary and nonstationary covariates. Journal of Business and Economic

Statistics 32, 416–429.



Hansen, P. R., Huang, Z., Shek, H. H., 2012. Realized GARCH: A joint model for return

and realized mesures of volatility. Journal of Applied Econometrics 27, 877–906.

Hansen, P. R., Lunde, A., 2006. Consistent ranking of volatility models. Journal of

Econometrics 131 (1-2), 97–121.

Hansen, P. R., Lunde, A., 2014. Estimating the persistence and the autocorrelation

function of a time series that is measured with error. Econometric Theory 30, 60–93.

Hansen, P. R., Lunde, A., Nason, J. M., 2011. The model confidence set. Econometrica

79 (2), 453–497.

Hillebrand, E., 2005. Neglecting parameter changes in GARCH models. Journal of

Econometrics 129, 121 – 138.

Huang, Z., Liu, H., Wang, T., 2016. Modeling long memory volatility using realized

measures of volatility: A realized HAR GARCH model. Economic Modelling 52,

812–821.

Huang, Z., Wang, T., Hansen, P. R., 2017. Option pricing with the Realized GARCH

model: An analytical approximation approach. Journal of Futures Markets 37 (4),

328–358.

32 CHAPTER 1.

Kim, T.-H., White, H., 2004. On more robust estimation of skewness and kurtosis.

Finance Research Letters 1, 56–73.

Kruse, R., Leschinski, C., Will, C., 2016. Comparing predictive ability under long

memory - with an application to volatility forecasting, CREATES Research Paper,

2016-17.


and the GARCH model. Journal of Business and Economic Statistics 8, 225–234.

Laursen, B., Jakobsen, J. S., 2017. Realized EGARCH model with time-varying uncon-

ditional variance, working Paper.

Louzis, D. P., Xanthopoulos-Sisinis, S., Refenes, A. P., 9 2013. The role of high-frequency

intra-daily data, daily range and implied volatility in multi-period Value-at-Risk

forecasting. Journal of Forecasting 32 (6), 561–576.

Louzis, D. P., Xanthopoulos-Sisinis, S., Refenes, A. P., 2014. Realized volatility models

and alternative Value-at-Risk prediction strategies. Economic Modelling 40, 101 –

116.

Lunde, A., Olesen, K. V., 2013. Modeling and forecasting the distribution of energy

forward returns, CREATES Research Paper, 2013-19.

McCloskey, A., Perron, P., 2013. Memory parameter estimation in the presence of level

shifts and deterministic trends. Econometric Theory 29 (6), 1196–1237.

Mikosch, T., Starica, C., 2004. Nonstationarities in financial time series, the long-range

dependence, and IGARCH effects. The Review of Economics and Statistics 86 (1),

378–390.

Miller, J. I., Park, J. Y., 2010. Nonlinearity, nonstationarity, and thick tails: How they

interact to generate persistence in memory. Journal of Econometrics 155, 83–89.

Müller, U. A., et al., 1993. Fractals and intrinsic time - a challenge to econometricians.

In: 39th International AEA Conference on Real Time Econometrics, 14-15 October

1993, Luxembourg.

Nelson, D. B., 1991. Conditional heteroskedasticity in asset returns: A new approach.

Econometrica 59 (2), 347–370.

Paparoditis, E., Politis, D. N., 2009. Resampling and subsampling for financial time

series. In "Handbook of financial time series", Springer, 983–999.

Parke, W. R., 1999. What is fractional integration? The Review of Economics and


1.6. REFERENCES 33

Patton, A. J., 2011. Volatility forecast comparison using imperfect volatility proxies.

Journal of Econometrics 160 (1), 246–256.

Perron, P., Qu, Z., 2007. An analytical evaluation of the log-periodogram: Estimate in

the presence of level shifts, working Paper.

Schennach, S. M., 2013. Long memory via networking, working paper.

Sharma, P., Vipul, 2016. Forecasting stock market volatility using Realized GARCH

model: International evidence. The Quarterly Review of Economics and Finance

59, 222–230.

Shephard, N., Sheppard, K., 2010. Realising the future: Forecasting with high-

frequency-based volatility (HEAVY) models. Journal of Applied Econometrics 25,

197–231.

Shimotsu, K., 2010. Exact local whittle estimation of fractional integration with un-

known mean and time trend. Econometric Theory 26 (2), 501–540.

Teräsvirta, T., Zhao, Z., 2011. Stylized facts of return series, robust estimates and three

popular models of volatility. Applied Financal Economics 21 (1-2), 67–94.

Tse, Y., 1998. The conditional heteroskedasticity of the yen-dollar exchange rate.

Journal of Applied Econometrics 13 (1), 49–55.

Vander Elst, H., 2015. FloGARCH: Realizing long memory and asymmetries in returns

volatility, working paper.

Varneskov, R., Perron, P., 2017. Combining long memory and level shifts in modeling

and forecasting the volatility of asset returns. Quantitative Finance, 1–23.

Wang, F., Ghysels, E., 2015. Econometric analysis of volatility component models.

Econometric Theory 32 (2), 362–393.

Watanabe, T., 3 2012. Quantile forecasts of financial returns using Realized GARCH

models. Japanese Economic Review 63 (1), 68–80.

Wenger, K., Leschinski, C., Sibbertsen, P., 2017. Long memory of volatility, working

paper.

Wintenberger, O., 2013. Continuous invertibility and stable qml estimation of the

EGARCH(1,1) model. Scandinavian Journal of Statistics 40, 846–867.

Zaffaroni, P., 2004. Contemporaneous aggregation of linear dynamic models in large

economies. Journal of Econometrics 120, 75–102.

34 CHAPTER 1.

Appendix

A.1 Derivation of score function

First, consider A(zt ) = ∂ loght+1/∂ loght and C (zt ) = ∂ loght+1/∂ log g t . From zt =rt−µσt

, it can easily be shown that

zt

loght= zt

log g t=−1

2zt . (A.1)

From ut = log xt −φ logσ2t −δ(zt ), we find

∂ut

∂ loght= −δ′ ∂b(zt )

∂zt

∂zt

loght−φ=−δ′bzt

∂zt

loght−φ, (A.2)

∂ut

∂ log g t= −δ′ ∂b(zt )

∂zt

∂zt

log g t−φ=−δ′bzt

∂zt

log g t−φ. (A.3)

Similarly, we have

∂τ(zt )

∂ loght= τ′

∂a(zt )

∂zt

∂zt

loght= τ′azt

∂zt

loght, (A.4)

∂τ(zt )

∂ log g t= τ′

∂a(zt )

∂zt

∂zt

log g t= τ′azt

∂zt

log g t. (A.5)

Inserting the above components in the following expressions for A(zt ) and C (zt )

A(zt ) = ∂ loght+1

∂ loght= β+ ∂τ(zt )

∂ loght+α ∂ut

∂ loght, (A.6)

C (zt ) = ∂ loght+1

∂ log g t= ∂τ(zt )

∂ log g t+α ∂ut

∂ log g t, (A.7)

yields

A(zt ) = (β−αφ)+ 1

2


)zt , (A.8)

C (zt ) = −αφ+ 1

2


)zt . (A.9)

Next, we turn to B(zt ,ut ) = ∂`t /∂ loght and D(zt ,ut ) = ∂`t /∂ log g t . The terms loght

and log g t enter the log-likelihood contribution at time t directly due to logσ2t =

loght + log g t and indirectly through z2t and u2

t . Thus, we have

B(zt ,ut ) = −1

2

[1+ ∂z2

t

∂ loght+ 1

σ2u

2ut∂ut

∂ loght

], (A.10)

D(zt ,ut ) = −1

2

[1+ ∂z2

t

∂ log g t+ 1

σ2u

2ut∂ut

∂ log g t

]. (A.11)

A.1. DERIVATION OF SCORE FUNCTION 35

We note that∂`t

∂ log g t= ∂`t

∂ loght=−z2

t . (A.12)

Combining the different expressions yields

B(zt ,ut ) = −1

2

[(1− z2

t )+ ut

σ2u

(δ′bzt zt −2φ

)], (A.13)

D(zt ,ut ) = −1

2

[(1− z2

t )+ ut

σ2u

(δ′bzt zt −2φ

)]. (A.14)

Now, we turn to the derivatives of loght+1 with respect to the different parameters.

For hµ,t+1 = ∂ht+1/∂µ, we have

hµ,t+1 =β∂ loght

∂µ+ ∂τ(zt )

∂µ+α∂ut

∂µ, (A.15)

where

∂τ(zt )

∂µ= ∂τ(zt )

∂zt

∂zt

∂µ= τ′azt

[−1

2zt∂ loght

∂µ− 1

σt

], (A.16)

∂ut

∂µ= −φ∂ loght

∂µ−δ′bzt

∂zt

∂µ

= −φ∂ loght

∂µ−δ′bzt

[−1

2zt∂ loght

∂µ− 1

σt

]. (A.17)

Inserting (A.16) and (A.17) in (A.15) and rearranging yields

hµ,t+1 =[(β−αφ)+ 1

2

[αδ′bzt −τ′azt

]zt

]∂ loght

∂µ+

[αδ′bzt −τ′azt

] 1

σt

= A(zt )hµ,t +[αδ′bzt −τ′azt

] 1

σt. (A.18)

For hθ1,t+1 = ∂ht+1/∂θ1, we have

hθ1,t+1 =β∂ loght

∂θ1+ ∂τ(zt )

∂θ1+α∂ut

∂θ1+ (loght , zt , z2

t −1,ut )′. (A.19)

However, we remember that τ(zt ) and ut only depend on θ1 through loght such that

we can reduce the first three terms into one


∂ loght

∂ loght

∂θ1+ (loght , zt , z2

t −1,ut )′

= A(zt )hθ1,t +mt . (A.20)

36 CHAPTER 1.

For hθ2,t+1 = ∂ht+1/∂θ2, hω,t+1 = ∂ht+1/∂ω and hη,t+1 = ∂ht+1/∂η, we obtain


∂ loght

∂ loght

∂θ2+α(1, logσ2

t , zt , z2t −1)′

= A(zt )hθ2,t +nt , (A.21)

hω,t+1 = ∂ loght+1

∂ loght

∂ loght

∂ω+ ∂ loght+1

∂ log g t

∂ log g t

∂ω

= A(zt )hω,t +C (zt ), (A.22)

hη,t+1 = ∂ loght+1

∂ loght

∂ loght

∂η+ ∂ loght+1

∂ log g t

∂ log g t

∂η

= A(zt )hη,t +C (zt )gη,t , (A.23)

respectively. Finally, we turn to the scores. The parameter µ enters the log-likelihood

contribution at time t through loght , zt , and u2t such that

∂`t

∂µ= −1

2hµ,t −

z2t

∂µ− 1

2

1

σ2u

∂u2t

∂µ

= ∂`t

∂ loght

∂ loght

∂µ−

[zt −δ′ ut

σ2u

bzt

]1

σt

= B(zt ,ut )hµ,t −[

zt −δ′ ut

σ2u

bzt

]1

σt. (A.24)

Since θ1 only enters the log-likelihood contribution at time t indirectly through loght ,

an application of the chain-rule yields

∂`t

∂θ1= B(zt ,ut )hθ1,t . (A.25)

The parameter vector θ2 also enters through u2t ,

∂`t

∂θ2= B(zt ,ut )hθ2,t +

ut

σ2u

nt . (A.26)

The parameters ω and η enter through loght and log g t ,

∂`t

∂ω= B(zt ,ut )hω,t +D(zt ,ut )gω,t ,

∂`t

∂η= B(zt ,ut )hη,t +D(zt ,ut )gη,t . (A.27)

The parameter σ2u only enters directly in the log-likelihood contribution such that

∂`t

∂σ2u

= 1

2

u2t −σ2

u

σ2u

. (A.28)

Stacking the above scores,

∂`t

∂θ=

(∂`t

∂µ,∂`t

∂θ′1,∂`t

∂θ′2,∂`t

∂ω,∂`t

∂η′,∂`t

∂σ2u

)′, (A.29)

yields the result in Proposition 1.

A.1. DERIVATION OF SCORE FUNCTION 37

A.1.1 Derivatives specific to the long-run component

In the REGARCH-HAR with f (·;η) given by (1.11), we have η= (γ1,γ2)′ such that

gη,t =(

15

∑5i=1 log xt−i−1

122

∑22i=1 log xt−i−1

). (A.30)

In the two-parameter REGARCH-MIDAS with f (·;η) given by (1.9), we have η =(λ,γ1,γ2)′ such that

gη,t =

∑Kk=1πk (γ1,γ2)yt−1,k∑K

k=1

(γ1−1)(1− k

K

)γ2−1(kK

)γ1−1 ∑Kj=1

(1− j

K

)γ2−1(

kK −

(j

K

)−1)(

jK

)γ1,i −1

[∑Kj=1

(j

K

)γ1−1(1− j

K

)γ2−1]2 yt−1,k

∑Kk=1

(γ2−1)(1− k

K

)γ2−1(kK

)γ1−1 ∑Kj=1

(1− j

K

)γ2−1(1− k

K −(1− j

K

)−1)(

jK

)γ1−1

[∑Kj=1

(j

K

)γ1−1(1− j

K

)γ2−1]2 yt−1,k

.

(A.31)

In the single-parameter REGARCH-MIDAS with f (·;η) given by (1.9), we have η =(λ,γ2)′ such that

gη,t =

∑K

k=1πk (γ1,γ2)yt−1,k∑Kk=1

(γ2−1)(1− k

K

)γ2−1 ∑Kj=1

(1− j

K

)γ2−1(1− k

K −(1− j

K

)−1)

[∑Kj=1

(1− j

K

)γ2−1]2 yt−1,k

. (A.32)

38 CHAPTER 1.

A.2 Figures

This page is intentionally left blank

A.2. FIGURES 39

Figure A.1: Standardized empirical distribution of estimated parametersThis figure depicts the standardized empirical distribution of a subset of the QML parametersusing a parametric bootstrap with resampling of the empirical residuals from the estimationon SPY (Paparoditis and Politis, 2009). We use 999 bootstrap replications and a sample size of2500 observations in the estimation. The left column depicts results for the original REGARCH,the middle column for the weekly, single-parameter REGARCH-MIDAS, and the right columnfor the REGARCH-HAR.

40 CHAPTER 1.

2002 2005 2008 2010 2013-0.15

-0.1

-0.05

0

0.05

0.1

0.15

2002 2005 2008 2010 20130

1

2

3

4

5

6

2002 2005 2008 2010 20130

0.5

1

1.5

2

2.5

3

50 100 150 200 2500

0.2

0.4

0.6

0.8

1

Figure A.2: Summary statistics for SPY daily returns and realized kernelThis figure depicts the evolution of SPY daily returns (upper-left panel), annualized squaredreturns (upper-right panel), annualized realized kernel (lower-left panel), and autocorrelationfunction of the logarithm of the realized kernel (lower-right panel). The solid line indicates theconventional autocorrelation function, whereas the dashed line indicates the instrumentedvariable autocorrelation function of Hansen and Lunde (2014) using their preferred instru-ments (four through ten) and optimal combination.

A.2. FIGURES 41

1 5 10 15 20 25 30 35 40

Lags (weeks)

0

0.1

0.2

0.3

0.4

0.5

0.6

K = 4K = 10K = 15K = 20K = 25K = 30K = 40K = 50K = 75K = 100

1 5 10 15 20 25 30 35 40

Lags (weeks)

0

0.1

0.2

0.3

0.4

0.5

0.6

K = 4K = 10K = 15K = 20K = 25K = 30K = 40K = 50K = 75K = 100

10 20 30 40 50 60 70 80 90 100

K

-5595

-5590

-5585

-5580

-5575

10 20 30 40 50 60 70 80 90 100

K

-5595

-5590

-5585

-5580

-5575

Figure A.3: Backward-looking horizon, K , for weekly REGARCH-MIDASThis figure depicts in the first row panel the estimated lag functions for SPY in the weeklytwo-parameter setting (left panel) and weekly single-parameter setting (right panel) for arange of values of K . The second row panel depicts the maximized log-likelihood values forK = 4, . . . ,104 weeks.

42 CHAPTER 1.

1 5 10 15 20 25

Lags (months)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K = 5K = 10K = 15K = 20K = 25

1 5 10 15 20 25

Lags (months)

0

0.1

0.2

0.3

0.4

0.5

0.6

K = 5K = 10K = 15K = 20K = 25

5 10 15 20 25

K

-5590

-5585

-5580

-5575

-5570

-5565

-5560

5 10 15 20 25

K

-5590

-5585

-5580

-5575

-5570

-5565

-5560

Figure A.4: Backward-looking horizon, K , for monthly REGARCH-MIDASThis figure depicts in the first row panel the estimated lag functions for SPY in the monthlytwo-parameter setting (left panel) and monthly single-parameter setting (right panel) for arange of values of K . The second row panel depicts the maximized log-likelihood values forK = 4, . . . ,26 months.

A.2. FIGURES 43

1 5 10 15 20 25

Lags (weeks)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Weekly REGARCH-MIDASWeekly REGARCH-MIDAS (single-parameter)Monthly REGARCH-MIDASMonthly REGARCH-MIDAS (single-parameter)REGARCH-HAR

Figure A.5: Estimated SPY weighting functionsThis figure depicts the estimated weighting functions in our proposed models for SPY withK = 52 and K = 12 in the weekly and monthly REGARCH-MIDAS, respectively. Blue lines relateto the weekly REGARCH-MIDAS, red lines relate to the monthly REGARCH-MIDAS, and thegreen line to the REGARCH-HAR. Solid lines refer to the two-parameter weighting function,whereas dashed lines refer to the restricted, single-parameter weighting function.

44 CHAPTER 1.

2002 2005 2008 2010 20130

0.2

0.4

0.6

0.8

1

1.2REGARCH-Spline

2002 2005 2008 2010 20130

0.2

0.4

0.6

0.8

1

1.2REGARCH-HAR

2002 2005 2008 2010 20130

0.2

0.4

0.6

0.8

1

1.2Weekly REGARCH-MIDAS (single-parameter)

2002 2005 2008 2010 20130

0.2

0.4

0.6

0.8

1

1.2Weekly REGARCH-MIDAS

2002 2005 2008 2010 20130

0.2

0.4

0.6

0.8

1

1.2Monthly REGARCH-MIDAS (single-parameter)

2002 2005 2008 2010 20130

0.2

0.4

0.6

0.8

1

1.2Monthly REGARCH-MIDAS

Figure A.6: Evolution of the conditional variance and the long-term componentThis figure depicts the evolution of the fitted conditional variance together with its long-term component from the multiplicative REGARCH modifications contained in Table A.3.The upper-left panel refers to the REGARCH-S, the upper-right panel to the REGARCH-HAR,the middle-left panel to the weekly single-parameter REGARCH-MIDAS, the middle-rightpanel to the weekly two-parameter REGARCH-MIDAS, the lower-left panel to the monthlysingle-parameter REGARCH-MIDAS, and lower-right panel to the monthly two-parameterREGARCH-MIDAS.

A.2. FIGURES 45

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleREGARCH

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleREGARCH(5,5)

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleWeekly REGARCH-MIDAS

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleWeekly REGARCH-MIDAS (single-parameter)

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleMonthly REGARCH-MIDAS

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleMonthly REGARCH-MIDAS (single-parameter)

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleREGARCH-HAR

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleFloEGARCH

Figure A.7: Simulated and sample autocorrelation function of logσ2t

This figure depicts the simulated (dashed line) and sample (solid line) autocorrelation functionof logσ2

t for the REGARCH, REGARCH(5,5), REGARCH-MIDAS, REGARCH-HAR and the FloE-GARCH. We use the estimated parameters for SPY reported in Table A.3 and K = 52 (K = 12)for the weekly (monthly) REGARCH-MIDAS. See Section 1.4.2 for additional details on theircomputation.

46 CHAPTER 1.

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleREGARCH

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleREGARCH(5,5)

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleWeekly REGARCH-MIDAS

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleWeekly REGARCH-MIDAS (single-parameter)

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleMonthly REGARCH-MIDAS

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleMonthly REGARCH-MIDAS (single-parameter)

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleREGARCH-HAR

50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SampleFloEGARCH

Figure A.8: Simulated and sample autocorrelation function of logRKtThis figure depicts the simulated (dashed line) and sample (solid line) autocorrelation functionof logRKt for the REGARCH, REGARCH(5,5), REGARCH-MIDAS, REGARCH-HAR and theFloEGARCH. We use the estimated parameters for SPY reported in Table A.3 and K = 52(K = 12) for the weekly (monthly) REGARCH-MIDAS. See Section 1.4.2 for additional details ontheir computation.

A.2. FIGURES 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220.9

1.0

1.1

1.2

1.3

1.4

1.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220.9

1.0

1.1

1.2

1.3

1.4

1.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220.9

1.0

1.1

1.2

1.3

1.4

1.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 220.9

1.0

1.1

1.2

1.3

1.4

1.5

Figure A.9: Forecast evaluation of REGARCH-MIDAS and REGARCH-HARThis figure depicts the ratio of forecast losses of the REGARCH-MIDAS and REGARCH-HARto the original REGARCH. Values exceeding unity indicate improvements in predictive abilityof our proposed models. Full circles indicate whether difference in forecast loss (for a givenforecast horizon) is significant on a 5% significance level using a Diebold-Mariano test forequal predictive ability. Empty circles indicate insignificance. See Section 1.4.3 for additionaldetails. The left panel uses the QLIKE loss function in (1.41), whereas the right panel uses theSPE loss function in (1.40). The upper panel reports results for the weekly single-parameterREGARCH-MIDAS and the lower panel for the REGARCH-HAR (results for the remainingREGARCH-MIDAS specifications are similar and are available upon request).

48 CHAPTER 1.

A.3 Tables

This page is intentionally left blank

A.3. TABLES 49

Tab

leA

.1:S

um

mar

yst

atis

tics

for

dai

lyre

turn

san

dre

aliz

edke

rnel

Th

ista

ble

rep

ort

ssu

mm

ary

stat

isti

csfo

rth

ed

aily

retu

rns

and

the

loga

rith

mo

fth

ere

aliz

edke

rnel

.D

aily

retu

rns

and

the

real

ized

kern

elar

ein

per

cen

tage

s.R

obu

stsk

ewn

ess

and

kurt

osis

are

from

Kim

and

Wh

ite

(200

4).T

he

frac

tion

alin

tegr

ated

par

amet

erd

ises

tim

ated

usi

ng

the

two-

step

exac

tlo

calW

hit

tle

esti

mat

or

ofS

him

ots

u(2

010)

and

ban

dw

idth

cho

ice

ofm

=bT

0.65c.

Ret

urn

Log(

RK

)

No.

of

ob

s.M

ean

Std

.D

ev.

Skew

Ro

bu

stSk

ewE

x.K

urt

.R

ob

ust

Ex.

Ku

rt.

Med

ian

Mea

nSt

d.

Dev

.M

edia

nd

SP50

03,

020

0.02

1.32

0.07

-0.0

811

.67

1.03

0.08

-0.3

51.

00-0

.50

0.66

AA

3,00

40.

002.

730.

230.

028.

920.

990.

001.

130.

860.

980.

64A

IG2,

999

-0.0

04.

551.

420.

0154

.58

1.17

-0.0

31.

071.

300.

880.

64A

XP

2,99

40.

072.

440.

550.

0411

.12

1.07

0.02

0.60

1.18

0.38

0.70

BA

2,99

60.

071.

890.

230.

014.

070.

840.

070.

600.

820.

470.

64C

AT2,

998

0.07

2.09

0.11

0.03

5.06

0.92

0.06

0.73

0.82

0.59

0.67

DD

2,99

50.

041.

78-0

.04

0.01

5.68

0.88

0.04

0.51

0.85

0.38

0.63

DIS

2,99

70.

071.

910.

51-0

.02

6.76

0.88

0.06

0.55

0.88

0.38

0.66

GE

3,00

50.

021.

990.

380.

0310

.30

1.06

0.00

0.40

1.05

0.22

0.68

IBM

2,99

60.

031.

530.

200.

016.

870.

870.

020.

100.

83-0

.05

0.65

INT

C3,

016

0.03

2.20

-0.2

2-0

.00

6.09

0.90

0.04

0.85

0.80

0.74

0.63

JNJ

2,99

70.

031.

16-0

.28

0.03

20.0

60.

950.

02-0

.28

0.86

-0.4

30.

68K

O2,

999

0.04

1.24

0.32

-0.0

211

.96

0.92

0.04

-0.1

00.

81-0

.22

0.63

MM

M2,

992

0.05

1.45

-0.0

60.

025.

540.

970.

060.

130.

800.

030.

64M

RK

2,99

40.

031.

80-1

.21

0.04

24.1

80.

870.

030.

380.

850.

260.

61M

SFT

3,01

60.

031.

810.

370.

028.

940.

960.

000.

460.

810.

320.

63P

G2,

998

0.04

1.14

-0.0

20.

016.

740.

920.

03-0

.18

0.77

-0.3

00.

61V

Z2,

995

0.03

1.57

0.34

-0.0

37.

370.

900.

050.

310.

890.

140.

67W

HR

2,99

20.

072.

520.

400.

035.

140.

96-0

.01

1.01

0.86

0.92

0.58

WM

T3,

001

0.03

1.31

0.30

-0.0

35.

570.

880.

020.

050.

80-0

.09

0.65

XO

M3,

001

0.05

1.60

0.34

-0.0

113

.37

0.81

0.07

0.24

0.86

0.14

0.68

50 CHAPTER 1.

Table A.2: Persistence parameters (π) and unit root tests (DF)This table reports estimated autoregressive persistence parameters, π, and unit roottests, DF. The first column contains the conventional least squares estimator, whereasthe following two columns contain the instrumented variables estimator from Hansenand Lunde (2014) using the first lag as instrument and their preferred specifica-tion (four through ten) with optimal combination, respectively. The following threecolumns contain the Dickey-Fuller unit root test using each estimate of the per-sistence parameter. The 1%, 5% and 10% critical values are -20.7, -14.1 and -11.3,respectively (see Fuller (1996), Table 10.A.1).

πOLS π1 π4:10 DFOLS DF1 DF4:10

SP500 0.883 0.959 0.985 -354.3 -124.8 -45.8

AA 0.865 0.961 0.985 -405.3 -116.6 -44.8AIG 0.919 0.966 0.990 -242.4 -103.1 -30.5AXP 0.926 0.980 0.992 -222.7 -59.0 -23.5BA 0.847 0.956 0.987 -458.6 -131.7 -37.4CAT 0.866 0.949 0.988 -400.6 -151.9 -35.6DD 0.856 0.952 0.983 -431.6 -143.8 -51.8DIS 0.866 0.956 0.986 -401.6 -132.3 -41.4GE 0.904 0.969 0.990 -287.6 -93.8 -30.6IBM 0.870 0.959 0.983 -389.5 -122.3 -52.0INTC 0.869 0.951 0.985 -395.4 -148.0 -45.5JNJ 0.852 0.955 0.988 -443.6 -134.3 -37.0KO 0.836 0.953 0.985 -492.7 -140.4 -45.7MMM 0.833 0.940 0.981 -499.5 -178.2 -57.3MRK 0.815 0.942 0.983 -552.7 -174.7 -49.6MSFT 0.857 0.951 0.981 -429.7 -146.7 -56.3PG 0.818 0.937 0.980 -546.2 -188.9 -58.5VZ 0.861 0.961 0.987 -414.7 -118.0 -38.5WHR 0.823 0.938 0.986 -528.4 -186.6 -41.5WMT 0.844 0.957 0.985 -467.4 -127.8 -43.8XOM 0.878 0.954 0.980 -366.7 -137.8 -59.4

A.3. TABLES 51

Tab

leA

.3:F

ull

sam

ple

resu

lts

for

SPY

Th

ista

ble

rep

orts

esti

mat

edp

aram

eter

s,ro

bu

stst

and

ard

erro

rs(u

sin

gth

esa

nd

wic

hfo

rmu

laan

dre

por

ted

inp

aren

thes

es),

nu

mb

erof

par

amet

ers

(p),

info

rmat

ion

crit

eria

,var

ian

cera

tio

fro

m(1

.34)

and

par

tial

,as

wel

las

full

max

imiz

edlo

g-lik

elih

oo

dva

lue

for

each

mo

del

un

der

con

sid

erat

ion

.Res

ult

sfo

rth

eR

EG

AR

CH

-MID

AS

are

for

K=

52(K

=12

)in

the

wee

kly

(mo

nth

ly)

case

.

EG

AR

CH

RE

GA

RC

HR

EG

AR

CH

-MID

AS

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-MID

AS

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-HA

RR

EG

AR

CH

-SFl

oE

GA

RC

H(w

eekl

y)(w

eekl

y)(m

on

thly

)(m

on

thly

)(s

ingl

e-p

aram

eter

)(s

ingl

e-p

aram

eter

)

µ0.

020(

0.01

35)

0.01

6(0.

0127

)0.

015(

0.01

44)

0.01

5(0.

0143

)0.

016(

0.01

43)

0.01

6(0.

0143

)0.

014(

0.01

44)

0.02

4(0.

0143

)0.

015(

0.01

01)

β0.

981(

0.00

25)

0.97

2(0.

0036

)0.

761(

0.01

66)

0.84

2(0.

0118

)0.

872(

0.00

98)

0.88

0(0.

0094

)0.

734(

0.01

80)

0.94

3(0.

0058

)0.

176(

0.02

74)

α0.

121(

0.01

44)

0.33

8(0.

0225

)0.

337(

0.02

74)

0.32

9(0.

0250

)0.

324(

0.02

39)

0.32

4(0.

0238

)0.

355(

0.02

70)

0.32

4(0.

0216

)0.

370(

0.02

26)

ξ−0

.265

(0.0

267)

−0.2

71(0

.026

9)−0

.270

(0.0

269)

−0.2

69(0

.026

9)−0

.269

(0.0

269)

−0.2

72(0

.026

8)−0

.264

(0.0

264)

−0.2

74(0

.026

7)σ

2 u0.

155(

0.00

58)

0.15

0(0.

0057

)0.

150(

0.00

57)

0.15

0(0.

0057

)0.

150(

0.00

57)

0.15

1(0.

0057

)0.

153(

0.00

57)

0.15

0(0.

0057

)τ

1−0

.138

(0.0

118)

−0.1

48(0

.007

4)−0

.170

(0.0

084)

−0.1

66(0

.008

1)−0

.164

(0.0

079)

−0.1

63(0

.007

9)−0

.171

(0.0

085)

−0.1

50(0

.007

5)−0

.170

(0.0

083)

τ2

0.04

0(0.

0049

)0.

047(

0.00

55)

0.04

5(0.

0053

)0.

044(

0.00

51)

0.04

4(0.

0051

)0.

047(

0.00

56)

0.04

1(0.

0051

)0.

051(

0.00

54)

δ1

−0.1

13(0

.008

3)−0

.115

(0.0

083)

−0.1

15(0

.008

3)−0

.115

(0.0

083)

−0.1

15(0

.008

3)−0

.114

(0.0

083)

−0.1

12(0

.008

4)−0

.115

(0.0

082)

δ2

0.04

9(0.

0059

)0.

051(

0.00

60)

0.05

0(0.

0059

)0.

050(

0.00

59)

0.05

0(0.

0059

)0.

051(

0.00

60)

0.05

0(0.

0062

)0.

051(

0.00

59)

φ0.

962(

0.02

53)

0.96

8(0.

0167

)0.

969(

0.01

87)

0.97

0(0.

0198

)0.

970(

0.02

01)

0.96

4(0.

0163

)0.

961(

0.02

32)

0.96

9(0.

0231

)ω

0.05

8(0.

1632

)−0

.089

(0.1

255)

0.24

3(0.

0397

)0.

225(

0.04

58)

0.22

2(0.

0499

)0.

213(

0.05

15)

0.23

5(0.

0386

)0.

175(

0.23

66)

−0.0

92(0

.167

0)λ

0.94

7(0.

0298

)0.

906(

0.03

27)

0.91

4(0.

0426

)0.

888(

0.03

97)

γ1

0.02

5(0.

4475

)−0

.518

(0.7

866)

0.29

4(0.

0429

)γ

26.

337(

6.33

25)

12.5

45(1

.834

9)2.

063(

2.84

31)

8.50

8(1.

4365

)0.

620(

0.04

61)

d0.

649(

0.01

42)

p5

1114

1314

1313

1812

logL

−5,6

23.5

5−5

,577

.52

−5,5

78.0

2−5

,577

.52

−5,5

78.2

3−5

,595

.10

−5,5

89.4

4−5

,584

.65

logL

p−4

,213

.84

−4,1

48.7

1−4

,159

.70

−4,1

57.4

3−4

,156

.60

−4,1

56.1

0−4

,160

.22

−4,1

48.6

2−4

,162

.54

AIC

11,2

69.1

011

,183

.04

11,1

82.0

311

,183

.04

11,1

82.4

611

,216

.20

11,2

14.8

711

,193

.29

BIC

11,3

35.2

411

,267

.23

11,2

60.2

011

,267

.22

11,2

60.6

211

,294

.37

11,3

23.1

111

,265

.45

HQ

IC11

,423

.39

11,3

79.4

111

,364

.37

11,3

79.4

011

,364

.79

11,3

98.5

311

,467

.34

11,3

61.6

0V

R0.

730.

650.

610.

590.

740.

40

52 CHAPTER 1.Tab

leA

.4:Differen

cein

maxim

izedlo

g-likeliho

od

relativeto

RE

GA

RC

HT

his

table

repo

rtsth

ed

ifferences

inth

em

aximized

log-likelih

oo

dvalu

esfo

ro

ur

pro

po

sedm

od

elsan

dth

eR

EG

AR

CH

-Splin

ean

dFlo

EG

AR

CH

relativeto

the

origin

alRE

GA

RC

H.P

ositive

values

ind

icateim

provem

ents

inem

piricalfi

t.We

repo

rtresu

ltsfo

rall20

ind

ividu

alstocks

and

inclu

de

SPY

for

com

parative

pu

rpo

ses.Gray

shad

edareas

ind

icateth

em

od

elwith

the

high

estlikeliho

od

gainrelative

toth

eR

EG

AR

CH

.

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-MID

AS

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-MID

AS

RE

GA

RC

H-H

AR

RE

GA

RC

H-S

FloE

GA

RC

H(w

eekly)(w

eekly)(m

on

thly)

(mo

nth

ly)(sin

gle-param

eter)(sin

gle-param

eter)

SP500

46.045.5

46.045.3

28.534.1

38.9

AA

46.545.0

41.340.9

35.928.4

40.1A

IG126.2

120.0116.3

112.5119.7

103.3123.2

AX

P41.6

40.941.6

41.427.4

35.840.5

BA

43.542.4

36.936.9

28.227.7

39.4C

AT50.6

47.334.8

34.441.3

23.542.6

DD

34.931.0

30.630.2

19.932.9

29.3D

IS53.0

50.737.6

36.838.4

34.943.7

GE

40.940.6

40.140.1

28.037.4

41.3IB

M31.1

27.322.5

22.519.5

19.916.9

INT

C63.4

60.450.8

50.845.6

43.352.7

JNJ

30.830.8

22.221.5

19.223.4

26.7K

O41.6

37.531.9

31.035.3

31.434.0

MM

M35.0

32.726.2

25.922.2

20.732.0

MR

K31.3

27.723.6

23.021.0

36.027.5

MSF

T47.6

43.641.4

41.135.9

36.436.5

PG

46.742.3

28.427.7

37.822.5

31.3V

Z29.0

28.923.7

23.714.7

17.320.4

WH

R71.8

69.169.2

68.967.0

48.058.4

WM

T40.6

36.129.4

29.328.5

24.128.4

XO

M44.0

40.326.0

25.335.8

22.816.2

SP500

46.045.5

46.045.3

28.534.1

38.9

Mean

47.444.8

39.138.5

35.733.5

39.0

A.3. TABLES 53

Tab

leA

.5:E

stim

ated

βac

ross

vari

ou

sR

EG

AR

CH

spec

ifica

tio

ns

Th

ista

ble

rep

orts

esti

mat

edβ

for

our

pro

pos

edm

odel

san

dth

eR

EG

AR

CH

-Sp

line

and

FloE

GA

RC

Hre

lati

veto

the

orig

inal

RE

GA

RC

H.W

ere

por

tres

ult

sfo

ral

l20

ind

ivid

ual

sto

cks

and

incl

ud

eSP

Yfo

rco

mp

arat

ive

pu

rpo

ses.

RE

GA

RC

HR

EG

AR

CH

-MID

AS

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-MID

AS

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-HA

RR

EG

AR

CH

-SFl

oE

GA

RC

H(w

eekl

y)(w

eekl

y)(m

on

thly

)(m

on

thly

)(s

ingl

e-p

aram

eter

)(s

ingl

e-p

aram

eter

)

SP50

00.

972

0.76

10.

842

0.87

20.

880

0.73

40.

943

0.17

6

AA

0.97

20.

649

0.71

10.

820

0.82

40.

638

0.93

30.

195

AIG

0.97

20.

577

0.62

30.

742

0.73

70.

578

0.86

40.

121

AX

P0.

987

0.71

00.

790

0.86

10.

865

0.81

10.

943

0.19

5B

A0.

977

0.60

60.

656

0.83

70.

836

0.58

30.

936

0.13

3C

AT0.

973

0.55

30.

584

0.82

50.

831

0.53

70.

941

0.10

4D

D0.

972

0.62

30.

710

0.86

40.

871

0.59

10.

932

0.17

9D

IS0.

981

0.52

00.

554

0.83

20.

841

0.50

80.

942

0.08

0G

E0.

982

0.81

70.

742

0.84

30.

845

0.80

10.

935

0.11

7IB

M0.

974

0.61

20.

638

0.89

10.

893

0.56

80.

948

0.16

1IN

TC

0.97

20.

602

0.65

40.

812

0.81

20.

565

0.91

30.

131

JNJ

0.97

60.

627

0.62

00.

850

0.85

60.

570

0.95

00.

093

KO

0.97

30.

562

0.61

70.

827

0.82

90.

558

0.93

20.

121

MM

M0.

967

0.58

70.

634

0.86

30.

856

0.55

40.

938

0.13

7M

RK

0.97

10.

552

0.65

70.

864

0.84

90.

494

0.92

60.

108

MSF

T0.

968

0.61

30.

689

0.82

60.

834

0.58

50.

917

0.15

8P

G0.

961

0.54

70.

567

0.83

10.

837

0.52

10.

930

0.13

8V

Z0.

979

0.64

90.

691

0.87

50.

871

0.59

90.

953

0.15

6W

HR

0.96

30.

554

0.63

10.

745

0.74

20.

531

0.90

50.

092

WM

T0.

979

0.53

60.

582

0.85

20.

850

0.50

00.

947

0.11

3X

OM

0.96

70.

598

0.61

00.

873

0.88

00.

561

0.94

70.

150

Mea

n0.

973

0.61

20.

657

0.83

80.

840

0.59

00.

932

0.13

6

54 CHAPTER 1.

Table

A.6:Fo

recastevaluatio

nfo

rin

divid

ualsto

cksT

his

table

repo

rtsa

sum

mary

ofth

ek

-steps

ahead

pred

ictiveab

ilityo

fthe

RE

GA

RC

H-H

AR

and

weekly

single-p

arameter

RE

GA

RC

H-M

IDA

Sb

ench

-m

arkedagain

stthe

originalR

EG

AR

CH

.Statisticalsignifi

cance

ofthe

differen

cesin

forecastlossesis

assessedby

mean

softh

eD

iebold

-Marian

otestfor

equ

alpred

ictiveab

ility.For

eachfo

recastho

rizon

and

stock

we

categorize

the

ou

tcom

eo

fthe

testaccord

ing

toth

esize

ofth

ep

-value

and

repo

rtthe

nu

mb

ero

fstocks

falling

into

eachcatego

ryin

the

table.Fo

rin

stance,th

elastrow

inth

eleftPan

elAin

dicates

thatfo

rth

e22-step

sah

eadfo

recast,the

weekly

RE

GA

RC

H-M

IDA

So

utp

erform

sR

EG

AR

CH

15/20tim

esatth

e1%

signifi

cance

leveland

1/20tim

esatth

e10%

signifi

cance

level(bu

tno

tatthe

5%an

d1%

level),wh

ereas2/20

times

the

differen

cein

forecastp

erform

ance

was

insign

ifican

tand

2/20tim

esth

eR

EG

AR

CH

was

perfo

rmin

gth

eb

est.A

san

oth

erm

easure

ofim

provem

ent,w

ealso

repo

rtth

em

edian

ratioo

fforecast

loss

ofo

ur

pro

po

sedm

od

elsrelative

toth

eo

riginalR

EG

AR

CH

.An

um

ber

above

un

ityin

dicates

sup

erior

perfo

rman

ceo

fou

rp

rop

osed

mo

del.See

Section

1.4.3fo

rad

ditio

nald

etails.

Pan

elA:Q

LIK

Elo

ssfu

nctio

n

Weekly

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-HA

R

Ho

rizon

1%5%

10%In

sign.

RE

GA

RC

HM

edian

loss

ratioH

orizo

n1%

5%10%

Insign

.R

EG

AR

CH

Med

ianlo

ssratio

k=

15

23

55

1.02k

=1

41

15

91.01

k=

26

32

54

1.03k

=2

32

17

71.02

k=

39

22

43

1.05k

=3

44

14

71.02

k=

49

11

63

1.05k

=4

70

32

81.02

k=

59

11

54

1.05k

=5

71

24

61.03

k=

1011

22

14

1.11k

=10

101

22

51.09

k=

1513

20

14

1.22k

=15

121

12

41.15

k=

2215

01

22

1.33k

=22

131

04

21.26

Pan

elB:Sq

uared

Pred

ition

Erro

rlo

ssfu

nctio

n

Weekly

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-HA

R

Ho

rizon

1%5%

10%In

sign.

RE

GA

RC

HM

edian

loss

ratioH

orizo

n1%

5%10%

Insign

.R

EG

AR

CH

Med

ianlo

ssratio

k=

16

21

47

1.01k

=1

34

11

111.00

k=

210

21

43

1.04k

=2

45

21

81.02

k=

39

41

42

1.08k

=3

65

02

71.03

k=

49

50

42

1.08k

=4

73

03

71.04

k=

59

41

33

1.09k

=5

82

12

71.04

k=

1013

30

22

1.17k

=10

121

00

71.11

k=

1514

12

12

1.30k

=15

121

01

61.19

k=

2215

20

12

1.48k

=22

111

11

61.32

A.3. TABLES 55

Tab

leA

.7:M

od

elC

on

fid

ence

Sete

valu

atio

nfo

rin

div

idu

alst

ock

san

dSP

YT

his

tab

lere

po

rts

the

per

cen

tage

of

sto

cks

and

SPY

for

wh

ich

agi

ven

mo

del

isin

clu

ded

inth

eM

CS

on

a10

%si

gnifi

can

cele

vela

nd

on

e-st

epan

dm

ult

i-st

epah

ead

fore

cast

ing.

For

inst

ance

,th

eR

EG

AR

CH

was

incl

ud

edin

the

MC

Sfo

r33

%o

fth

est

ock

sw

ith

afo

reca

sth

ori

zon

of

k=

22w

hen

eval

uat

edu

sin

gth

eQ

LIK

Elo

ssfu

nct

ion

,wh

erea

sth

em

on

thly

RE

GA

RC

H-M

IDA

Sw

asin

clu

ded

inth

eM

CS

for

90%

of

the

sto

cks.

We

use

ab

lock

bo

ots

trap

wit

h10

5in

the

imp

lem

enta

tio

n.S

eeSe

ctio

n1.

4.3

for

add

itio

nal

det

ails

.

Pan

elA

:QL

IKE

loss

fun

ctio

n

Ho

rizo

nR

EG

AR

CH

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-MID

AS

RE

GA

RC

H-M

IDA

SR

EG

AR

CH

-MID

AS

RE

GA

RC

H-H

AR

RE

GA

RC

H-S

Flo

EG

AR

CH

(wee

kly)

(wee

kly)

(mo

nth

ly)

(mo

nth

ly)

(sin

gle-

par

amet

er)

(sin

gle-

par

amet

er)

k=

10.

570.

670.

810.

520.

570.

710.

380.

76k

=2

0.29

0.67

0.71

0.52

0.52

0.67

0.38

0.05

k=

30.

330.

670.

710.

520.

570.

670.

430.

10k

=4

0.38

0.76

0.81

0.71

0.67

0.67

0.43

0.14

k=

50.

430.

710.

760.

810.

760.

670.

520.

38k

=10

0.29

0.67

0.76

0.81

0.76

0.67

0.52

0.76

k=

150.

290.

710.

860.

900.

760.

710.

520.

81k

=22

0.33

0.76

0.76

0.90

0.90

0.71

0.76

0.86

Pan

elB

:Sq

uar

edP

red

itio

nE

rro

rlo

ssfu

nct

ion

k=

10.

430.

570.

710.

570.

670.

520.

520.

86k

=2

0.29

0.76

0.81

0.67

0.71

0.62

0.48

0.19

k=

30.

330.

760.

810.

670.

670.

670.

430.

10k

=4

0.33

0.76

0.81

0.67

0.57

0.62

0.52

0.19

k=

50.

290.

760.

810.

760.

760.

710.

520.

33k

=10

0.10

0.71

0.81

0.81

0.86

0.76

0.62

0.76

k=

150.

240.

710.

860.

900.

860.

760.

760.

81k

=22

0.14

0.71

0.76

0.86

0.86

0.67

0.67

0.86

56 CHAPTER 1.

A.4 In-sample results for individual stocks

Table A.8: REGARCHThis table reports full-sample estimated parameters, information criteria as well as full maxi-mized log-likelihood value for the original REGARCH.

AA AIG AXP BA CAT DD DIS GE IBM INTC

µ 0.015 -0.017 0.051 0.074 0.074 0.041 0.053 0.021 0.029 0.026β 0.972 0.972 0.987 0.977 0.973 0.972 0.981 0.982 0.974 0.972α 0.355 0.606 0.394 0.322 0.376 0.413 0.356 0.418 0.438 0.478ξ -0.518 -0.296 -0.385 -0.446 -0.590 -0.207 -0.327 -0.342 -0.375 -0.285σ2

u 0.136 0.201 0.148 0.135 0.130 0.147 0.146 0.153 0.129 0.129τ1 -0.054 -0.086 -0.085 -0.062 -0.056 -0.076 -0.076 -0.063 -0.072 -0.051τ2 0.039 0.039 0.039 0.036 0.016 0.023 0.022 0.029 0.014 0.021δ1 -0.061 -0.049 -0.065 -0.054 -0.069 -0.071 -0.079 -0.045 -0.063 -0.038δ2 0.063 0.042 0.060 0.076 0.043 0.048 0.047 0.049 0.037 0.036φ 1.055 0.855 0.993 1.046 1.105 0.961 0.981 0.985 0.962 0.924ω 1.544 1.513 1.135 1.041 1.155 0.770 1.067 0.865 0.549 1.353

logL -7,883.37 -8,493.81 -7,122.76 -7,000.25 -7,237.72 -6,751.76 -6,965.93 -6,832.49 -6,176.02 -7,343.71AIC 15,788.74 17,009.63 14,267.52 14,022.51 14,497.45 13,525.52 13,953.85 13,686.98 12,374.05 14,709.42BIC 15,854.83 17,075.70 14,333.57 14,088.57 14,563.51 13,591.57 14,019.91 13,753.08 12,440.11 14,775.55

HQIC 15,942.92 17,163.77 14,421.62 14,176.62 14,651.58 13,679.63 14,107.98 13,841.17 12,528.17 14,863.68

JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM

µ 0.030 0.029 0.044 0.023 0.035 0.026 0.028 0.074 0.017 0.042β 0.976 0.973 0.967 0.971 0.968 0.961 0.979 0.963 0.979 0.967α 0.359 0.399 0.357 0.257 0.447 0.373 0.334 0.287 0.307 0.357ξ -0.123 -0.150 -0.342 -1.017 -0.381 -0.159 -0.185 -1.112 -0.259 -0.291σ2

u 0.151 0.144 0.147 0.191 0.137 0.151 0.153 0.171 0.137 0.123τ1 -0.066 -0.065 -0.080 -0.034 -0.044 -0.059 -0.064 -0.040 -0.035 -0.087τ2 0.038 0.027 0.010 0.001 0.013 0.025 0.039 0.024 0.028 0.041δ1 -0.031 -0.053 -0.071 -0.057 -0.036 -0.053 -0.061 -0.043 -0.028 -0.105δ2 0.058 0.064 0.041 0.010 0.033 0.056 0.059 0.068 0.058 0.050φ 0.961 0.949 1.067 1.450 1.003 1.084 1.009 1.344 1.120 1.082ω -0.071 0.112 0.437 0.974 0.900 0.021 0.530 1.538 0.350 0.554

logL -5,426.50 -5,694.49 -6,291.30 -7,470.15 -6,832.84 -5,651.66 -6,419.04 -8,210.50 -5,951.18 -6,113.27AIC 10,875.01 11,410.98 12,604.59 14,962.29 13,687.67 11,325.33 12,860.09 16,443.00 11,924.36 12,248.54BIC 10,941.07 11,477.05 12,670.64 15,028.34 13,753.80 11,391.39 12,926.14 16,509.04 11,990.44 12,314.61

HQIC 11,029.13 11,565.12 12,758.68 15,116.40 13,841.94 11,479.46 13,014.20 16,597.08 12,078.52 12,402.69

A.4. IN-SAMPLE RESULTS FOR INDIVIDUAL STOCKS 57

Table A.9: Weekly REGARCH-MIDASThis table reports full-sample estimated parameters, information criteria, variance ratio from(1.34) as well as full maximized log-likelihood value for the weekly two-parameter REGARCH-MIDAS. Results are for K = 52.


µ 0.011 -0.016 0.056 0.078 0.071 0.044 0.063 0.025 0.033 0.031β 0.649 0.577 0.710 0.606 0.553 0.623 0.520 0.817 0.612 0.602α 0.392 0.589 0.410 0.360 0.432 0.455 0.429 0.452 0.479 0.531ξ -0.487 -0.291 -0.392 -0.441 -0.572 -0.202 -0.322 -0.333 -0.378 -0.276σ2

u 0.133 0.191 0.144 0.132 0.126 0.144 0.142 0.149 0.127 0.125τ1 -0.062 -0.094 -0.096 -0.076 -0.062 -0.090 -0.089 -0.068 -0.086 -0.061τ2 0.044 0.050 0.048 0.047 0.022 0.025 0.029 0.035 0.015 0.018δ1 -0.063 -0.048 -0.065 -0.054 -0.069 -0.071 -0.080 -0.046 -0.064 -0.040δ2 0.061 0.047 0.059 0.075 0.042 0.047 0.045 0.050 0.037 0.032φ 1.039 0.872 0.999 1.044 1.093 0.959 0.976 0.978 0.963 0.920ω 0.538 0.361 0.402 0.448 0.547 0.224 0.340 0.362 0.392 0.331λ 0.900 1.117 0.973 0.908 0.873 0.999 0.993 0.945 0.981 1.038γ1 -0.156 -0.577 -0.041 -0.214 -0.822 -0.758 -0.866 2.003 -0.971 -0.296γ2 6.538 1.481 6.938 6.994 2.134 1.000 1.130 27.928 1.004 4.883

logL -7,836.88 -8,367.58 -7,081.18 -6,956.80 -7,187.14 -6,716.86 -6,912.97 -6,791.63 -6,144.97 -7,280.34AIC 15,701.75 16,763.15 14,190.36 13,941.60 14,402.28 13,461.72 13,853.93 13,611.26 12,317.94 14,588.67BIC 15,785.86 16,847.24 14,274.42 14,025.67 14,486.36 13,545.79 13,938.01 13,695.38 12,402.01 14,672.84

HQIC 15,897.98 16,959.33 14,386.49 14,137.75 14,598.44 13,657.86 14,050.09 13,807.49 12,514.09 14,785.01VR 0.82 0.87 0.89 0.85 0.85 0.82 0.89 0.80 0.83 0.84


µ 0.034 0.032 0.045 0.025 0.038 0.028 0.031 0.067 0.021 0.051β 0.627 0.562 0.587 0.552 0.613 0.547 0.649 0.554 0.536 0.598α 0.411 0.453 0.415 0.297 0.492 0.416 0.384 0.315 0.342 0.382ξ -0.128 -0.149 -0.340 -1.002 -0.365 -0.158 -0.177 -1.055 -0.254 -0.288σ2

u 0.147 0.141 0.143 0.188 0.133 0.147 0.150 0.164 0.133 0.120τ1 -0.076 -0.071 -0.086 -0.039 -0.048 -0.068 -0.071 -0.048 -0.045 -0.102τ2 0.049 0.036 0.018 0.002 0.015 0.031 0.048 0.032 0.036 0.046δ1 -0.029 -0.053 -0.070 -0.057 -0.038 -0.053 -0.060 -0.045 -0.027 -0.105δ2 0.058 0.065 0.041 0.010 0.033 0.055 0.058 0.067 0.056 0.049φ 0.958 0.936 1.060 1.437 0.986 1.071 1.000 1.315 1.117 1.084ω 0.103 0.149 0.324 0.712 0.391 0.132 0.192 0.861 0.225 0.279λ 0.949 1.017 0.873 0.647 0.951 0.863 0.938 0.701 0.855 0.851γ1 1.471 -0.875 -0.638 -0.563 -0.477 -0.957 0.001 -0.034 -0.776 -1.116γ2 48.950 1.000 3.334 4.365 3.124 1.000 11.393 7.918 2.321 1.000

logL -5,395.67 -5,652.84 -6,256.25 -7,438.83 -6,785.22 -5,604.99 -6,390.01 -8,138.73 -5,910.60 -6,069.31AIC 10,819.34 11,333.68 12,540.50 14,905.66 13,598.45 11,237.98 12,808.02 16,305.46 11,849.19 12,166.62BIC 10,903.42 11,417.77 12,624.56 14,989.73 13,682.61 11,322.06 12,892.09 16,389.52 11,933.29 12,250.72

HQIC 11,015.50 11,529.86 12,736.61 15,101.79 13,794.78 11,434.14 13,004.16 16,501.58 12,045.39 12,362.82VR 0.83 0.84 0.80 0.83 0.82 0.80 0.84 0.81 0.87 0.80

58 CHAPTER 1.

Table A.10: Weekly REGARCH-MIDAS (single-parameter)This table reports full-sample estimated parameters, information criteria, variance ratio from(1.34) as well as full maximized log-likelihood value for the weekly single-parameter REGARCH-MIDAS. Results are for K = 52.


µ 0.011 -0.017 0.057 0.078 0.071 0.044 0.064 0.025 0.033 0.032β 0.711 0.623 0.790 0.656 0.584 0.710 0.554 0.742 0.638 0.654α 0.390 0.592 0.408 0.359 0.433 0.451 0.431 0.459 0.482 0.532ξ -0.491 -0.301 -0.392 -0.442 -0.573 -0.204 -0.321 -0.333 -0.378 -0.276σ2

u 0.133 0.191 0.144 0.132 0.126 0.144 0.142 0.149 0.127 0.125τ1 -0.062 -0.095 -0.095 -0.076 -0.061 -0.089 -0.089 -0.068 -0.085 -0.060τ2 0.043 0.049 0.046 0.046 0.022 0.025 0.029 0.036 0.016 0.018δ1 -0.063 -0.048 -0.065 -0.054 -0.069 -0.071 -0.079 -0.046 -0.063 -0.040δ2 0.061 0.047 0.059 0.075 0.043 0.048 0.045 0.049 0.037 0.032φ 1.041 0.876 1.000 1.045 1.093 0.959 0.975 0.978 0.962 0.920ω 0.563 0.406 0.409 0.459 0.571 0.256 0.358 0.356 0.398 0.353λ 0.878 1.078 0.959 0.889 0.844 0.945 0.964 0.964 0.942 1.014γ2 22.519 26.382 17.205 27.678 40.966 27.351 39.739 19.960 44.999 25.944

logL -7,838.40 -8,373.86 -7,081.88 -6,957.85 -7,190.46 -6,720.73 -6,915.24 -6,791.92 -6,148.72 -7,283.34AIC 15,702.80 16,773.72 14,189.76 13,941.71 14,406.93 13,467.46 13,856.48 13,609.84 12,323.44 14,592.68BIC 15,780.90 16,851.80 14,267.82 14,019.78 14,485.01 13,545.53 13,934.55 13,687.95 12,401.51 14,670.83

HQIC 15,885.01 16,955.88 14,371.88 14,123.85 14,589.08 13,649.59 14,038.63 13,792.06 12,505.58 14,774.99VR 0.80 0.86 0.87 0.83 0.84 0.78 0.88 0.84 0.82 0.82


µ 0.034 0.032 0.045 0.024 0.038 0.028 0.031 0.068 0.021 0.050β 0.620 0.617 0.634 0.657 0.689 0.567 0.691 0.631 0.582 0.610α 0.411 0.453 0.414 0.293 0.492 0.420 0.381 0.315 0.343 0.388ξ -0.129 -0.150 -0.343 -1.012 -0.364 -0.157 -0.177 -1.052 -0.256 -0.287σ2

u 0.147 0.141 0.144 0.188 0.134 0.148 0.150 0.164 0.134 0.120τ1 -0.076 -0.071 -0.086 -0.039 -0.048 -0.068 -0.071 -0.048 -0.045 -0.103τ2 0.049 0.036 0.017 0.002 0.015 0.031 0.048 0.032 0.036 0.046δ1 -0.029 -0.053 -0.070 -0.057 -0.038 -0.054 -0.060 -0.044 -0.027 -0.105δ2 0.058 0.064 0.041 0.010 0.033 0.055 0.058 0.067 0.056 0.048φ 0.958 0.939 1.064 1.446 0.986 1.073 1.001 1.313 1.119 1.082ω 0.104 0.148 0.332 0.723 0.407 0.125 0.197 0.871 0.229 0.288λ 0.953 0.973 0.838 0.625 0.920 0.820 0.923 0.691 0.828 0.820γ2 37.685 35.949 35.219 28.038 24.932 46.647 28.890 20.578 39.253 53.373

logL -5,395.73 -5,657.00 -6,258.62 -7,442.47 -6,789.21 -5,609.35 -6,390.15 -8,141.40 -5,915.05 -6,072.96AIC 10,817.46 11,340.01 12,543.24 14,910.94 13,604.43 11,244.70 12,806.31 16,308.79 11,856.10 12,171.92BIC 10,895.53 11,418.09 12,621.29 14,989.00 13,682.59 11,322.78 12,884.37 16,386.85 11,934.20 12,250.01

HQIC 10,999.61 11,522.17 12,725.34 15,093.06 13,786.74 11,426.86 12,988.44 16,490.90 12,038.29 12,354.10VR 0.83 0.83 0.78 0.79 0.79 0.78 0.83 0.79 0.86 0.79


Table A.11: Monthly REGARCH-MIDASThis table reports full-sample estimated parameters, information criteria, variance ratio from(1.34) as well as full maximized log-likelihood value for the monthly two-parameter REGARCH-MIDAS. Results are for K = 12.


µ 0.011 -0.018 0.057 0.078 0.071 0.044 0.064 0.025 0.034 0.032β 0.820 0.742 0.861 0.837 0.825 0.864 0.832 0.843 0.891 0.812α 0.383 0.590 0.404 0.345 0.414 0.432 0.390 0.445 0.451 0.515ξ -0.492 -0.306 -0.391 -0.441 -0.567 -0.201 -0.335 -0.333 -0.379 -0.276σ2

u 0.133 0.193 0.144 0.132 0.127 0.144 0.143 0.149 0.127 0.126τ1 -0.062 -0.092 -0.094 -0.073 -0.063 -0.086 -0.087 -0.069 -0.078 -0.060τ2 0.041 0.048 0.044 0.042 0.020 0.023 0.026 0.034 0.014 0.018δ1 -0.063 -0.048 -0.065 -0.054 -0.070 -0.071 -0.080 -0.046 -0.063 -0.040δ2 0.061 0.049 0.059 0.075 0.044 0.048 0.044 0.050 0.036 0.033φ 1.042 0.881 0.999 1.044 1.089 0.957 0.987 0.978 0.966 0.920ω 0.577 0.405 0.404 0.467 0.565 0.241 0.361 0.355 0.390 0.349λ 0.864 1.080 0.959 0.869 0.842 0.951 0.951 0.952 0.897 1.009γ1 -0.583 -1.605 0.025 1.398 -0.865 -0.266 -0.878 0.424 0.800 1.028γ2 4.112 1.000 5.262 13.651 3.036 2.728 1.933 8.113 6.714 9.921

logL -7,842.06 -8,377.55 -7,081.13 -6,963.39 -7,202.89 -6,721.18 -6,928.36 -6,792.40 -6,153.55 -7,292.92AIC 15,712.13 16,783.10 14,190.26 13,954.77 14,433.77 13,470.37 13,884.72 13,612.79 12,335.11 14,613.85BIC 15,796.24 16,867.19 14,274.33 14,038.85 14,517.86 13,554.44 13,968.80 13,696.91 12,419.18 14,698.01

HQIC 15,908.35 16,979.28 14,386.39 14,150.92 14,629.94 13,666.51 14,080.88 13,809.03 12,531.26 14,810.18VR 0.72 0.82 0.83 0.73 0.70 0.65 0.76 0.78 0.60 0.74


µ 0.034 0.032 0.045 0.024 0.038 0.028 0.031 0.067 0.021 0.049β 0.850 0.827 0.863 0.864 0.826 0.831 0.875 0.745 0.852 0.873α 0.388 0.434 0.380 0.277 0.476 0.400 0.359 0.314 0.325 0.361ξ -0.127 -0.149 -0.344 -1.006 -0.367 -0.159 -0.180 -1.032 -0.256 -0.294σ2

u 0.148 0.142 0.144 0.188 0.134 0.149 0.151 0.165 0.134 0.121τ1 -0.074 -0.069 -0.084 -0.038 -0.050 -0.065 -0.069 -0.048 -0.040 -0.095τ2 0.045 0.033 0.013 0.001 0.014 0.028 0.044 0.031 0.032 0.044δ1 -0.030 -0.053 -0.070 -0.057 -0.039 -0.053 -0.060 -0.045 -0.028 -0.105δ2 0.058 0.064 0.041 0.010 0.033 0.056 0.058 0.067 0.057 0.049φ 0.958 0.941 1.071 1.439 0.988 1.080 1.005 1.301 1.120 1.089ω 0.082 0.141 0.330 0.730 0.407 0.118 0.205 0.874 0.225 0.297λ 0.914 0.962 0.783 0.595 0.906 0.794 0.877 0.686 0.807 0.761γ1 -1.427 -1.388 2.524 3.325 0.013 -0.931 1.769 3.777 1.456 -0.651γ2 1.000 1.000 18.021 21.725 5.094 1.783 13.707 33.579 13.546 2.319

logL -5,404.34 -5,662.62 -6,265.13 -7,446.52 -6,791.44 -5,623.29 -6,395.34 -8,141.29 -5,921.83 -6,087.31AIC 10,836.68 11,353.24 12,558.27 14,921.04 13,610.88 11,274.58 12,818.68 16,310.58 11,871.65 12,202.61BIC 10,920.76 11,437.32 12,642.32 15,005.10 13,695.05 11,358.66 12,902.75 16,394.64 11,955.75 12,286.71

HQIC 11,032.84 11,549.41 12,754.38 15,117.17 13,807.22 11,470.75 13,014.82 16,506.70 12,067.85 12,398.81VR 0.67 0.70 0.58 0.63 0.69 0.59 0.68 0.74 0.72 0.54

60 CHAPTER 1.

Table A.12: Monthly REGARCH-MIDAS (single-parameter)This table reports full-sample estimated parameters, information criteria, variance ratiofrom (1.34) as well as full maximized log-likelihood value for the monthly single-parameterREGARCH-MIDAS. Results are for K = 12.


µ 0.012 -0.017 0.057 0.078 0.071 0.045 0.065 0.025 0.034 0.032β 0.824 0.737 0.865 0.836 0.831 0.871 0.841 0.845 0.893 0.812α 0.383 0.591 0.404 0.345 0.413 0.431 0.387 0.445 0.450 0.515ξ -0.492 -0.311 -0.391 -0.441 -0.568 -0.201 -0.337 -0.333 -0.378 -0.276σ2

u 0.134 0.192 0.144 0.132 0.127 0.144 0.143 0.149 0.127 0.126τ1 -0.062 -0.093 -0.094 -0.073 -0.063 -0.085 -0.087 -0.069 -0.078 -0.060τ2 0.041 0.048 0.044 0.042 0.019 0.023 0.026 0.034 0.014 0.018δ1 -0.063 -0.047 -0.065 -0.054 -0.070 -0.071 -0.080 -0.046 -0.063 -0.040δ2 0.061 0.048 0.059 0.075 0.044 0.048 0.044 0.050 0.036 0.033φ 1.042 0.883 0.999 1.045 1.090 0.957 0.988 0.978 0.966 0.920ω 0.587 0.431 0.406 0.466 0.576 0.252 0.373 0.356 0.390 0.349λ 0.855 1.054 0.955 0.871 0.828 0.930 0.933 0.950 0.894 1.009γ2 12.772 21.575 9.843 11.378 13.169 8.224 10.921 11.252 7.447 9.787

logL -7,842.51 -8,381.29 -7,081.41 -6,963.39 -7,203.34 -6,721.53 -6,929.11 -6,792.42 -6,153.56 -7,292.92AIC 15,711.02 16,788.57 14,188.81 13,952.79 14,432.67 13,469.06 13,884.22 13,610.84 12,333.13 14,611.85BIC 15,789.13 16,866.65 14,266.87 14,030.86 14,510.75 13,547.13 13,962.29 13,688.95 12,411.20 14,690.00

HQIC 15,893.23 16,970.74 14,370.94 14,134.93 14,614.83 13,651.19 14,066.36 13,793.06 12,515.27 14,794.16VR 0.72 0.82 0.83 0.73 0.70 0.64 0.76 0.78 0.59 0.74


µ 0.034 0.033 0.045 0.024 0.038 0.028 0.031 0.067 0.021 0.049β 0.856 0.829 0.856 0.849 0.834 0.837 0.871 0.742 0.850 0.880α 0.387 0.434 0.381 0.277 0.474 0.399 0.360 0.314 0.326 0.361ξ -0.127 -0.149 -0.345 -1.016 -0.367 -0.159 -0.180 -1.037 -0.256 -0.293σ2

u 0.148 0.142 0.144 0.188 0.134 0.149 0.151 0.165 0.134 0.121τ1 -0.073 -0.069 -0.085 -0.038 -0.050 -0.065 -0.069 -0.048 -0.041 -0.095τ2 0.045 0.033 0.013 0.001 0.014 0.028 0.044 0.030 0.032 0.044δ1 -0.030 -0.052 -0.070 -0.058 -0.039 -0.053 -0.060 -0.044 -0.028 -0.105δ2 0.058 0.064 0.041 0.010 0.033 0.056 0.058 0.067 0.057 0.049φ 0.958 0.940 1.072 1.448 0.988 1.080 1.005 1.304 1.120 1.089ω 0.077 0.140 0.329 0.728 0.409 0.116 0.203 0.874 0.224 0.304λ 0.889 0.938 0.792 0.602 0.901 0.772 0.883 0.686 0.809 0.739γ2 14.140 14.689 10.161 10.076 9.315 11.923 9.798 15.579 11.051 10.146

logL -5,405.05 -5,663.48 -6,265.40 -7,447.15 -6,791.78 -5,623.97 -6,395.39 -8,141.55 -5,921.85 -6,087.94AIC 10,836.10 11,352.96 12,556.81 14,920.31 13,609.55 11,273.94 12,816.77 16,309.10 11,869.70 12,201.87BIC 10,914.18 11,431.04 12,634.86 14,998.37 13,687.71 11,352.02 12,894.84 16,387.16 11,947.79 12,279.96

HQIC 11,018.25 11,535.13 12,738.91 15,102.43 13,791.87 11,456.10 12,998.90 16,491.21 12,051.89 12,384.05VR 0.65 0.70 0.59 0.65 0.69 0.58 0.68 0.74 0.72 0.52


Table A.13: REGARCH-HARThis table reports full-sample estimated parameters, information criteria, variance ratio from(1.34) as well as full maximized log-likelihood value for the REGARCH-HAR.


µ 0.011 -0.017 0.056 0.077 0.071 0.043 0.063 0.024 0.032 0.030β 0.638 0.578 0.811 0.583 0.537 0.591 0.508 0.801 0.568 0.565α 0.396 0.585 0.420 0.367 0.434 0.465 0.438 0.459 0.486 0.538ξ -0.495 -0.310 -0.391 -0.445 -0.576 -0.203 -0.322 -0.337 -0.377 -0.280σ2

u 0.134 0.191 0.145 0.133 0.127 0.145 0.143 0.150 0.127 0.126τ1 -0.062 -0.094 -0.095 -0.076 -0.061 -0.089 -0.089 -0.069 -0.086 -0.060τ2 0.045 0.050 0.046 0.048 0.022 0.026 0.029 0.036 0.016 0.019δ1 -0.063 -0.047 -0.065 -0.055 -0.069 -0.070 -0.080 -0.046 -0.064 -0.040δ2 0.061 0.048 0.060 0.075 0.043 0.048 0.044 0.051 0.037 0.033φ 1.043 0.881 0.998 1.047 1.096 0.958 0.976 0.980 0.959 0.922ω 0.572 0.418 0.429 0.465 0.571 0.256 0.360 0.377 0.398 0.366γ1 0.321 0.391 0.033 0.373 0.465 0.497 0.506 0.001 0.597 0.453γ2 0.552 0.677 0.898 0.510 0.381 0.453 0.457 0.920 0.356 0.551

logL -7,847.46 -8,374.10 -7,095.31 -6,972.04 -7,196.44 -6,731.86 -6,927.53 -6,804.48 -6,156.54 -7,298.11AIC 15,720.92 16,774.20 14,216.63 13,970.07 14,418.88 13,489.73 13,881.07 13,634.95 12,339.08 14,622.23BIC 15,799.03 16,852.28 14,294.69 14,048.14 14,496.96 13,567.79 13,959.14 13,713.06 12,417.15 14,700.38

HQIC 15,903.13 16,956.37 14,398.75 14,152.21 14,601.04 13,671.86 14,063.21 13,817.17 12,521.22 14,804.54VR 0.82 0.86 0.84 0.85 0.85 0.82 0.88 0.80 0.84 0.84


µ 0.034 0.032 0.044 0.025 0.039 0.028 0.030 0.068 0.020 0.049β 0.570 0.558 0.554 0.494 0.585 0.521 0.599 0.531 0.500 0.561α 0.419 0.458 0.423 0.303 0.502 0.421 0.394 0.317 0.349 0.389ξ -0.130 -0.149 -0.339 -0.998 -0.362 -0.157 -0.180 -1.066 -0.254 -0.288σ2

u 0.148 0.141 0.145 0.189 0.134 0.148 0.152 0.165 0.135 0.120τ1 -0.076 -0.072 -0.085 -0.039 -0.046 -0.068 -0.072 -0.047 -0.045 -0.103τ2 0.050 0.037 0.018 0.002 0.016 0.031 0.049 0.032 0.037 0.046δ1 -0.029 -0.053 -0.069 -0.056 -0.038 -0.053 -0.061 -0.044 -0.028 -0.105δ2 0.059 0.065 0.042 0.010 0.033 0.055 0.059 0.066 0.056 0.049φ 0.959 0.936 1.058 1.433 0.982 1.069 1.001 1.321 1.117 1.082ω 0.106 0.148 0.331 0.719 0.408 0.126 0.200 0.881 0.228 0.287γ1 0.480 0.499 0.457 0.356 0.423 0.500 0.437 0.246 0.472 0.556γ2 0.474 0.480 0.388 0.275 0.501 0.329 0.485 0.435 0.361 0.273

logL -5,407.30 -5,659.23 -6,269.06 -7,449.17 -6,796.98 -5,613.81 -6,404.37 -8,143.54 -5,922.70 -6,077.50AIC 10,840.60 11,344.46 12,564.13 14,924.33 13,619.96 11,253.63 12,834.74 16,313.08 11,871.40 12,181.01BIC 10,918.68 11,422.54 12,642.18 15,002.40 13,698.11 11,331.71 12,912.80 16,391.13 11,949.49 12,259.10

HQIC 11,022.75 11,526.62 12,746.23 15,106.46 13,802.27 11,435.78 13,016.87 16,495.18 12,053.58 12,363.19VR 0.84 0.84 0.80 0.84 0.82 0.80 0.85 0.81 0.87 0.81

62 CHAPTER 1.

Table A.14: REGARCH-SplineThis table reports full-sample estimated parameters, information criteria as well as full maxi-mized log-likelihood value for the REGARCH-Spline. Results are for K = 6.


µ -0.017 0.048 0.059 0.088 0.075 0.047 0.068 0.023 0.035 0.017β 0.933 0.864 0.943 0.936 0.941 0.932 0.942 0.935 0.948 0.913α 0.359 0.588 0.389 0.342 0.376 0.417 0.355 0.428 0.440 0.487ξ -0.510 -0.274 -0.379 -0.399 -0.591 -0.199 -0.328 -0.326 -0.375 -0.275σ2

u 0.134 0.195 0.144 0.133 0.128 0.144 0.143 0.150 0.128 0.127τ1 -0.058 -0.075 -0.088 -0.069 -0.058 -0.079 -0.081 -0.065 -0.073 -0.055τ2 0.039 0.045 0.040 0.043 0.016 0.023 0.024 0.031 0.013 0.018δ1 -0.063 -0.039 -0.064 -0.055 -0.069 -0.069 -0.078 -0.045 -0.062 -0.037δ2 0.062 0.048 0.058 0.076 0.042 0.047 0.046 0.050 0.036 0.034φ 1.052 0.864 0.994 1.000 1.108 0.953 0.980 0.973 0.959 0.919ω 1.684 1.435 1.726 2.015 1.375 1.398 2.009 1.860 1.632 2.769

logL -7,854.94 -8,390.56 -7,086.92 -6,972.56 -7,214.19 -6,718.88 -6,931.05 -6,795.12 -6,156.17 -7,300.44AIC 15,745.89 16,817.12 14,209.84 13,981.11 14,464.37 13,473.76 13,898.10 13,626.24 12,348.33 14,636.88BIC 15,854.03 16,925.23 14,317.92 14,089.21 14,572.48 13,581.85 14,006.20 13,734.39 12,456.43 14,745.10

HQIC 15,998.18 17,069.35 14,462.00 14,233.30 14,716.59 13,725.94 14,150.30 13,878.54 12,600.52 14,889.32VR 0.51 0.79 0.75 0.62 0.50 0.56 0.63 0.68 0.40 0.61


µ 0.035 0.033 0.047 0.039 0.050 0.021 0.033 0.067 0.020 0.049β 0.950 0.932 0.938 0.926 0.917 0.930 0.953 0.905 0.947 0.947α 0.353 0.404 0.353 0.283 0.449 0.371 0.343 0.307 0.310 0.343ξ -0.123 -0.147 -0.351 -0.902 -0.383 -0.143 -0.177 -0.988 -0.252 -0.295σ2

u 0.148 0.142 0.144 0.190 0.134 0.149 0.152 0.167 0.135 0.121τ1 -0.068 -0.067 -0.081 -0.031 -0.047 -0.059 -0.064 -0.045 -0.036 -0.087τ2 0.040 0.028 0.012 0.002 0.013 0.025 0.040 0.027 0.028 0.043δ1 -0.030 -0.052 -0.070 -0.052 -0.037 -0.053 -0.060 -0.044 -0.028 -0.104δ2 0.057 0.064 0.040 0.011 0.032 0.055 0.058 0.069 0.057 0.050φ 0.964 0.941 1.077 1.349 1.002 1.087 1.004 1.273 1.113 1.094ω 0.962 0.738 0.936 1.168 2.001 0.169 1.366 1.490 1.164 1.040

logL -5,403.06 -5,663.08 -6,270.65 -7,434.19 -6,796.46 -5,629.11 -6,401.72 -8,162.50 -5,927.05 -6,090.50AIC 10,842.13 11,362.17 12,577.29 14,904.38 13,628.92 11,294.23 12,839.43 16,361.01 11,890.10 12,217.00BIC 10,950.23 11,470.28 12,685.37 15,012.46 13,737.14 11,402.34 12,947.52 16,469.08 11,998.23 12,325.13

HQIC 11,094.33 11,614.40 12,829.44 15,156.55 13,881.35 11,546.45 13,091.61 16,613.15 12,142.36 12,469.26VR 0.46 0.59 0.40 0.63 0.57 0.42 0.44 0.61 0.53 0.35


Table A.15: FloEGARCHThis table reports full-sample estimated parameters, information criteria as well as full maxi-mized log-likelihood value for the FloEGARCH.


µ 0.017 -0.009 0.045 0.061 0.072 0.035 0.042 0.021 0.025 0.010β 0.195 0.121 0.195 0.133 0.104 0.179 0.080 0.117 0.161 0.131α 0.400 0.589 0.418 0.373 0.436 0.460 0.424 0.473 0.484 0.536ξ -0.476 -0.293 -0.389 -0.440 -0.566 -0.205 -0.327 -0.332 -0.378 -0.272σ2

u 0.134 0.192 0.144 0.132 0.127 0.144 0.142 0.149 0.128 0.126τ1 -0.065 -0.096 -0.098 -0.078 -0.066 -0.089 -0.095 -0.072 -0.084 -0.063τ2 0.041 0.048 0.045 0.045 0.020 0.025 0.029 0.037 0.015 0.020δ1 -0.063 -0.047 -0.067 -0.057 -0.069 -0.071 -0.082 -0.047 -0.064 -0.042δ2 0.060 0.048 0.060 0.076 0.043 0.047 0.045 0.050 0.036 0.033φ 1.036 0.872 0.999 1.041 1.092 0.964 0.985 0.979 0.975 0.923ω 1.393 1.076 1.299 1.365 0.950 0.961 1.510 0.980 0.825 1.898d 0.633 0.620 0.678 0.658 0.673 0.645 0.673 0.678 0.672 0.644

logL -7,843.29 -8,370.57 -7,082.26 -6,960.83 -7,195.15 -6,722.47 -6,922.21 -6,791.17 -6,159.11 -7,291.01AIC 15,710.58 16,765.13 14,188.52 13,945.66 14,414.29 13,468.95 13,868.42 13,606.34 12,342.22 14,606.02BIC 15,782.68 16,837.21 14,260.57 14,017.72 14,486.36 13,541.01 13,940.48 13,678.44 12,414.28 14,678.16

HQIC 15,878.78 16,933.28 14,356.63 14,113.79 14,582.44 13,637.07 14,036.55 13,774.54 12,510.34 14,774.31


µ 0.030 0.029 0.042 0.029 0.029 0.024 0.028 0.088 0.014 0.040β 0.093 0.121 0.137 0.108 0.158 0.138 0.156 0.092 0.113 0.150α 0.430 0.458 0.413 0.298 0.498 0.425 0.396 0.317 0.361 0.413ξ -0.130 -0.147 -0.343 -1.010 -0.368 -0.159 -0.178 -1.062 -0.258 -0.292σ2

u 0.147 0.141 0.144 0.188 0.134 0.148 0.151 0.165 0.134 0.122τ1 -0.078 -0.072 -0.088 -0.041 -0.051 -0.068 -0.071 -0.051 -0.045 -0.101τ2 0.049 0.035 0.015 0.002 0.015 0.031 0.049 0.030 0.035 0.049δ1 -0.030 -0.053 -0.070 -0.057 -0.039 -0.055 -0.060 -0.044 -0.029 -0.106δ2 0.058 0.065 0.041 0.010 0.033 0.056 0.059 0.065 0.057 0.050φ 0.955 0.939 1.073 1.448 0.998 1.080 1.001 1.325 1.116 1.078ω 0.300 0.367 0.466 0.827 1.307 0.307 0.726 1.171 0.643 0.832d 0.692 0.674 0.655 0.666 0.641 0.643 0.671 0.618 0.681 0.656

logL -5,399.76 -5,660.47 -6,259.27 -7,442.64 -6,796.38 -5,620.41 -6,398.68 -8,152.11 -5,922.81 -6,097.09AIC 10,823.52 11,344.93 12,542.53 14,909.29 13,616.76 11,264.82 12,821.36 16,328.22 11,869.62 12,218.18BIC 10,895.59 11,417.01 12,614.58 14,981.34 13,688.90 11,336.89 12,893.42 16,400.27 11,941.70 12,290.26

HQIC 10,991.66 11,513.09 12,710.63 15,077.40 13,785.05 11,432.97 12,989.49 16,496.32 12,037.79 12,386.35

C H A P T E R 2REALIZED EGARCH MODELS WITH

TIME-VARYING UNCONDITIONAL VARIANCE

Bo LaursenAarhus University and CREATES


Abstract

In this paper, we suggest four new parametric alternatives to the Realized EGARCH

model by allowing the unconditional variance to exhibit time-variation. The con-

ditional variance is decomposed into a stationary and a non-stationary part. The

stationary part is specified by a zero mean standard Realized EGARCH model. The un-

conditional variance is modeled by the non-stationary component. We propose four

functional alternatives to the non-stationary part: a smooth time-varying structure, a

flexible Fourier form, a quadratic spline and a cubic spline.

An empirical application to the exchange-traded index fund SPY that tracks the

S&P 500 Index including both a forecasting and Value-at-Risk exercise illustrates the

performance of the models in practice. The results show that the introduction of a

time-varying unconditional variance improves the in-sample fit of the models, but

does generally not lead to any improvements out-of-sample.

65

66 CHAPTER 2.

2.1 Introduction

The introduction of the AutoRegressive Conditional Heteroskedasticity (ARCH) model

of Engle (1982) and the Generalized ARCH (GARCH) model of Bollerslev (1986)

sparked a huge and successful literature on modeling time-varying volatility. An

astronomic number of generalizations and new models have been suggested and

studied; see Teräsvirta (2009) for an overview. The popularity of GARCH type models

is mainly due to their ability to describe stylized facts of financial return series such

as volatility clustering.

It is a well-known stylized fact that most daily and high-frequency financial

time series exhibit quite persistent autocorrelation in their squared return, realized

variance, and other measures of volatility. This has motivated a strand of the GARCH

literature to model long range dependence explicitly using long-memory models such

as the Fractionally Integrated GARCH (FIGARCH) model of Baillie, Bollerslev, and

Mikkelsen (1996). However, another strand of the literature is essentially arguing that

long-memory and the so-called ’Integrated GARCH effect’ in volatility may merely

be a statistical artifact. The observed long-memory behaviour may be spurious and

caused by structural changes in the volatility dynamics bound to occur in sufficiently

long time series. Diebold (1986) was the first to suggest that occasional level shifts in

the intercept of the GARCH model can bias the estimates towards the parameters of

an integrated GARCH model; see also Mikosch and Starica (2004) and Lamoureux

and Lastrapes (1990). Theoretical explanations have been provided by Hillebrand

(2005) and Morana (2002). However, it may be noted that Morana and Beltratti (2004)

argue that both long-memory and structural changes are important.

The idea of structural changes in volatility dynamics has promoted the develop-

ment of new and more flexible GARCH type models allowing for changing parameters,

state-dependence and time-varying unconditional variance, etc. In this paper, we

build upon the literature that explicitly model structural changes in the parameters.

One possibility is to assume that the volatility process is smoothly non-stationary

and model it accordingly. There is an expanding literature taking this approach begin-

ning with Bellegem and von Sachs (2004) and Feng (2004). Dahlhaus and Subba Rao

(2006) introduced a smoothly time-varying parameter ARCH model. Bellegem and

von Sachs (2004), Engle and Rangel (2008) and Amado and Teräsvirta (2013) all con-

sidered a multiplicative decomposition of the volatility process into a stationary and

non-stationary component. However, while Bellegem and von Sachs (2004) nonpara-

metrically estimated the non-stationary component, Engle and Rangel (2008) and

Amado and Teräsvirta (2013) parametrically estimated the non-stationary compo-

nent using splines and generalized logistic functions respectively. See also Brownless

and Gallo (2010) who use splines to fit a time-varying component in the Multiplica-

tive Error Model (MEM) of Engle and Gallo (2006) and Mishra, Su, and Ullah (2010)

who correct potential misspecification due to a ’rough’ parametric GARCH specifica-

tion by a smooth nonparametric component. Baillie and Morana (2009) suggested

2.1. INTRODUCTION 67

the additively decomposed Adaptive FIGARCH model that accounts for both long-

memory and structural changes in the volatility processes using the Fourier flexible

form; see also Mazur and Pipien (2012). We refer the reader to Teräsvirta (2012) and

Van Bellegem (2012) for a more comprehensive survey.

Since French, Schwert, and Stambaugh (1987) and due to the availability of

databases providing intra daily prices of financial assets, a literature focusing on

using data sampled at very high frequency to compute ex-post measures of volatility

at a lower frequency has emerged. A wide range of realized measures of volatility has

been proposed in the literature since Andersen and Bollerslev (1998) showed that

such measures can be very useful for the evaluation of volatility models. Barndorff-

Nielsen and Shephard (2002) proposed the Realized Variance (RV). However, this

measure is sensitive to market microstructure noise. This has motivated the devel-

opment of robust estimators such as the two-scale and multi-scale estimator by

Zhang, Mykland, and Aït-Sahalia (2005) and Zhang (2006), the Realized Kernel (RK)

by Barndorff-Nielsen, Hansen, Lunde, and Shephard (2008), and the Realized Range

by Christensen and Podolskij (2007). The economic and statistical gains from in-

corporating realized measures in volatility models are typically found to be large;

see Christoffersen, Feunou, Jacobs, and Meddahi (2014) and Dobrev and Szerszen

(2010). Andersen et al. (2003) find that classical GARCH models are poorly suited for

situations where volatility jumps to a new level. The basic intuition is that realized

measures constitute a stronger and less noisy signal of latent volatility than squared

daily returns. Hence, models utilizing realized measures can adapt faster to a new

level of volatility.

A number of volatility models incorporating realized measures has been suggested

in the literature. Models such as the Heteroskedastic AutoRegressive of Realized

Variance (HAR-RV) model of Corsi (2009) and its extensions seek to model realized

variance directly. On the other hand, a part of the literature models latent return

volatility by utilizing realized measures in conventional GARCH type models. The

simplest way of incorporating realized measures is by using GARCH-X type models;

see e.g. Engle (2002). However, a GARCH-X model is incomplete as it does not specify

the dynamic properties of the realized measures. Thus, multi-period forecasting

is infeasible. This motivated the MEM of Engle and Gallo (2006) and the HEAVY

model of Shephard and Sheppard (2010) who operate with multiple latent volatility

processes. The realized measures are modeled with additional GARCH type models.

Another possibility is to consider the Realized (E)GARCH framework that is based

on measurement equations that tie the realized measure to the conditional return

variance; see Hansen, Huang, and Shek (2012) and Hansen and Huang (2016). In this

paper, we utilize the Realized EGARCH framework.

The purpose of this paper is to combine the two strands of the GARCH litera-

ture described above by introducing a low-frequency or non-stationary component

into the Realized EGARCH framework of Hansen and Huang (2016). We suggest four

68 CHAPTER 2.

different specifications of the low-frequency component inspired by the existing

GARCH literature. The low-frequency component is deterministically time-varying

and proxies all factors that affect the unconditional variance. The addition of this

component has the potential to capture the long-run dynamic behaviour of volatility

such as structural changes that are bound to happen in sufficiently long time se-

ries. Thus, the model keeps the attractive features of GARCH type models in fitting

and forecasting at high and medium frequency while allowing for a low frequency

component associated with volatility dynamics at a lower frequency.

It is important to note that the low-frequency component, of course, is esti-

mated based on the in-sample observations and that it needs to be extrapolated

out-of-sample. Typically, it is fixed for the forcasting horizon. In-sample, the de-

terministically time-varying component is a convenient tool to allow for changing

amplitudes of volatility clusters often observed in financial time series (see e.g. the

empirical section of this paper or Amado and Teräsvirta (2013)) and it should help us

get the level of volatility correct. However, in an out-sample exercise, the new models

are prone to level-shift in volatility in the same way as the benchmark counterpart.

It is an empirical question whether the attempt to get the level right will help the

modeller out-of-sample or an estimated benchmark without this feature, maybe with

spuriously induced persistence, will outperform.

The rest of the paper proceeds as follows. In Section 2.2, we introduce the four

new Realized EGARCH models. Estimation and inference are discussed in Section 2.3.

Section 2.4 and 2.5 describe techniques used for forecasting and Value-at-Risk (VaR)

calculations. In Section 2.6, an empirical application to the exchange-traded index

fund SPY is presented. Finally, concluding remarks are given in Section 2.7.

2.2 Four new Realized EGARCH models

In this paper, we seek to model the volatility of an asset return series by extending the

Realized EGARCH framework of Hansen and Huang (2016). Let Ft−1 be the informa-

tion set containing the historical information at time t −1 and define the conditional

mean, µt = E(rt |Ft−1

), and the conditional variance, σ2

t = Var(rt |Ft−1

), of the asset

returns{rt

}. The conditional variance,σ2

t = ht g t , is multiplicatively decomposed into

a stationary (high-) and a non-stationary (low-frequency) component as in Engle and

Rangel (2008) and Amado and Teräsvirta (2013), among others. The idea is to model

the low-frequency component, g t , as a deterministic function of time to take changes

in the unconditional variance into account. The general framework is defined by the

2.2. FOUR NEW REALIZED EGARCH MODELS 69

following equations

rt = µt +√σ2

t zt , (2.1)

loght = β loght−1 +τ(zt−1

)+αut−1, (2.2)

log xt = ξ+φ log(σ2

t

)+δ(

zt)+ut , (2.3)

log g t = ωt . (2.4)

We refer to the equations as the return equation, the GARCH equation, the measure-

ment equation, and the low-frequency component, respectively. It is assumed that

zt ∼ i.i.d.(0,1

)and ut ∼ i.i.d.

(0,σ2

u

)are mutually and serially independent. xt is a

realized measure, e.g. the realized variance, bipower variation, daily range, or a robust

measure such as the RK.

In GARCH applications, the return equation is standard. The conditional mean

can be modeled in different ways including popular approaches such as GARCH-in-

mean, a constant or an autoregressive process. However, as the focus is on modeling

volatility, we assume that µt = 0. It can be noted that Hansen and Huang (2016)

empirically find that this restriction may improve out-of-sample fit relative to a model

using an unrestricted conditional mean, when considering the exchange-traded index

fund SPY also used in the financial application in Section 2.6.

We see that the GARCH equation is an AR(1) model for loght with the innovation

τ(zt−1

)+αut−1. Thus, β < 1 is a measure of persistence of the stationary part of

the conditional variance. The parameter α tells us how the realized measure affects

future volatility. The intercept of the GARCH equation is normalized to zero such

that E[loght

] = 0 in order to deal with an identification problem that emerges if

both loght and log g t are allowed to a have a free constant. This implies that log g t =E[

logσ2t

]models the now potentially time-varying unconditional variance.

The measurement equation defines the process for the realized measure. The

realized measure, xt , is an ex-post estimator for σ2t . Hence, it is natural to assume a

link between this ex-post measure of volatility and the the ex-ante conditional vari-

ance. However, discrepancies between the two measures are expected for numerous

reasons. In this paper, the conditional variance is a measure of close-to-close return

volatility while the realized measure only measures the volatility during trading hours.

As the open-to-close volatility empirically accounts for approximately 75% of daily

volatility, it is necessary to include the intercept, ξ, in (2.3). The parameter φ is an

exponential scaling factor that often is estimated to be close to one. In addition, the

error term, ut , is included to account for discrepancies in the two measures stemming

from noise and sampling errors in realized measures.

The leverage functions τ(zt

)and δ

(zt

)are defined by

τ (z) = τ1z +τ2

(z2 −1

), (2.5)

δ (z) = δ1z +δ2

(z2 −1

). (2.6)

70 CHAPTER 2.

The choice of the leverage functions follows Hansen and Huang (2016) who find that

this specification makes the independence assumption between zt and ut realistic

in an application to the exchange-traded index fund SPY. This specification allows

for a polynomial effect of zt on the realized measure and future volatility. The four

equations fully define the dynamic properties of returns and the realized measure of

volatility.

The non-stationary component, log g t , is a deterministic function of time that

proxies all factors that affect the unconditional variance. The addition of this compo-

nent has the potential to capture the long-run dynamic behaviour of volatility such

as structural changes that are bound to happen in sufficiently long financial time

series. Thus, the model keeps the attractive features of GARCH type models in fitting

and forecasting at high and medium frequency while allowing for a low-frequency

component associated with volatility dynamics at longer horizons.

We consider five different specifications of ωt in (2.4) inspired by the existing

literature on GARCH models with time-varying unconditional variance. First, we

consider the special case of a standard Realized EGARCH model by setting

ωt =ω. (2.7)

Second, we consider the Realized TV-EGARCH model inspired by the TV-GJR-GARCH

model of Amado and Teräsvirta (2013) by defining

ωt =ω+K∑

k=1γkG

(t

T;λk ,ck

)(2.8)

as a linear combination of bounded transition functions. The so-called transition

functions, G(

tT ;λk ,ck

)for k = 1, ...,K , are generalized logistic functions

G

(t

T;λk ,ck

)=

1+exp

(−λk

(t

T− ck

))−1

(2.9)

satisfying the identification restrictions λk > 0 and 0 ≤ c1 < c2 < ... < cK ≤ 1. The

transition functions allow the unconditional variance to change smoothly as a func-

tion of time, t/T . It should be noted that the specification differs from the one in

Amado and Teräsvirta (2013) by restricting the transition functions to only have one

transition each. This is done to simplify model specification. The parameters ck and

λk determine respectively the location and speed of the transition between different

regimes. The larger λk , the faster the transition between states. When λk →∞, the

transition function becomes a step function. In general, (2.8) and (2.9) can generate

very flexible parametrizations capable of modeling changes in the unconditional

variance. A difference between this specification and the following specifications is

that the location parameters are estimated. Estimation of the location makes the nu-

merical optimization quite tedious, but the additional flexibility most likely reduces

the number of transition functions needed.

2.2. FOUR NEW REALIZED EGARCH MODELS 71

Third, we also suggest a Realized Adaptive-EGARCH model using the flexible

functional form

ωt =ω+K∑

k=1

(γk sin

(2πk

t

T

)+λk cos

(2πk

t

T

))(2.10)

used in the Adaptive-FIGARCH model of Baillie and Morana (2009); see also Andersen

and Bollerslev (1997) and Mazur and Pipien (2012). Baillie and Morana (2009) argue

that advantages of the Fourier flexible form, specified by Gallant (1984), are efficient

modeling of smooth structural changes without requiring pretesting to determine

the actual location of break points and relatively straightforward estimation. We also

find the model easily estimable. Although the functional form is smooth, it has been

shown to accommodate quite abrupt shifts.

Finally, inspired by the Spline-GARCH model of Engle and Rangel (2008), we

consider two spline specifications: the quadratic spline defined by

ωt =ω+γ0t

T+

K∑k=1

γ1k

((t

T− ck−1

)+

)2

, (2.11)

and the cubic spline defined by

ωt =ω+γ0t

T+γ1

(t

T

)2

+K∑

k=1γ2k

((t

T− ck−1

)+

)3

. (2.12)

The knot-points, ck , are assumed to be equidistant as in Engle and Rangel (2008). The

models are labelled the Realized Q-Spline-EGARCH and Realized C-Spline-

EGARCH model, respectively. The assumption about equidistant knot-points makes

the estimation quite straightforward, but may have the consequence that one sud-

denly needs to increase K when increasing the sample size merely because the

locations of the knot-points have changed.

Clearly, all the new models add a high degree of flexibility in fitting the level

of volatility in-sample. The larger K , the more flexibility. Thus in order to avoid

overfitting, determining K is an important part of modeling the deterministically

time-varying component. Amado and Teräsvirta (2017) propose a modeling strat-

egy for the TV-GARCH model of Amado and Teräsvirta (2013) based on Lagrange

Multiplier (LM) misspecification tests. The idea is to keep adding transition function

until the null of no additional time-variation in the unconditional variance cannot

be rejected. For the Adaptive-FIGARCH model of Baillie and Morana (2009) and the

Spline-GARCH of Engle and Rangel (2008), K was selected using information criteria.

In this paper, we will apply information criteria to select K , because we are covering

multiple specifications and want to have an uniform way of selecting K .

72 CHAPTER 2.

2.3 Estimation and inference

In this section, we briefly discuss estimation and inference within a Quasi-Maximum

Likelihood (QML) framework obtained by assuming that zt ∼ i.i.d.N(0,1

)and ut ∼

i.i.d.N(0,σ2

u

). Write the leverage functions as τ

(zt

) = τ′a(zt

)and δ

(zt

) = δ′b(zt

)with a

(zt

)= b(zt

)= (zt , z2

t −1)′

. The initial value of the logarithm of the conditional

variance, log h1, is set equal to its unconditional mean, E[loght ] = 0. Define the

parameter vector

θ1 =(β,τ′,α,ξ,φ,δ′,σ2

u

), (2.13)

and let the parameter vector θ2 contain the parameters used to model the low-

frequency component, log g t . The log-likelihood function reads

L(r, x;θ1,θ2

)= T∑t=1

`t(rt , xt ;θ1,θ2

)(2.14)

with the log-likelihood contribution at time t given by

`t(rt , xt ;θ1,θ2

)=−1

2

[2log(2π)+ log

(ht

)+ log(g t

)+ z2t + log

(σ2

u

)+ u2

t

σ2u

], (2.15)

where zt = zt(θ1,θ2

)= rt /√

ht g t and ut(θ1,θ2

)= log(xt

)−ξ−φ log(σ2

t

)−δ(

zt). The

QML Estimator (QMLE), θ =(θ′1, θ′2

)′, is obtained by maximizing the log-likelihood

function. We now derive the score that defines the first-order conditions for the

QMLE. Key components are the derivatives stated in the following lemma.

Lemma 1. The derivatives of loght+1 with respect to loght and log g t are given by

respectively,

∂ loght+1

∂ loght= (

β−αφ)+ 1

2

(αδ

∂b(zt

)∂zt

−τ∂a(zt

)∂zt

)zt , (2.16)

∂ loght+1


2

(αδ

∂b(zt

)∂zt

−τ∂a(zt

)∂zt

)zt , (2.17)

and the derivatives of `t with respect to loght and log g t are given by respectively,

∂`t

∂ loght=−1

2

(1− z2

t

)+ ut

σ2u

(δ∂b

(zt

)∂zt

zt −2φ

) , (2.18)

∂`t

∂ log g t=−1

2

(1− z2

t

)+ ut

σ2u

(δ∂b

(zt

)∂zt

zt −2φ

) . (2.19)

Proof: See Appendix A.1.

2.3. ESTIMATION AND INFERENCE 73

Next, we define the score of loght+1 with respect to θ.

Lemma 2. The derivatives of loght+1 with respect to θ are given from the stochastic

recursions

∂ loght+1

∂(β,τ′,α

)′ = ∂ loght+1

∂ loght

∂ loght

∂(β,τ′,α

)′ + (loght , zt , z2

t −1,ut

)′, (2.20)

∂ loght+1

∂(ξ,φ,δ′

)′ = ∂ loght+1

∂ loght

∂ loght

∂(ξ,φ,δ′

)′ +α(1, log

(σ2

t

), zt , z2

t −1

)′, (2.21)

∂ loght+1

∂θ2= ∂ loght+1

∂ loght

∂ loght

∂θ2+ ∂ loght+1

∂ log g t

∂ log g t

∂θ2, (2.22)

where ∂ log g t∂θ2

differs between the specifications for the non-stationary component and

is therefore presented in Appendix A.1.


Finally, we present a theorem summarizing the score.

Theorem 1. The score with respect to the parameters in θ1 =(β,τ′,α,ξ,φ,δ′,σ2

u

)is

given by the components

∂`t

∂(β,τ′,α

)′ = ∂`t

∂ loght

∂ loght

∂(β,τ′,α

)′ (2.23)

∂`t

∂(ξ,φ,δ′

)′ = ∂`t

∂ loght

∂ loght

∂(ξ,φ,δ′

)′ + ut

σ2u

(1, log

(σ2

t

), zt , z2

t −1

)′(2.24)

∂`t

∂σ2u= 1

2

u2t −σ2

u

σ4u

(2.25)

and the score with respect to θ2 is given by

∂`t

∂θ2= ∂`t

∂ loght

∂ loght

∂θ2+ ∂`t

∂ log g t

∂ log g t

∂θ2, (2.26)


differs between the specifications for the non-stationary component; see

Appendix A.1.


Following Hansen and Huang (2016), we note that the score is a martingale differ-

ence sequence if E(zt |Ft−1

)= 0, E(z2

t |Ft−1

)= 1, E

(ut |zt ,Ft−1

)= 0, and E(u2

t |zt ,Ft−1

)=

σ2u . The first two conditions are related to the correct specification of the conditional

mean and variance of rt . The third condition is basically a requirement for δ(zt

)being flexible enough to model the conditional mean of ut . The last condition is a

homoskedasticity assumption on ut .

74 CHAPTER 2.

2.3.1 Asymptotic distribution of estimators

The development of asymptotic theory for conventional GARCH models has been

and continue to be a demanding journey stretching more than two decades. So far, we

only have a limited number of asymptotic results for the simplest models; see Francq

and Zakoïan (2010) and references therein. It is only recently that the asymptotic

theory for the EGARCH(1,1) model was established by Wintenberger (2013) and to

our knowledge nothing has been established for Realized EGARCH models yet. The

introduction of a low frequency component complicates the matter further. Chen

and Hong (2016) develop the asymptotic theory for a time-varying parameter GARCH

model by imposing a smoothness condition such that the process displays locally

stationary behaviour; see also Dahlhaus (1996a), Dahlhaus (1996b), and Dahlhaus

(1997). Likeωt in this paper, the parameters in Chen and Hong (2016) are functions of

rescaled time, t/T , rather than t . A similar approach is taken in Amado and Teräsvirta

(2013), who develop the asymptotic theory for the TV-GJR-GARCH model. This is a

common scaling scheme in the time series literature; see Robinson (1989), Hillebrand,

Medeiros, and Xu (2013) and Dahlhaus and Subba Rao (2006), among others. To

understand the necessity of this approach, we refer to Hillebrand et al. (2013), who

develop the asymptotic theory in Smooth Transition Regression (STR) models with

rescaled time as the state variable. However, the basic idea of the rescaling is to keep

the parameters fixed as a fraction of the sample size when T →∞ such as the amount

of local information increases suitably. For instance, the transitions are assumed to

happen after a certain fraction of the sample and not at a certain point in time. We

leave the development of the asymptotic theory for the estimators of the four new

Realized EGARCH models for future research, but conjecture as in Hansen and Huang

(2016) that

pT

(θ−θ0

)d→N

(0,T I−1JI−1

), (2.27)

where J is the limit of the outer-product of the scores and I is (minus) the limit of the

Hessian matrix for the log-likelihood functions.

2.3.2 Partial log-likelihood function

The partial log-likelihood function is defined by the time t contribution

`pt

(rt , xt ;θ1,θ2

)=−1

2

[log(2π)+ log

(ht

)+ log(g t

)+ z2t

], (2.28)

and it is the Kullback-Leibler measure associated with the conditional distribution

of returns. Consequently, it is directly comparable with the log-likelihood obtained

from conventional GARCH models, such as the EGARCH model.

2.4. FORECASTING 75

2.3.3 Numerical issues related to the Realized TV-EGARCH model

For the TV-GARCH model, Amado and Teräsvirta (2013) find that parameter estima-

tion often is numerically difficult. We encounter similar problems in the Realized

TV-EGARCH model. In practice, it can be very useful to improve the reliability of the

estimates by using non-gradient based optimization routines and relevant parameter

transformations. As in the case of STR models, see Teräsvirta, Tjøstheim, and Granger

(2010), one problem is related to the estimation of the slope parameters, λk , when

it is very large. In that case, the switches in the intercept are rather abrupt which in

turn implies that large changes of λk only affect the transition function in a close

neighbourhood of the location parameter, ck . Therefore, determining the curvature

of the transition function requires a large number of observations of the transition

variable, t/T , in a small neighbourhood of ck . To alleviate numerical issues, we simply

restrict λk to not exceed a certain threshold. In the empirical application, we set the

threshold to 250. In addition, as originally suggested by Goodwin, Holt, and Preste-

mon (2011), we apply the transformation λk = exp(νk

)and estimate νk . Writing the

parameter as an exponential mapping scales the parameter used in the optimization

to a smaller range that is comparable to the other parameters. It is well known that

badly scaled parameters, i.e. parameters that differ considerably in magnitude, can

result in numerical convergence issues. An additional advantage is that no positivity

constraint is needed since the transformation is positively monotone.

As a result of the numerical difficulties often encountered, the choice of initial

values is of utmost importance. We apply a strategy similar to the one suggested

by Teräsvirta et al. (2010) in the context of STR models. In the case of the Realized

TV-EGARCH model, we consider a grid of starting values for the location parameters,

ck ,k = 1, ...,K . Then, conditional on the combination of parameter values in the grid,

we estimate the remaining parameters and choose the starting values that maximize

the QML criterion. Finally, all parameters are estimated based on the chosen starting

values. This strategy is computationally feasible when the number of transitions is

limited and the required identification restrictions, 0 ≤ c1 < ... < cK ≤ 1, are imposed.

2.4 Forecasting

The Realized EGARCH and the extensions presented in this paper can easily be used

to perform multi-period ahead forecasting of both the conditional return variance

and the realized measure. Recursive substitution of the autoregressive equation (2.2)

implies that

loght+h =βh loght +h∑

i=1βi−1

(τ(zt+h−i

)+αut+h−i

). (2.29)

76 CHAPTER 2.

As the logarithm of the conditional return variance is the sum of its stationary part,

loght , and its non-stationary part, log g t , we have that

logσ2t+h =ωt+h +βh loght +

h∑i=1

βi−1(τ(zt+h−i

)+αut+h−i

). (2.30)

We are not required to generate auxiliary future values of the realized measure, xt ,

when the objective is to forecast future values of logσ2t . The reason is that the innova-

tions zt and ut are sufficient for generating the volatility path. Due to the assumption

E[ut

] = E[zt

] = 0 and the specification of the leverage functions, we may obtain a

forecast of logσ2t+h , h > 0 by using

logσ2t+h|t = E

[logσ2

t+h |Ft

]=ωt+h +βh loght +βh−1

(τ(zt

)+αut

). (2.31)

One of the major issues, when forecasting using models with a time-varying structure,

is how to forecast the low-frequency component out-of-sample. As the low-frequency

component models the unconditional variance and may be assumed to be relatively

stable, we have chosen to set it equal to its value at time t , i.e. ωt+h =ωt . This is the

standard approach taken in the literature.

It should be noted that the deterministic component is based on in-sample ob-

servations. Therefore, the new models with a deterministic component are prone

to sudden changes in the level of volatility in the same way as the benchmark coun-

terparts. The idea is to get the level of volatility ’right’ at the end of the sample. As

forecast converge to this level with the forecast horizon, it may be crucial for the

out-of-sample performance. However, it may be empirical preferred to get the level

wrong, but have more persistent (β close on 1) volatility dynamics such that the

convergence to the unconditional volatility is slower.

One issue related to forecasting logσ2t+h is the need to account for the distribu-

tional aspects of logσ2t+h in order to produce an unbiased forecast of σ2

t+h . To deal

with this problem in our forecasting application, we resort to simulations. Thus, we

consider

σ2t+h|t =

1

S

S∑s=1

exp(logσ2

t+h|t ,s

), (2.32)

where logσ2t+h|t ,s is a simulated path obtained using (2.30) and S denotes the num-

ber of simulated paths. In our application, we use the empirical distribution of the

innovations.

2.5 A VaR framework

A key application for volatility modeling is in the field of financial risk management.

New financial regulation, such as the Basel Accords, has dramatically increased the

need for risk measurement. According to Basel Accords, banks are required to hold

an amount of regulatory capital that can be thought of as the amount of capital that

2.5. A VAR FRAMEWORK 77

must be held to make the risk acceptable to regulators. Value at Risk is arguably the

most widely used risk measure in financial institutions and hundreds of academic

papers on VaR have been published.

Numerous methods for forecasting VaR are found in the literature; see Kuester,

Mittnik, and Paolella (2006) for a comparison. In this paper, the 100(1−p

)% VaR is

simply defined as the p-quantile of the return distribution. We consider a method

called "filtered historical simulation" suggested by Barone-Adesi, Bourgoin, and

Giannopoulos (1998) that is easily applicable in a GARCH framework. Let F denote the

zero mean, unit variance and possibly skewed and leptokurtic cumulative distribution

function of the i.i.d. innovations{

zt}

. Our modeling framework implies that the close-

to-close return, rt =√σ2

t zt , conditional on information available at time t − 1 is

distributed as

rt /√σ2

t |Ft−1 ∼ F(0,1

). (2.33)

Thus, the one-day ahead 100(1−p

)% VaR is simply defined by

VaRpt |t−1 ≡ F−1 (

p)√

σ2t . (2.34)

In a QML framework, F is, however, generally unknown and needs to be estimated

either parametrically or nonparametrically. A simple nonparametric method involves

taking for F the empirical distribution of the standardized residuals, rt /√σ2

t . Another

issue is the lack of an explicit formula for the h-day ahead return distribution condi-

tional on the information available at time t −1. This implies that we need to resort

to simulations. We apply the following recipe suggested by Barone-Adesi et al. (1998);

see also Francq and Zakoïan (2010):

1. Fit a model using the observed returns rt , t = 1, ...,n, and compute the esti-

mated volatility σ2t for t = 1, ...,n +1.

2. Simulate a large number S of scenarios for rn+1, ...,rn+h by iterating the follow-

ing three steps for each simulation, s:

a) simulate the i.i.d. innovations zn+1,s , ..., zn+h,s using the empirical distri-

bution F ;

b) set σn+1,s =σn+1 and rn+1,s =σn+1,s zn+1,s ;

c) for k = 2, ...,h, set σn+k,s equal to the value obtained from recursively

applying the volatility model of interest.

3. Determine the h-period VaR as the p-quantile of the simulations rn+1,s + ...+rn+h,s , s = 1, ...,S.

78 CHAPTER 2.

2.6 Empirical application to stock market volatility

In this section, we present an empirical application using returns and realized mea-

sures for the exchange-traded index fund SPY, which tracks the S&P 500 Index. The

series was also considered in Hansen and Huang (2016) and Hansen et al. (2012).

We apply the restriction φ= 1 also used in Hansen and Huang (2016). Generally, the

empirical results indicate that introducing a low-frequency component helps us to

more accurately model volatility in-sample. The computational and practical burden

of estimating the new models and the comparable forecasting and VaR performance

limit the desire to implement the new models unless explicitly called for. However, as

the results from the forecasting and the VaR application seem to differ between sub-

samples, it will be of interest to consider more empirical applications and investigate

forecast combination.

2.6.1 Data and realized measures

Our full sample spans the time period from January 2, 1998 to December 31, 2013,

which amounts to 4025 daily observations. We consider the model fit on the full

sample, but we do also reserve the period until December 31, 2004 (1760 observations)

for initial estimation of the models in the forecasting and VaR application. The only

realized measure that we consider is the RK.1 The motivation for this is twofold. First,

the main goal of this paper is to examine the effect of including a low-frequency

component in Realized EGARCH models. Second, the RK is found to perform very

well in Hansen and Huang (2016), where a range of realized measures are considered,

and in the VaR application by Brownless and Gallo (2010); see also the comparison of

realized measures in Gatheral and Oomen (2010).

The RK by Barndorff-Nielsen et al. (2008) is one of several robust measures of

volatility. In this paper, we consider the variant derived in Barndorff-Nielsen, Hansen,

Lunde, and Shephard (2011) which is given by RK =∑Hh=−H k

(h

H+1

)κh ,where k(x) is

the Parzen kernel and κh =∑ni=|h|+1 r i ,nri−h,n . Here ri ,n = pti ,n −pti−1,n is the intraday

return. See also Barndorff-Nielsen, Hansen, Lunde, and Shephard (2009).

Figure A.1 depicts returns, squared returns (annualized), RK (annualized) and the

autocorrelation function of the RK. The dotted line separates the initial estimation

window from the window used for the forecasting and VaR application. As common

for financial return series, we recognize volatility clustering that has motivated the

GARCH literature. There is a period of relative tranquillity in the middle of the sample

while volatility seems to be higher in the beginning and end of the sample. We

also observe the well-known stylized fact of long-memory looking behaviour in

the different measures of volatility; the autocorrelation function of the RK is slowly

decaying and significant for very high lags. The latter observation motivates either an

1The data was kindly provided by Asger Lunde.

2.6. EMPIRICAL APPLICATION TO STOCK MARKET VOLATILITY 79

explicit modeling of the long-memory property or allowing for structural changes

that can generate spurious long-memory behaviour if not accounted for.


2.6.2 Full sample estimates

When applying the models to the full period from January 2, 1998 to December

31, 2013, we simply select the number of transitions, trigonometric functions and

splines using the Hannan-Quinn information criteria.2 It should be noted that the

number of parameters selected in a given model is quite sensitive to the choice of

information criteria. However, we leave a thorough investigation of model selection

for future research. Table A.1 contains the parameter estimates for the full period. The

parameters in ωt are generally not comparable between the specifications and not

particularly interesting, so we leave them out. The parameter estimates for the five

models are quite similar, but in line with our expectations the persistence of volatility

measured by β is reduced when allowing for a time-varying unconditional variance.


The estimated conditional variance processes, σ2t = g t ht , for the five models are very

hard to discriminate from each other based on a visual inspection. The same picture

is seen from the information contained in Table A.2, which summarizes some basic

descriptive statistics and presents the correlation between the estimated conditional

variance processes. This is interesting compared to the differences between the TV-

GJR-GARCH and GJR-GARCH model found in Amado and Teräsvirta (2013). In this

case, there is a substantial difference between the conditional variance processes

due to the extreme persistence required in classical GARCH model to fit the observed

long-memory behaviour. This supports the usefulness of using realized measures in

GARCH type models as it helps the model to adapt quickly to a new level of volatility.

We only plot the process for the Realized EGARCH in Figure A.2. Evidently, there

is a period of high baseline volatility in the beginning of the sample followed by a

period of relative tranquillity. In the second part of the sample, we see a big spike

in volatility around the financial crisis and some smaller spikes afterwards that may

be contributed to increased uncertainty in financial markets due to the European

sovereign debt crisis.


Figure A.2 also depicts the unconditional variance for the five Realized EGARCH

models. The Realized TV-, Adaptive-, and Q-Spline-EGARCH models seem to capture

2The Hannan-Quinn information criteria penalizes additional parameters less than the Bayesianinformation criteria, but more than the Akaike information criteria.

80 CHAPTER 2.

similar movements in the baseline volatility, i.e. the high volatility in the beginning of

the crisis and spikes during the financial crisis and the sovereign debt crisis. The ωt

component of the Realized C-Spline-EGARCH model, however, is quite different from

the three other models and not intuitive. A similar picture is seen, when we in Figure

A.3 plot the stationary components for the four non-stationary Realized EGARCH

models. While ht looks ’stationary’ for the Realized TV-, Adaptive-, and Q-Spline-

EGARCH model, it does not seem to be the case for the Realized C-Spline-EGARCH

model. This indicates that using a cubic spline may be problematic, if we want to

separate the conditional variance into a stationary and non-stationary part.



Even though the QML approach allows for possible misspecification of the distribu-

tional assumptions about{

zt ,ut}

, it is of interest assessing whether the assumptions

seem reasonable. Our findings are very similar to Hansen and Huang (2016) across

all five models. Hence, we only present diagnostics for the Realized TV-EGARCH

model. In order for the score to be a martingale difference sequence, we require

the absence of autocorrelation in zt , z2t ,ut and u2

t . The first 40 autocorrelations are

reported in Figure A.4 with 95% confidence bands. In general, many autocorrelations

are significant for all four series. One concern is that the first order autocorrelations

all are significant at a 5% level. For the case of zt , z2t and u2

t , we do not find more

violations than may be explained by pure chance. In the case of ut , a large proportion

of the first 40 autocorrelations are significant. Hansen and Huang (2016) explain this

with time-variation in σ2u and examine a GARCH structure for σ2

u , but do find little

impact on parameter estimates of introducing this feature. However, based on the

autocorrelation functions, it is more likely that misspecification of the conditional

mean of ut is the issue.

The independence assumption between zt and ut is supported by the finding

that the empirical correlation is numerically equal to zero for all models.


Now, we turn to the normality for zt and ut . Figure A.5 depicts QQ-plots for the

two series. The normality assumption for zt seems fairly reasonable, but negative

skewness is observed. One culprit is the extreme outlier that occurred on February 27,

2007. The outlier has been associated with a computer glitch on the New York Stock

Exchange on that day. On the other hand, the empirical distribution of ut exhibits

severe excess kurtosis. Again, this may be related to time-variation in the conditional

mean of ut or alternatively be explained by outliers in zt that affect ut through the

leverage function, δ (z). It should be noted that the degree of time-variation in the

mean of ut is more pronounced for the standard Realized EGARCH model which


is supporting the necessity of explicitly modeling a low-frequency component. In

fact, implementing the CUSUM test by Brown, Durbin, and Evans (1975) only results

in rejection of the null hypothesis of no break in the mean of ut in the case of the

Realized EGARCH model; see Appendix A.2.

To sum up, the Realized EGARCH model and the extensions presented in this

paper model the returns reasonably, but the models including a low-frequency com-

ponent are more in line with the assumptions about ut .


2.6.3 Forecasting exercise

In this section, we present the results from the forecasting exercise. For both the

forecasting and the Value-at-Risk application we reserve the time period from January

2, 1998 to December 31, 2004 for initial estimation of the models. For every 20th

business day hereafter, we reestimate the model using all data up to that date. As

in the full sample, we select the number of transitions, trigonometric functions and

splines using the Hannan-Quinn information criteria. Reestimation of the model is

necessary as it allows the time-varying component to update. We believe that this

approach is appropriate from a practitioner’s perspective; it is reasonable to assume

that asset managers, etc. update models at least on a monthly basis.

In order to evaluate forecasting performance, we need a proxy for latent volatility.

As the RK is a measure of open-to-close volatility, it will be a biased measure of close-

to-close latent volatility. To deal with this issue, we follow a strategy similar to Sharma

and Vipul (2016). As a proxy for the latent close-to-close volatility, we use σ2t = ηRKt

with

η=1T

∑Tt=1 r 2

t1T

∑Tt=1 RK t

. (2.35)

We have chosen to only use data for the out-of-sample period to calculate η due to

observed time variation in the scaling factor.

It is common in the forecasting literature to implement a variety of evaluation

criteria or loss functions. However, Hansen and Lunde (2006) show that when the

target is observed with error, the choice of criteria becomes critical. In fact, some

criteria may be non-robust, i.e. the ranking based on such a criterion will depend

on whether the proxy or the true latent volatility are the target. Patton (2011) pro-

vides necessary and sufficient conditions on the functional form of the loss function

ensuring consistency of the ordering when using a proxy. Two robust measures are

the Squared Forecasting Error (SFE) and the Quasi-Likelihood (QLIKE) loss function.

82 CHAPTER 2.

Considering h-step ahead forecasts, the two loss functions are defined as respectively

SFEt+h =(σ2

t+h −σ2t+h|t

)2,

QLIKEt+h = σ2t+h

σ2t+h|t

− logσ2

t+h

σ2t+h|t

−1.

The SFE penalizes the forecasting errors in a symmetrical way while the QLIKE is

an asymmetric loss function that penalizes underprediction more heavily than over-

prediction. The latter may be suitable if the application involves risk management and

VaR forecasting, where under-prediction of volatility is more of a concern than over-

prediction. Since the QLIKE loss function is related to the Gaussian log-likelihood

function, it should be comparable with the likelihood based inference in Hansen et al.

(2012).

Tabel A.3 presents the results based on the full out-of-sample period from January

2, 2005 to December 31, 2013. By looking at the median forecast error, we see that

all models tend to overpredict latent volatility measured by rescaled RK. In general,

the Realized EGARCH model, followed by the EGARCH model, performs best using

both robust loss functions, i.e. the MSFE and the QLIKE loss function. However, it

should be noted that the Realized TV-, C-Spline-, and Q-Spline-Q-EGARCH model

perform better when considering the Median Squared Forecast Error (MedSFE).

This indicates that the poor performance of the new models may be related to a

limited number of large forecast errors. The upper panel of Figure A.6 depicts the

MSFE of the models relative to the MSFE of the Realized EGARCH model and a filled

marker indicates that the forecasting model is included in the Model Confidence

Set (MCS) by Hansen et al. (2011) at a 10% significance level. The Realized EGARCH,

the Realized Adaptive-EGARCH and the EGARCH model are all included in the MCS

at all forecasting horizons while the rest of the models are included only at some

horizons. As expected, it seems that using a Realized EGARCH model is preferred

at short horizons. However, a standard EGARCH model seems suitable at longer

horizons. The picture is the same when considering the QLIKE loss function in Figure

A.7.

We find that the forecasting performance is very sensitive to the choice of out-of-

sample period. In fact, the new models tend to perform much better before and after

the global financial and European sovereign debt crisis. This may be contributed to

estimation difficulties related to fitting the unconditional variance during extreme

market conditions. To investigate this issue further, we present in Table A.4 the

results from a forecasting exercise when excluding the one year period from August 1,

2008 to July 31, 2009 covering the most volatile parts of the financial crisis. Clearly,

the non-stationary models now perform much better and, in fact, better than the

Realized EGARCH and EGARCH model for many horizons. Considering the MCS

for the shortened sample presented in the lower panel of Figure A.6, the Realized

Adaptive-EGARCH model is now the only one included at all horizons.


¿ Insert Table A.3-A.4 about here À

¿ Insert Figure A.6-A.7 about here À

2.6.4 VaR application

We consider three different values for the VaR coverage rate p, 0.01, 0.05 and 0.1.

As a benchmark, we also consider VaR forecasts from the EGARCH model and a

Historical Simulation (HS) method. In HS, the VaR forecasts are made based solely

on the empirical distribution of historical returns. Figure A.8 shows the empirical

coverage rates compared to the theoretical coverage rates for all forecast horizons.

It is clear that all models incorporating time-varying volatility are generally very

close and tend to produce conservative VaR forecasts. Except for one-period VaR the

models generally underestimate the coverage rate. The overprediction of the latent

volatility presented in the previous section gives some support for this result. HS is

different since it implicitly assumes a constant volatility. The model overestimates

the coverage rate for 1% VaR and underestimates for 5% and 10% VaR.


Methods for backtesting VaR forecasts are usually based on two key statistical proper-

ties: unconditional coverage and independence. The first property is obviously related

to whether the model provides an empirical coverage rate in line with the theoretical.

The independence property states that VaR violations should be independent over

time. A good model is expected to satisfy both properties. Generally, the tests are

based on the sequence

Hi tt+h|t =1{

r t+h|t < VaRpt+h|t

}(2.36)

of VaR violations where r t+h|t = rt+h+rt+h−1+...+rt+1. Coverage tests include Kupiec

(1995), Pearsons-Q in Cambell (2007), and Pérignon and Smith (2008)(multivariate).

Independence tests are presented in Christoffersen (1998), Berkowitz, Christoffersen,

and Pelletier (2011) and Christoffersen and Pelletier (2004). Joint tests for both cover-

age and independence include Engle and Manganelli (2004) and Hurlin and Tokpavi

(2007)(multivariate). Here multivariate refers to tests that consider several coverage

rates jointly. We have implemented all aforementioned tests using a 5% significance

level.3 Despite the conservative VaR forecasts presented in Figure A.8, the coverage

tests do not reject the null of a correctly specified model more often than what may

be explained by chance. The independence tests rarely reject the null for all mod-

els incorporating time-varying volatility. For HS, the null is rejected for almost all

3Generally the tests assume that the sequence of Hi t variables cannot be based on overlappingforecasts horizons. In our forecasting exercise this is not satisfied when h ≥ 2. In these cases, the tests arebased on Bonferroni sub samples of length T /h and we reject the null if the p-value from any sub sampleis smaller than 0.05/h.

84 CHAPTER 2.

coverage rates and forecast horizons. In periods of high volatility, HS often leads to

violation clusters and thereby violates the assumption of independence. This is a

general finding and supports the idea that time-varying volatility is crucial when

modeling VaR. The joint test for both coverage and independence rejects the null

more often. However, for the time-varying volatility models there are no systematic

rejections and it is impossible to announce a ’winner’ based on these tests.

One problem with many of the tests is that they may exhibit low power in realistic

finite sample sizes since the amount of information in a binary sequence is limited

(Cambell (2007)). An alternative is to specify a loss function that also depends on the

magnitude of VaR violations. The tick loss function

T Lpt =

(p −1

{r t+h|t < VaRp

t+h|t})(

r t+h|t −VaRpt+h|t

)(2.37)

examined in Giacomini and Komunjer (2005) is an example of such a function. The

function is motivated by the loss functions used in quantile regression. If a model

provides a VaR that is indeed the p-quantile, it would minimize the average tick loss

function. Figure A.9 presents the ratio of the average tick loss functions relative to

the standard Realized EGARCH model for the three coverage rates and all forecasts

horizons. Filled out markers indicate that the model is part of the MCS at a 10%

significance level. HS performs poorly and for visualization we choose omit it. It is

only a part of the MCS for long forecasting horizons when p = 0.01. The plots show

that we only rarely are able to exclude any of the time- varying volatility models from

the MCS especially for long forecasting horizons. The performance of the models

is generally very similar. However, for very short forecasting horizons, we find the

EGARCH model to be the worst performing model across all coverage rates. This

is expected since the Realized EGARCH models are adapting faster to a changing

volatility environment than conventional GARCH models. The effect is, however,

not always large enough to exclude the EGARCH model from the MCS. For medium

forecast horizons and a coverage rate of 0.05 or 0.1 the EGARCH model actually

performs better than the models based on realized measures.

Similar to the forecasting analysis, the VaR measures are also heavily affected

by the financial crises. Figure A.10 presents the ratio of average tick loss functions

excluding the period during the financial crises. For short term forecasting horizons,

we find no significant differences compared to the full sample. The EGARCH model

is still the worst performing model and all Realized EGARCH models generally show

similar behaviour. For long term horizons we observe a different picture. For many

horizons it is possible to exclude the Realized C-Spline-EGARCH model from the

MCS. For p = 0.01, the standard Realized EGARCH model can also be excluded. While

the Realized TV-EGARCH model was generally the worst performing model in the full

sample, it is now the best model closely followed by the Realized Q-Spline-EGARCH

model.


2.7. CONCLUSION 85

2.7 Conclusion

In this paper, we suggest four new Realized EGARCH models by introducing a low-

frequency or non-stationary component into the Realized EGARCH framework of

Hansen and Huang (2016). We suggest four different specifications of the low-frequency

component inspired by the existing GARCH literature. The low-frequency component

is deterministically time-varying and proxies all factors that affect the unconditional

variance. The addition of this component has the potential to capture the long-run

dynamic behaviour of volatility such as structural changes that are bound to occur in

sufficiently long time series. Thus, the model keeps the attractive features of GARCH

type models in fitting and forecasting at high and medium frequency while allow-

ing for a low-frequency component associated with volatility dynamics at longer

horizons.

We consider an empirical application to the exchange-traded index fund SPY

examining both the performance in respect to forecasting and VaR calculations. In

general, models utilizing the RK outperform the standard EGARCH model at short

horizons but not necessarily at longer horizons. The new Realized EGARCH models

deliver a better in-sample fit, but in general fail to deliver superior performance

out-of-sample in regards to forecasting and VaR calculations. It seems that a Realized

EGARCH model does quite a good job. The initial suggestion for the practitioner is to

apply the Realized EGARCH model and only model a low-frequency component if

it seems reasonable, e.g. in a situation where the GFC is included in the estimation

window and not in the forecasting window. It will be of interest to consider more

empirical studies and see whether forecast combination can improve the out-of-

sample fit.

Acknowledgement

We thank Timo Teräsvirta, Asger Lunde, Peter Reinhardt Hansen, and Esther Ruiz

Ortega for insightful comments. We also extend our gratitude to the participant at the

27th New Zealand Econometric Study Group meeting and the Brown-bag seminar

at the School of Economics and Finance, Queensland University of Technology for

feedback on this paper.

The authors acknowledge support from CREATES - Center for Research in Econo-

metric Analysis of Time Series (DNRF78), funded by the Danish National Research

Foundation.

86 CHAPTER 2.

2.8 References



Amado, C., Teräsvirta, T., 2017. Specification and testing of multiplicative time-

varying GARCH models with applications. Econometric Reviews 36 (4), 421–446.

Andersen, T., Bollerslev, T., 1997. Heterogeneous information arrivals and return

volatility dynamics: Uncovering the long-run in high frequency returns. Journal of

Finance 52, 975–1005.

Andersen, T., Bollerslev, T., 1998. Answering the skeptics: Yes, standard volatility

models do provide accurate forecasts. International Economic Review 39, 885–905.


realized volatility. Econometrica 71, 579–625.

Baillie, R. T., Bollerslev, T., Mikkelsen, H. O., 1996. Fractionally Integrated Generalized

Autoregressive Conditional Heteroskedasticity. Journal of Econometrics 74, 3–30.

Baillie, R. T., Morana, C., 2009. Modelling long memory and structural breaks in condi-

tional variances: An Adaptive FIGARCH approach. Journal of Economic Dynamics

and Control 33, 1577–1592.



noise. Econometrica 76, 1481–1536.

Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2009. Realized kernels

in practice: Trades and quotes. The Econometrics Journal 12, 1–32.

Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2011. Multivariate

realised kernels: Consistent positive semi-definite estimators of the covariation of

equity prices with noise and non-synchronous trading. Journal of Econometrics

162, 149 – 169.

Barndorff-Nielsen, O. E., Shephard, N., 2002. Econometric analysis of realized volatil-

ity and its use in estimating stochastic volatility models. Journal of the Royal Statis-

tical Society: Series B (Statistical Methodology) 64, 253–280.

Barone-Adesi, G., Bourgoin, F., Giannopoulos, K., 1998. Don’t Look Back. Risk 11,

100–104.

Bellegem, S. V., von Sachs, R., 2004. Forecasting economic time series with uncondi-

tional time-varying variance. International Journal of Forecasting 20, 611–627.

2.8. REFERENCES 87

Berkowitz, J., Christoffersen, P., Pelletier, D., 2011. Evaluating Value-at-Risk models

with desk-level data. Management Science 57, 2213–2227.

Bollerslev, T., 1986. Generalized Autoregressive Conditional Heteroscedasticity. Jour-


Brown, R., Durbin, J., Evans, J., 1975. Techniques for testing the constancy of re-

gression relationships over time. Journal of the Royal Statistical Society. Series B

(Methodological) 37, 149–192.

Brownless, C. T., Gallo, G. M., 2010. Comparison of volatility measures: A risk man-

agement perspective. Journal of Financial Econometrics 8, 29–56.

Cambell, S. D., 2007. A review of backtesting and backtesting procedures. Journal of

Risk 9, 1–17.

Chen, B., Hong, Y., 2016. Detecting for smooth structural changes in GARCH models.

Econometric Theory 32, 740–791.

Christensen, K., Podolskij, M., 2007. Realized range-based estimation of integrated

variance. Journal of Econometrics 141, 323 – 349.

Christoffersen, P., 1998. Evaluating interval forecasts. International Economic Review

30, 841–862.

Christoffersen, P., Feunou, B., Jacobs, K., Meddahi, N., 2014. The economic value of

realized volatility: Using high-frequency returns for option valuation. Journal of

Financial and Quantitative Analysis 49, 663–697.

Christoffersen, P., Pelletier, D., 2004. Backtesting Value-at-Risk: A duration-based

approach. Journal of Empirical Finance 2, 84–108.

Corsi, F., 2009. A simple approximate long-memory model of realized volatility. Jour-

nal of Financial Econometrics 7, 174–196.

Dahlhaus, R., 1996a. Maximum likelihood estimation and model selection for locally

stationary processes. Journal of Nonparametric Statistics 6, 171–191.

Dahlhaus, R., 1996b. On the Kullback-Leibler information divergence of locally sta-

tionary processes. Stochastic Processes and their Applications 62, 139–168.

Dahlhaus, R., 1997. Fitting time series models to nonstationary processes. Annals of


Dahlhaus, R., Subba Rao, S., 2006. Statistical inference for time-varying ARCH pro-

cesses. Annals of Statistics 34, 1075–1114.

88 CHAPTER 2.

Diebold, F. X., 1986. Modeling the persistence of conditional variances: A comment.

Econometric Reviews 5, 51–56.

Dobrev, D., Szerszen, P., 2010. The information content of high-frequency data for

estimating equity return models and forecasting risk. Finance and economics

discussion series. Federal Reserve Board.


425–446.

Engle, R., Manganelli, S., 2004. CAViaR: Conditional autoregressive Value at Risk by

regression quantiles. American Statistical Association 22, 367–381.

Engle, R. F., 1982. Autoregressive conditional heteroscedasticity with estimates of the

variance of United Kingdom inflation. Econometrica 50, 987–1007.




and its global macroeconomic causes. Review of Financial Studies 21, 1187–1222.

Feng, Y., 2004. Simultaneously modeling conditional heteroskedasticity and scale

change. Econometric Theory 20, 563–596.

Francq, C., Zakoïan, J.-M., 2010. GARCH models: Structure, statistical inference and

financial applications. John Wiley & Sons, Ltd.

French, K., Schwert, G., Stambaugh, R., 1987. Expected stock returns and volatility.

Journal of Financial Economics 19, 3–29.

Gallant, R., 1984. The Fourier flexible form. American Journal of Agricultural Eco-

nomics 66, 204–208.

Gatheral, J., Oomen, R. C. A., 2010. Zero-intelligence realized variance estimation.

Finance and Stochastics 14, 249–283.

Giacomini, R., Komunjer, I., 2005. Evaluating and combination of conditional quantile

forecasts. American Statistical Association 23, 416–431.

Goodwin, B. K., Holt, M. T., Prestemon, J. P., 2011. North american oriented strand

board markets, arbitrage activity, and market price dynamics: A smooth transition

approach. American Journal of Agricultural Economics 93, 993–1014.


of volatility. Journal of Business and Economic Statistics 34:2, 269–287.

2.8. REFERENCES 89

Hansen, P. R., Huang, Z., Shek, H. H., 2012. Realized GARCH: A joint model for returns

and realized measures of volatility. Journal of Applied Econometrics 27, 877–906.




79, 453–497.



Hillebrand, E., Medeiros, M. C., Xu, J., 2013. Asymptotic theory for regressions with

smoothly changing parameters. Journal of Time Series Econometrics 5, 133–162.

Hurlin, C., Tokpavi, S., 2007. Backtesting Value-at-Risk accuracy: A simple new test.

Journal of Risk 9, 19–37.

Kuester, K., Mittnik, S., Paolella, M. S., 2006. Value-at-Risk prediction: A comparison

of alternative strategies. Journal of Financial Econometrics 4, 53–89.

Kupiec, P., 1995. Techniques for verifying the accuracy of risk management models.

Journal of Derivatives 3, 73–84.


and the GARCH Model. Journal of Business & Economic Statistics 8, 225–234.

Mazur, B., Pipien, M., 2012. On the empirical importance of periodicity in the volatility

of financial returns - time varying GARCH as a second order APC(2) process. Central

European Journal of Economic Modelling and Econometrics 4, 95–116.

Mikosch, T., Starica, C., 2004. Nonstationarities in financial time series, the long-

range dependence, and the IGARCH effects. Review of Economics and Statistics 86,

378–390.

Mishra, S., Su, L., Ullah, A., 2010. Semiparametric estimator of time series conditional

variance. Journal of Business & Economic Statistics 28, 256–274.

Morana, C., 2002. IGARCH effects: An interpretation. Applied Economics Letters 9,

745–748.

Morana, C., Beltratti, A., 2004. Structural change and long-range dependence in

volatility of exchange rates: Either, neither or both? Journal of Empirical Finance

11, 629–658.


Journal of Econometrics 160, 246 – 256, realized Volatility.

90 CHAPTER 2.

Pérignon, C., Smith, D. R., 2008. A new approach to comparing VaR estimation meth-

ods. The Journal of Derivatives 16, 54–66.

Robinson, P. M., 1989. Nonparametric estimation of time-varying parameters.

Springer Berlin Heidelberg.

Sharma, P., Vipul, 2016. Forecasting stock market volatility using Realized GARCH


59, 222 – 230.

Shephard, N., Sheppard, K., 2010. Realising the future: forecasting with high-


197–231.

Teräsvirta, T., 2009. An introduction to univariate GARCH models. In: Handbook of

Financial Time Series. Springer Berlin Heidelberg.

Teräsvirta, T., 2012. Nonlinear models for autoregressive conditional heteroskedastic-

ity. John Wiley & Sons, Inc.

Teräsvirta, T., Tjøstheim, D., Granger, C. W. J., 2010. Modelling nonlinear economic

time series. Oxford University Press.

Van Bellegem, S., 2012. Locally stationary volatility modeling. John Wiley & Sons, Inc.

Wintenberger, O., 2013. Continuous invertibility and stable QML estimation of the


Zhang, L., 2006. Efficient estimation of stochastic volatility using noisy observations:

A multi-scale approach. Bernoulli 12, 1019–1043.

Zhang, L., Mykland, P. A., Aït-Sahalia, Y., 2005. A tale of two time scales: Determin-

ing integrated volatility with noisy high-frequency data. Journal of the American

Statistical Association 100, 1394–1411.

A.1. DERIVATION OF SCORES 91

Appendix

A.1 Derivation of scores

A.1.1 Proof of Lemma 1

First, consider ∂ loght+1/∂ loght . Using zt = rt

h1/2t g 1/2

t, it can easily be shown that

∂zt

∂ loght=−1

2zt . (A.1)

For ut = log xt −φ logσ2t −δ

(zt

), we find that

∂ut

∂ loght=− ∂δ

(zt

)∂ loght

−φ, (A.2)

where∂δ

(zt

)∂ loght

= δ′ ∂b(zt

)∂zt

∂zt

∂ loght. (A.3)

Similarly, we have∂τ

(zt

)∂ loght

= τ′ ∂a(zt

)∂zt

∂zt

∂ loght. (A.4)

Inserting the above components in the following expression for the ∂ loght+1/∂ loght

∂ loght+1

∂ loght=β+ ∂τ

(zt

)∂ loght

+α ∂ut

∂ loght(A.5)

yields∂ loght+1

∂ loght= (

β−αφ)+ 1

2

(αδ

∂b(zt

)∂zt

−τ∂a(zt

)∂zt

)zt . (A.6)

Using the exactly the same arguments, we can derive

∂ loght+1


2

(αδ

∂b(zt

)∂zt

−τ∂a(zt

)∂zt

)zt . (A.7)

Next, we turn to the derivates of `t with respect to loght and log g t . As the two

expressions will be identical, we only derive it with respect to loght . loght enters the

log-likelihood contribution at time t directly due to the loght term and indirectly

through z2t and u2

t . Thus, we have

∂`t

∂ loght=−1

2

[1+ ∂z2

t

∂ loght+ 1

σ2u

2ut∂ut

∂ loght

]. (A.8)

We note that∂z2

t

∂ loght=−z2

t . (A.9)

92 CHAPTER 2.

Combing the different expressions yields,

∂`t

∂ loght=−1

2

(1− z2

t

)+ ut

σ2u

(δ∂b

(zt

)∂zt

zt −2φ

) . (A.10)

�


First, consider ∂ loght+1/∂(β,τ′,α

)′. We note that

∂ loght+1

∂(β,τ′,α

)′ =β ∂ loght

∂(β,τ′,α

) + ∂τ(zt

)∂(β,τ′,α

) +α ∂ut

∂(β,τ′,α

) + (loght , zt , z2

t −1,ut

)′. (A.11)

However, we remember that τ(zt

)and ut only depends on

(β,τ′,α

)′ through loght

such that the three first terms can be collapsed to one. Thus, we can instead write

∂ loght+1

∂(β,τ′,α

)′ = ∂ loght+1

∂ loght

∂ loght

∂(β,τ′,α

) + (loght , zt , z2

t −1,ut

)′. (A.12)

Exactly, what we wanted to show. For ∂ loght+1/∂(ξ,φ,δ′

)′, we have

∂ loght+1

∂(β,τ′,α

)′ =β ∂ loght

∂(ξ,φ,δ′

) + ∂τ(zt

)∂(ξ,φ,δ′

) +α(1, log

(σ2

t

), zt , z2

t −1

)′. (A.13)

Using the same arguments as before, we obtain

∂ loght+1

∂(ξ,φ,δ′

)′ = ∂ loght+1

∂ loght

∂ loght

∂(ξ,φ,δ′

)′ +α(1, log

(σ2

t

), zt , z2

t −1

)′. (A.14)

Finally, for ∂ loght+1/∂θ2, we can write

∂ loght+1

∂θ2=β∂ loght

∂θ2+ ∂τ

(zt

)∂θ2

+α∂ut

∂θ2. (A.15)

We note that∂zt

∂θ2=−1

2zt

[∂ loght

∂θ2+ ∂ log g t

∂θ2

](A.16)

and that θ2 does not enter directly in loght+1, but only indirectly through loght and

log g t . Thus, we can additively separate the score in the following two contributions

∂ loght+1

∂θ2= ∂ loght+1

∂ loght

∂ loght

∂θ2+ ∂ loght+1

∂ log g t

∂ log g t

∂θ2. (A.17)

Exactly, what we wanted to show.

�


A.1.3 Proof of Theorem 1

As(β,τ′,α

)′ only enters the log-likelihood contribution at time t indirectly through

loght , an application of the chain-rule yields

∂`t

∂(β,τ′,α

)′ = ∂`t

∂ loght

∂ loght

∂(β,τ′,α

)′ , (A.18)

and as(ξ,φ,δ′

)′ enters through loght and u2t , an application of the chain-rule yields

∂`t

∂(ξ,φ,δ′

)′ = ∂`t

∂ loght

∂ loght

∂(ξ,φ,δ′

)′ + ut

σ2u

(1, log

(σ2

t

), zt , z2

t −1

)′. (A.19)

The parameter σ2u only enters directly into the log-likelihood contribution such that

we obtain

∂`t

∂σ2u= 1

2

u2t −σ2

u

σ4u

. (A.20)

Finally, θ2 enters the log-likelihood contribution both through loght and log g t . We

obtain∂`t

∂θ2= ∂`t

∂ loght

∂ loght

∂θ2+ ∂`t

∂ log g t

∂ log g t

∂θ2, (A.21)


differs between the different specifications for the non-stationary com-

ponent.

�

Derivatives of log g t with respect to θ2

In the case of the Realized EGARCH model, we have

∂ log g t

∂ω= 1. (A.22)

In the case of the Realized TV-EGARCH model, we have

∂ log g t

∂ω= 1, (A.23)

∂ log g t

∂γi=G

(t

T;λi ,ci

)i = 1, ...,K , (A.24)

∂ log g t

∂λi= γi G

(t

T;λi ,ci

)(1−G

(t

T;λi ,ci

))(t/T − ci

)i = 1, ...,K , (A.25)

∂ log g t

∂ci=−γi G

(t

T;λi ,ci

)(1−G

(t

T;λi ,ci

))i = 1, ...,K . (A.26)

94 CHAPTER 2.

In the case of the Realized Adaptive-EGARCH model, we have

∂ log g t

∂ω= 1, (A.27)

∂ log g t

∂γi= sin

(2πi

t

T

)i = 1, ...,K , (A.28)

∂ log g t

∂λi= cos

(2πi

t

T

)i = 1, ...,K . (A.29)

In the case of the Realized Spline-Q-EGARCH model, we have

∂ log g t

∂ω= 1, (A.30)

∂ log g t

∂γ0= t i = 1, ...,K , (A.31)

∂ log g t

∂γ1i=

((t − tk−1

)+)2

i = 1, ...,K . (A.32)

In the case of the Realized Spline-Q-EGARCH model, we have

∂ log g t

∂ω= 1, (A.33)

∂ log g t

∂γ0= t , (A.34)

∂ log g t

∂γ1= t 2, (A.35)

∂ log g t

∂γ2k=

((t − tk−1

)+)3

i = 1, ...,K . (A.36)

A.2. FIGURES 95

A.2 Figures

1998 2002 2006 2009 2013

-0.1

0

0.1

1998 2002 2006 2009 20130

2

4

6

1998 2002 2006 2009 20130

0.5

1

1.5

2

20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Figure A.1: Data plotsThe upper-left panel: SPY daily return. The lower-left panel: SPY daily RK (annualized). Theupper-right panel: SPY daily squared returns (annualized). The lower-right panel: SPY daily RKautocorrelation function.

96 CHAPTER 2.

1998 2002 2006 2009 20130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure A.2: Illustration of non-stationery componentAnnualized volatility for Realized EGARCH (grey). Annualized unconditional volatility gtfor Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH(yellow), Realized C-Spline-EGARCH (green) and Realized Q-Spline-EGARCH (purple).

A.2. FIGURES 97

1998 2002 2006 2009 20130

5

10

15

20

25

30

1998 2002 2006 2009 20130

5

10

15

20

25

30

1998 2002 2006 2009 20130

5

10

15

20

25

30

1998 2002 2006 2009 20130

5

10

15

20

25

30

Figure A.3: illustration of stationary part of volatility processIllustration of ht for Realized TV EGARCH (red), Realized Adaptive-EGARCH (yel-low), Realized C-Spline-EGARCH (green) and Realized Q-Spline-EGARCH (purple).

98 CHAPTER 2.

Lag10 20 30 40

Autocorrelationofzt

-0.1

-0.05

0

0.05

0.1

Lag10 20 30 40

Autocorrelationofz2 t

-0.1

-0.05

0

0.05

0.1

Lag10 20 30 40

Autocorrelationofut

-0.1

-0.05

0

0.05

0.1

Lag10 20 30 40

Autocorrelationofu2 t

-0.1

-0.05

0

0.05

0.1

Figure A.4: Autocorrelations for z, z2,u and u2

A.2. FIGURES 99

Standard Normal Quantiles-4 -2 0 2 4

EmpiricalQuantilesofzt

-8

-6

-4

-2

0

2

4

Standard Normal Quantiles-4 -2 0 2 4

EmpiricalQuantilesofut

-3

-2

-1

0

1

2

3

Figure A.5: QQ-plots for z and u

100 CHAPTER 2.

Full sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.90

1.00

1.10

1.20

1.30

Excl. Financial Crisis

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.90

1.00

1.10

1.20

1.30

Figure A.6: Ratio of MSFE relative to the stationary Realized EGARCHFilled out markers indicate that the model is part of the MCS at a 10% significance level.Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow),Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple) and EGARCH (blue).

A.2. FIGURES 101

Full sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.75

1.00

1.25

1.50

1.75


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.75

1.00

1.25

1.50

1.75

Figure A.7: Ratio of MSFE relative to the stationary Realized EGARCHFilled out markers indicate that the model is part of the MCS at a 10% significance level.Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow),Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple) and EGARCH (blue).

102 CHAPTER 2.

Forecast Horizon - h

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Em

piric

al C

over

age

Rat

e

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

Figure A.8: Empirical coverage ratesSignificance levels are indicated in the following way: p = 0.01 (circle), p = 0.05 (triangle) andp = 0.1 (square). Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow), Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple),EGARCH (blue) and Historical Simulation (grey).

A.2. FIGURES 103

Full sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.92

0.97

1.03

1.09

1.14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.97

0.98

0.99

1.00

1.01

1.02

1.03

1.04

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.97

0.98

0.99

1.01

1.02

1.03

Figure A.9: Ratio of average tick loss functions relative to the stationary Realized EGARCHFilled out markers indicate that the model is part of the MCS at a 10% significance level.Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow),Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple) and EGARCH (blue).

104 CHAPTER 2.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.92

0.97

1.03

1.09

1.14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.97

0.98

0.99

1.00

1.01

1.02

1.03

1.04

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.97

0.98

0.99

1.01

1.02

1.03

Figure A.10: Ratio of average tick loss functions relative to the stationary Realized EGARCHFilled out markers indicate that the model is part of the MCS at a 10% significance level.Realized EGARCH (black), Realized TV-EGARCH (red), Realized Adaptive-EGARCH (yellow),Realized C-Spline-EGARCH (green), Realized Q-Spline-EGARCH (purple) and EGARCH (blue).

A.2. FIGURES 105

Stationary

500 1000 1500 2000 2500 3000 3500 4000

Iteration

-200

-150

-100

-50

0

50

100

150

CU

SU

M

Figure A.11: CUSUM test

TV Adaptive

1000 2000 3000 4000

Iteration

-150

-100

-50

0

50

100

150

CU

SU

M

1000 2000 3000 4000

Iteration

-150

-100

-50

0

50

100

150

CU

SU

M

Q-Spline C-Spline

1000 2000 3000 4000

Iteration

-150

-100

-50

0

50

100

150

CU

SU

M

1000 2000 3000 4000

Iteration

-150

-100

-50

0

50

100

150

CU

SU

M

Figure A.12: CUSUM test.

106 CHAPTER 2.

A.3 Tables

Table A.1: Maximum likelihood parameter estimates for the five different modelsStandard errors are given in parenthesis. p denotes the total number of parameters in the givenmodel.

Stationary TV Adaptive Q-Spline C-Spline

ω -9.105 (0.0717) -8.788 (0.0553) -9.126 (0.0424) -8.837 (0.2069) -8.797 (0.1482)β 0.969 (0.0028) 0.921 (0.0053) 0.936 (0.0042) 0.943 (0.0039) 0.966 (0.0026)α 0.301 (0.0113) 0.284 (0.0116) 0.289 (0.0113) 0.290 (0.0114) 0.281 (0.0106)ξ -0.278 (0.0193) -0.274 (0.0198) -0.277 (0.0199) -0.277 (0.0198) -0.280 (0.0194)σ2

u 0.160 (0.0026) 0.154 (0.0025) 0.156 (0.0025) 0.156 (0.0026) 0.158 (0.0026)τ1 -0.149 (0.0053) -0.154 (0.0056) -0.154 (0.0056) -0.153 (0.0056) -0.147 (0.0053)τ2 0.032 (0.0026) 0.033 (0.0024) 0.033 (0.0023) 0.033 (0.0024) 0.032 (0.0024)δ1 -0.124 (0.0061) -0.122 (0.0061) -0.123 (0.0061) -0.122 (0.0061) -0.121 (0.0061)δ2 0.047 (0.0031) 0.047 (0.0031) 0.048 (0.0030) 0.048 (0.0030) 0.045 (0.0031)

p 9 27 21 18 14logL 10,683.87 10,758.99 10,737.32 10,728.02 10,698.84

logLp 12,705.82 12,712.61 12,707.88 12,706.45 12,705.93

Table A.2: Descriptive statistics for the annualized volatility of the five models

Stationary TV Adaptive Q-Spline C-Spline

Mean 18.04 18.02 18.02 18.01 18.06Std. 9.47 9.61 9.37 9.32 9.45Min 5.93 6.11 6.34 5.72 5.68Max 97.68 98.41 94.55 94.17 95.68

CorrelationsStationary 1.000

TV 0.996 1.000Adaptive 0.998 0.998 1.000Q-Spline 0.998 0.997 0.999 1.000C-Spline 1.000 0.996 0.998 0.999 1.000

A.3. TABLES 107

Table A.3: Full sample forecasting results

Realized EGARCH

Horizon MedFE MSFE MedSFE QLIKE Lp

h = 1 −8.145 ·10−6 1.078 ·10−7 4.293 ·10−10 1.084 ·10−1 7331.76h = 5 −1.587 ·10−5 1.509 ·10−7 1.079 ·10−9 2.150 ·10−1

h = 10 −2.559 ·10−5 1.885 ·10−7 1.772 ·10−9 3.030 ·10−1

h = 20 −4.270 ·10−5 2.149 ·10−7 3.522 ·10−9 4.315 ·10−1

Realized TV-EGARCH


h = 1 −6.622 ·10−6 1.181 ·10−7 4.001 ·10−10 1.100 ·10−1 7326.89h = 5 −1.289 ·10−5 1.757 ·10−7 9.336 ·10−10 2.284 ·10−1

h = 10 −1.682 ·10−5 2.090 ·10−7 1.251 ·10−9 3.350 ·10−1

h = 20 −2.165 ·10−5 2.389 ·10−7 1.775 ·10−9 4.986 ·10−1

Realized Adaptive-EGARCH


h = 1 −9.237 ·10−6 1.137 ·10−7 4.503 ·10−10 1.089 ·10−1 7327.28h = 5 −1.967 ·10−5 1.654 ·10−7 1.178 ·10−9 2.217 ·10−1

h = 10 −2.841 ·10−5 2.013 ·10−7 1.919 ·10−9 3.188 ·10−1

h = 20 −4.164 ·10−5 2.270 ·10−7 3.230 ·10−9 4.501 ·10−1

Realized Q-Spline-EGARCH


h = 1 −6.430 ·10−6 1.118 ·10−7 4.277 ·10−10 1.093 ·10−1 7326.36h = 5 −1.254 ·10−5 1.616 ·10−7 9.827 ·10−10 2.246 ·10−1

h = 10 −1.595 ·10−5 1.979 ·10−7 1.379 ·10−9 3.142 ·10−1

h = 20 −1.902 ·10−5 2.293 ·10−7 1.812 ·10−9 4.499 ·10−1

Realized C-Spline-EGARCH


h = 1 −7.255 ·10−6 1.121 ·10−7 4.329 ·10−10 1.098 ·10−1 7329.44h = 5 −1.347 ·10−5 1.619 ·10−7 1.009 ·10−9 2.273 ·10−1

h = 10 −1.784 ·10−5 1.989 ·10−7 1.509 ·10−9 3.274 ·10−1

h = 20 −2.176 ·10−5 2.306 ·10−7 1.981 ·10−9 4.715 ·10−1

EGARCH


h = 1 −7.214 ·10−6 1.212 ·10−7 6.967 ·10−10 1.624 ·10−1 7275.93h = 5 −1.177 ·10−5 1.599 ·10−7 1.252 ·10−9 2.560 ·10−1

h = 10 −1.795 ·10−5 1.879 ·10−7 1.748 ·10−9 3.284 ·10−1

h = 20 −2.846 ·10−5 2.128 ·10−7 2.588 ·10−9 4.363 ·10−1

108 CHAPTER 2.

Table A.4: Forecasting results excl. financial crises

Realized EGARCH


h = 1 −9.281 ·10−6 9.644 ·10−9 3.455 ·10−10 1.097 ·10−1 6738.80h = 5 −1.751 ·10−5 1.800 ·10−8 8.925 ·10−10 2.122 ·10−1

h = 10 −2.732 ·10−5 1.910 ·10−8 1.512 ·10−9 2.764 ·10−1

h = 20 −4.572 ·10−5 2.153 ·10−8 3.267 ·10−9 3.669 ·10−1

Realized TV-EGARCH


h = 1 −7.011 ·10−6 9.820 ·10−9 2.966 ·10−10 1.084 ·10−1 6737.02h = 5 −1.294 ·10−5 1.729 ·10−8 7.028 ·10−10 2.141 ·10−1

h = 10 −1.615 ·10−5 1.956 ·10−8 9.310 ·10−10 2.848 ·10−1

h = 20 −2.080 ·10−5 2.318 ·10−8 1.316 ·10−9 3.766 ·10−1

Realized Adaptive-EGARCH


h = 1 −1.055 ·10−5 9.464 ·10−9 3.779 ·10−10 1.097 ·10−1 6736.28h = 5 −2.106 ·10−5 1.684 ·10−8 9.811 ·10−10 2.142 ·10−1

h = 10 −3.008 ·10−5 1.843 ·10−8 1.656 ·10−9 2.798 ·10−1

h = 20 −4.417 ·10−5 2.093 ·10−8 2.902 ·10−9 3.570 ·10−1

Realized Q-Spline-EGARCH


h = 1 −6.857 ·10−6 9.550 ·10−9 3.050 ·10−10 1.087 ·10−1 6735.88h = 5 −1.238 ·10−5 1.729 ·10−8 7.416 ·10−10 2.156 ·10−1

h = 10 −1.576 ·10−5 1.928 ·10−8 1.044 ·10−9 2.776 ·10−1

h = 20 −1.932 ·10−5 2.361 ·10−8 1.308 ·10−9 3.635 ·10−1

Realized C-Spline-EGARCH


h = 1 −7.508 ·10−6 9.661 ·10−9 3.072 ·10−10 1.091 ·10−1 6739.19h = 5 −1.346 ·10−5 1.812 ·10−8 7.382 ·10−10 2.188 ·10−1

h = 10 −1.743 ·10−5 2.043 ·10−8 1.074 ·10−9 2.935 ·10−1

h = 20 −2.194 ·10−5 2.614 ·10−8 1.469 ·10−9 4.019 ·10−1

EGARCH


h = 1 −1.028 ·10−5 1.213 ·10−8 5.517 ·10−10 1.616 ·10−1 6691.30h = 5 −1.451 ·10−5 1.783 ·10−8 1.011 ·10−9 2.510 ·10−1

h = 10 −2.037 ·10−5 1.932 ·10−8 1.340 ·10−9 3.052 ·10−1

h = 20 −3.274 ·10−5 2.120 ·10−8 2.330 ·10−9 3.688 ·10−1

C H A P T E R 3INTRODUCING MACRO-FINANCE VARIABLES INTO

THE REALIZED EGARCH FRAMEWORK


Abstract

We propose two ways of including macro-finance variables into the Realized EGARCH

model of Hansen and Huang (2016). Firstly, an additive decomposition, where the

exogenous variables are added directly to the GARCH equation. Secondly, a multi-

plicative component model that separates the latent volatility into a part describing

the conditional heteroskedasticity and a part modeling the baseline volatility as a

function of exogenous variables. An empirical application to the exchange-traded

index fund SPY that tracks the S&P 500 Index and 20 individual stocks involving

three macro-finance variables shows promising in-sample gains but only modest

out-of-sample gains. Furthermore, our results corroborate that the additional infor-

mation content from including exogenous covariates is much smaller when working

in a framework utilizing realized measures of volatility compared to one that solely

exploits squared returns.

109

110 CHAPTER 3.

3.1 Introduction

The seminal papers of Engle (1982) and Bollerslev (1986) introduced the GARCH

framework able to model the time-varying conditional heteroskedasticity inherent to

financial return series. The GARCH model and a plethora of extensions are capable

of describing salient features of financial returns series such as volatility clustering

(Mandelbrot, 1963) and the leverage effect (Black, 1976). Quickly, a big literature uti-

lizing in particular the GARCH framework for research in linking macroeconomic and

financial variables to volatility emerged and numerous studies have since explored

different macro-finance variables’ ability to improve in-sample and out-of-sample

volatility predictions.1 Although much research in this area continues to use clas-

sical GARCH models that utilize squared returns as the signal about volatility, the

benchmark volatility model arguably ought to exploit realized measures that have

revolutionized the modeling of latent volatility of financial markets in the last couple

of decades.

In this paper, we are interested in specifying a model that extracts information

from exogenous covariates while exploiting realized measures of volatility. To this end,

we apply the successful Realized EGARCH (REGARCH) model of Hansen and Huang

(2016). The REGARCH model is basically an EGARCH-X model with the realized mea-

sure as the covariate combined with a measurement equation. By linking the realized

measure and the conditional variance, the measurement equation ensures dynami-

cal completeness and thereby facilitates multi-step ahead volatility predictions. We

consider both an additive and a multiplicative extension of the REGARCH model.

The former adds the exogenous variables directly to the GARCH equation while the

latter is a multiplicative component model that separates the latent volatility into a

part describing the conditional heteroskedasticity and a part modeling the baseline

volatility as a function of the exogenous variables.

We present an empirical application using the exchange-traded index fund SPY

tracking the S&P 500 Index and 20 individual stocks. We consider three daily macro-

finance indicators commonly applied in the literature: the Chicago Board Options

Exchange’s volatility index typically known by its ticker symbol VIX, the Economic

Policy Uncertainty index (EPU), and the Arouba-Diebold-Scotti business condition

index (ADS). The variables are selected since they cover three interesting cases. VIX

is the risk neutral expectation of the integrated variance of the S&P 500 Index and

1A non-exhaustive list of considered variables includes interest levels (Glosten, Jagannathan, andRunkle, 1993; Brenner, Harjes, and Kroner, 1996; Gray, 1996), interest spreads (Dominguez, 1998; Hagiwaraand Herce, 1999), bid-ask spread (Bollerslev and Melvin, 1994), forward-premiums (Hodrick, 1989), volume(Lamoureux and Lastrapes, 1990; Wagner and Marsh, 2005; Fleming, Kirby, and Ostdiek, 2008; Gallo andPacini, 2000; Girma and Mougoué, 2002), period effects such as day-of-week effect (Connolly, 1989; Baker,Rahman, and Saadi, 2008; Alagidede, 2008; Charles, 2010), realized measures (Engle, 2002), VIX (Day andLewis, 1992; Amado and Laakkonen, 2014; Kambouroudis and McMillan, 2016), the Policy UncertaintyIndex (Asgharian, Christiansen, and Hou, 2015; Liu and Zhang, 2015), the Arouba-Diebold-Scotti businesscondition index (Dorion, 2016), low-frequency macro-economic time series (Engle and Rangel, 2008;Engle, Ghysels, and Sohn, 2013; Conrad and Loch, 2015; Paye, 2012).

3.2. REALIZED MEASURES OF VOLATILITY 111

therefore a natural predictor of realized variance. ADS tells us about the state of the

economy and helps us examining whether financial market volatility is countercycli-

cal. EPU allows us to investigate whether economic policy uncertainty can explain

financial market volatility. Furthermore, the variables are available at daily frequency

such that there is no need for mixed data sampling or alternative approaches. Several

interesting results emerge from the in- and out-of-sample evaluation of the models

including exogenous covariates and the comparison with their benchmark counter-

parts. For the multiplicative decomposition, we realize large in-sample and modest

short horizon out-of-sample gains from including VIX as a covariate, while somewhat

smaller gains for the additive specification. This stipulates that a multiplicative speci-

fication may be the preferred avenue when incorporating implied volatility in GARCH

type models. For the ADS and EPU, we also find more modest evidence of superior

in-sample performance, but close to non out-of-sample gains. Expectedly, we find

that including additional covariates is less beneficial in the REGARCH model than

in the EGARCH model, which does not include realized measures. In the estimated

EGARCH models, we find that VIX subsumes the information in squared returns.

The paper is structured as follows. Section 3.2 introduces the concept of Realized

Volatility. The modeling framework and estimation strategy are introduced in Section

3.3 and Section 3.4, respectively. We discuss forecasting methodology and evaluation

in Section 3.5 and Section 3.6, respectively. The empirical applications are presented

in Section 3.7. Finally, concluding remarks are given in Section 3.8.

3.2 Realized measures of volatility

Assume that there exists a representation of the log-price of an asset or index, Yt ,

such that for all t ∈ [0,T

]Yt =

T∫0

µudu +T∫

0

σudWu , (3.1)

where µu is the instantaneous drift, σu is the instantaneous volatility, and dWu is a

standard Brownian motion.

We are interested in daily compounded returns defined by rt = Yt−Yt−1. Andersen

et al. (2003) and Barndorff-Nielsen and Shephard (2002) showed that rt is Gaussian

conditional on Ft =σ(Ys , s ≤ t

), the σ-algebra generated by the sample paths of Y .

In particular,

rt |Ft ∼ N

t∫t−1

µt−1+udu,

t∫t−1

σ2udu

, (3.2)

where the term

IVt =t∫

t−1

σ2udu, (3.3)

112 CHAPTER 3.

is known as the Integrated Variance (IV) and measures the ex-post variance of day t .

The integrated variance is latent. Thus, in empirical applications, we need to esti-

mate the quantity using prices observed at discrete and possibly irregularly spaced

intervals. One possibility is to sample n intra-daily observations equidistant in calen-

dar time and calculate the Realized Variance (RV):

RVt =n∑

i=1r 2

t ,i , (3.4)

where rt ,i = Yt−1+i 1n−Yt−1+(i−1) 1

n. Realized volatility is the square root of RVt . Under

the stated assumption of a semi-martingale in (3.1), Barndorff-Nielsen and Shephard

(2002) showed that the RV is ap

n consistent estimator of the IV. However, this as-

sumption is often at odds with empirical evidence due to issues related to market

microstructure noise. Research in estimation of IV in the presence of microstructure

noise has become an flourishing area in financial econometrics. The early literature,

e.g. Andersen, Bollerslev, Diebold, and Ebens (2001), focused on sparse sampling. The

idea is to sample at an arbitrary lower frequency, e.g. at 5 min intervals, to balance

the accuracy and bias introduced by market microstructure noise. The more recent

literature focuses on the development of estimators robust to market microstructure

noise. Consistent estimation methods for IV include the Realized Kernel (RK) esti-

mators of Barndorff-Nielsen et al. (2008), the modified MA filter of Hansen, Large,

and Lunde (2008), the two time scales realized volatility estimator of Zhang et al.

(2005), the Multi-Scale approach of Zhang (2006), and the Range-Based estimator of

Christensen and Podolskij (2007).

In the empirical applications of this paper, we only consider the RK defined by

RKt =H∑

h=−HK

(h

H +1

)γh , γh =

n∑i=|h|+1

rt ,i rt ,i−|h|, (3.5)

where K (·) is the Parzen kernel function. The RK was found to perform very well in

Hansen and Lunde (2006), where a range of measures was considered, and in the

VaR application in Brownless and Gallo (2010); see also the comparison of realized

measures in Gatheral and Oomen (2010).

3.3 Modeling framework

In this paper, we model the latent volatility of the daily return series of an asset or

index{rt

}by including exogenous variables in the REGARCH framework of Hansen

and Huang (2016). We start by introducing the original framework. Let Ft−1 be the

information set containing the historical information at time t −1. Define the condi-

tional mean, µt = E[rt |Ft−1], and the conditional variance, σ2t = Var

[rt |Ft−1

], of the

return series.

3.3. MODELING FRAMEWORK 113

3.3.1 The Realized EGARCH model

The REGARCH model of Hansen and Huang (2016) with a single realized measure is

defined by the following three equations:

rt =µt +σt zt , (3.6)

logσ2t =ω+β logσ2

t−1 +τ(zt−1

)+αut−1, (3.7)

log xt = ξ+φ logσ2t +δ

(zt

)+ut . (3.8)

The equations are known as the return equation, the GARCH equation, and the

measurement equation, respectively. It is typically assumed that zt ∼ i.i.dN(0,1

)and

ut ∼ i.i.d.N(0,σ2

u

)are mutually and serially independent. The realized measure, xt ,

is an estimate of IV such as RV or RK.

The return equation, which decomposes the returns into a time-varying condi-

tional mean and an error term with conditional heteroskedasticity, and the GARCH

equation, modeling the conditional heteroskedasticity, are standard in GARCH type

models. The focus is typically on modeling σ2t as daily returns often are close to

serially uncorrelated for many financial assets. Therefore, we simply model the con-

ditional mean as a constant, i.e. µt =µ. The GARCH equation has an autoregressive

structure for logσ2t with the innovation term τ

(zt−1

)+αut−1. Thus, β< 1 measures

the persistence of the conditional variance and α tells us how shocks to the realized

measure affect the conditional variance. One reason for choosing a log specification

is that it guarantees positivity of the conditional variance. Another advantages is a

more reasonable fit to the distributional assumption applied in the quasi-maximum-

likelihood estimation. However, one disadvantage to be noted is the necessity to

account for distributional aspects when performing multi-period ahead forecasting

due to a Jensen’s inequality term.

The idea of a measurement equation discriminates the REGARCH model from

other volatility models incorporating realized measures such as the GARCH-X model

of Engle (2002), the HEAVY model of Shephard and Sheppard (2010), and the MEM

model of Engle and Gallo (2006). The measurement equation makes the model com-

plete in the sense that it specifies the dynamic structure of the realized measure. As

the conditional variance is an ex-ante measure of volatility and the realized measure

is an ex-post measure of volatility, it is natural to assume a link between the two.

However, some discrepancies between the two measures are expected. One source

is that we are modeling close-to-close return volatility while the realized measure

is calculated using open-to-close information. Thus, we would expect the realized

measure to be lower than the conditional variance on average. Therefore, it is neces-

sary to include the parameters ξ and φ. If the two measures are almost proportional,

we should expect to find ξ≈ 0 and φ≈ 1. Another way to motivate the link between

the conditional variance and the realized measure follows from the mean equation

which implies log(rt −µt

)2 = logσ2t + log z2

t . Because xt is a realized measure just as

114 CHAPTER 3.

(rt −µt

)2, albeit less noisy, we would expect that log xt ≈ logσ2t + f

(zt

)+et , where et

is an error term. This also motivates the logarithmic measurement equation.

The so-called leverage functions τ (z) and δ (z) are defined by

τ (z) = τ1z +τ2

(z2 −1

),

δ (z) = δ1z +δ2

(z2 −1

).

The notion of leverage is related to the well-known leverage effect (Black, 1976). The

term τ (z) can be related to the news impact curve introduced by Engle and Ng

(1993) as it models how positive and negative returns impact the log conditional

variance. Hansen et al. (2012) considered a class of leverage functions constructed

using Hermite polynomials. They and later Hansen and Huang (2016) found the

quadratic functional form to be satisfactory in different empirical applications. One

advantage of this particular leverage function is that E[τ (z)

] = E[δ (z)

] = 0 when

E [z] = 0 and Var[z] = 1. Together with the logarithmic volatility specification, this

particular choice of leverage function is empirically found to make the assumption

about independence between zt and ut realistic.

3.3.2 Additive decomposition

We now turn to the inclusion of exogenous variables in the volatility dynamics speci-

fied by (3.6)-(3.8). One possibility often considered in classical GARCH models is to

additively include the exogenous variables in the GARCH equation:

rt =µt +σt zt , (3.9)


t−1 +τ(zt−1

)+γ′vt−1 +αut−1, (3.10)


(zt

)+ut , (3.11)

where vt−1 is a vector of exogenous variables that must be lagged one period to be

included in the information set at time t −1. We note that the latent volatility can be

written as

logσ2t+1 =βk+1 logσ2

t−k +k∑

i=0βi [

ω+γ′vt−i +τ(zt−i )+αut−i]

. (3.12)

This implies that weight put on the information of the exogenous variables is decaying

at the same speed as the weight put on the realized measure, which is the main

difference compared to the multiplicative specification presented below. Thus, if the

persistence is large and the coefficients on the exogenous covariates are large, we also

put large weight on past values of the exogenous covariates. This may be problematic,

if only the latest values are relevant for predicting latent volatility while we need a

large persistence for incorporating the information from the realized measure.

From the literature on classical GARCH models, we know that the additive inclu-

sion of exogenous covariates may help to overcome shortcominings of the benchmark

3.3. MODELING FRAMEWORK 115

counterparts. Han (2015) discusses the asymptotic properties of GARCH-X processes

and shows that such processes more adequately explain stylized facts of financial time

series such as the long-memory and leptokurtosis. The properties of the GARCH-X

process heavily depend on the degree of persistence of the exogenous covariate.

3.3.3 Multiplicative decomposition

Another way to include covariates is to specify a multiplicative component model by

assuming that the latent volatility can be decomposed into a component describing

the conditional heteroskedasticity, ht , and a component describing the baseline

volatility, g t : σ2t = ht g t . This modeling approach has become increasingly popular in

the GARCH literature (see e.g. Engle and Lee (1999), Engle and Rangel (2008), Amado

and Teräsvirta (2013), and Engle et al. (2013)). The reason is that component models

offer a parsimonious way to model the often complex dynamics of financial time se-

ries and explain stylized facts that cannot be captured by classical, stationary GARCH

models. Conrad and Kleen (2016) discuss the statistical properties of multiplicative

GARCH models and show that these models are better able to match stylized facts of

financial return series than their benchmark GARCH model.

One particular empirical issue is the presence of structural breaks that may cause

spurious long-memory; see Mikosch and Starica (2004) and Hillebrand (2005), among

others. Several component models in the literature address non-stationarity by de-

composing the volatility into a high-frequency (short-run) and low-frequency (long-

run) component. In the GARCH literature, Engle and Rangel (2008), Morana (2002),

and Amado and Teräsvirta (2013), among others, model the low-frequency compo-

nent as a deterministic function of time. One appealing feature of these models is

that the amplitude of volatility clusters now is allowed to vary over time in line with

empirical observations but in contradiction to stationary GARCH models. Laursen

and Jakobsen (2017) extend the idea to the REGARCH framework.

In order to link, in particular low-frequency, macroeconomic variables to financial

market volatility, Engle et al. (2013) propose the GARCH-MIDAS model that specifies

the low-frequency component as a MIDAS-filter of the macroeconomic variables.

This modeling framework has been applied in numerous empirical applications

(Asgharian, Hou, and Javed, 2013; Asgharian et al., 2015; Conrad and Loch, 2015;

Dorion, 2016). In the Realized GARCH framework, Dominicy and Vander Elst (2015)

include a MIDAS component.

In this paper, we restrict ourselves to daily macro-finance indicators. Thus, there

is no explicit need for a MIDAS type specification. Actually, Conrad and Schienle

(2015) admit that their attempt to estimate a GARCH-MIDAS specification using VIX

and RV resulted in weights decaying so fast to zero that it resembled a GARCH-X

specification. However, this should not discourage the use of MIDAS specifications

with daily indicators in general. It is likely that a MIDAS filter may help smooth noisy

signals such as the EPU.

116 CHAPTER 3.

In this paper, we consider the following specification:

rt =µt +σt zt , (3.13)

loght =β loght−1 +τ(zt−1

)+αut−1, (3.14)


(zt

)+ut , (3.15)

log g t =ω+γ′vt−1. (3.16)

Due to identification issues, a constant is only included in the baseline volatility

component such that E[loght

]= 0.2

For the multiplicative specification, we have

logσ2t+1 = log g t+1 + loght+1

= ω+γ′vt +βk+1 loght−k +k∑

i=0βi [

αut−i +τ(zt−i )]

. (3.17)

Thus, compared to the additive case, the information content in the most recent

observations of the exogenous covariates is used to model the time-varying baseline

volatility and not as an additional variable in the GARCH filter.

3.3.4 The EGARCH model

The EGARCH model of Nelson (1991) is defined by the return equation in (3.6) with


t−1 +α(|zt−1|−

p2/π

)+τ1zt−1. (3.18)

In the empirical application, we consider both an additive and a multiplicative version

with exogenous covariates similar to the ones specified in the REGARCH framework.

3.4 Estimation

We now turn to estimation and inference within a Quasi-Maximum-Likelihood

(QML) framework. We assume that zt ∼ i.i.d.N(0,1

)and ut ∼ i.i.d.N

(0,σ2

u

)and write

the leverage functions as τ(zt

) = τ′a(zt

)and δ

(zt

) = δ′b(zt

)with a

(zt

) = b(zt

) =(zt , z2

t −1)′

. Define the parameter vector

θ =(µ,ω,β,τ′,α,ξ,φ,δ′,σ2

u ,γ′)′

. (3.19)

The log-likelihood function reads

L(r, x, v ;θ

)= T∑t=1

`t(rt , xt , vt ;θ

), (3.20)

2The GARCH equation implies that loght = β j loght− j +∑ j−1

i=0 βi [τ(zt−1−i )+αut−1−i

]such that

loght has a stationary representation if |β| < 1. Hence, the result immediately follows.

3.4. ESTIMATION 117

where `t(rt , xt , vt ;θ

)is the log-likelihood contribution at time t . As the log-likelihood

contribution and therefore also the score differ depending on whether an additive or

a multiplicative decomposition are applied, we present the two cases separately. We

now derive the score since it defines the first order conditions for the QML Estimator

(QMLE) and is necessary to obtain standard errors. For notional convenience, let

azt = ∂a(zt )/∂zt and bzt = ∂b(zt )/∂zt .

3.4.1 Additive decomposition

The log-likelihood contribution at time t is given by

`t(rt , xt , vt ;θ

)=−1

2

[2log2π+ logσ2

t + z2t + logσ2

u + u2t

σ2u

], (3.21)

with zt = zt(θ)= (

rt −µ)

/σt and ut(θ)= log xt −ξ−φ logσ2

t −δ(zt

). Key components

are the derivatives stated in the following lemma.

Lemma 1. The derivatives A(zt ) = ∂ logσ2t+1/∂ logσ2

t and B(zt ,ut ) = ∂`t /∂ logσ2t are

given from respectively

A(zt ) = (β−αφ)+ 1

2


)zt , (3.22)

and

B(zt ,ut ) =−1

2

[(1− z2

t

)+ ut

σ2u

(δ′bzt zt −2φ

)]. (3.23)


Next, we define the score of logσ2t+1 with respect to θ.

Lemma 2. Let θ1 = (ω,β,τ′,α)′ and θ2 = (ξ,φ,δ′)′. Furthermore, define mt =(1, logσ2

t , zt ,

z2t −1,ut

)′and nt =

(1, logσ2

t , zt , z2t −1

)′. The derivatives ∂ logσ2

t+1/∂θ are given from

the stochastic recursions

hµ,t+1 = ∂ logσ2t+1

∂µ= A(zt )hµ,t +


) 1

σt, (3.24)

hθ1,t+1 = ∂ logσ2t+1

∂θ1= A(zt )hθ1,t +mt , (3.25)

hθ2,t+1 = ∂ logσ2t+1

∂θ2= A(zt )hθ2,t +αnt , (3.26)

hγ,t+1 = ∂ logσ2t+1

∂γ= A(zt )hγ,t +vt . (3.27)


118 CHAPTER 3.

Finally, we present the score.

Theorem 1. Utilizing components from Lemma 1 and Lemma 2, the score with respect

to the parameters, θ =(µ,ω,β,τ′,α,ξ,φ,δ′,σ2

u ,γ′)′

, is given from

∂`t

∂µ= B(zt ,ut )hµ,t +

[zt −δ′ ut

σ2u

bzt

]1

σt, (3.28)

∂`t

∂θ2= B(zt ,ut )hθ1,t , (3.29)

∂`t


ut

σ2u

nt , (3.30)

∂`t

∂γ= B(zt ,ut )hγ,t , (3.31)

∂`t

∂σ2u

= 1

2

u2t −σ2

u

σ4u

. (3.32)


3.4.2 Multiplicative decomposition

The initial value of the logarithm of the conditional variance, log h0, is set equal to its

unconditional mean, E[loght ] = 0.3

`t(rt , xt , vt ;θ

)=−1

2

[2log2π+ loght + log g t + z2

t + logσ2u + u2

t

σ2u

], (3.33)

with zt = zt(θ)= (

rt −µ)

/σt and ut(θ)= log xt −ξ−φ

[loght + log g t

]−δ(zt

).

Lemma 3. The derivatives C (zt ) = ∂ loght+1/∂ loght , D(zt ) = ∂ loght+1/∂ log g t , and

E(zt ,ut ) = ∂`t /∂ loght = ∂`t /∂ log g t are given from respectively

C (zt ) = (β−αφ)+ 1

2


)zt , (3.34)

D(zt ) = −αφ+ 1

2


)zt , (3.35)

and

E(zt ,ut ) = −1

2

[(1− z2

t

)+ ut

σ2u

(δ′bzt −2φ

)]. (3.36)


Next, we define the score of loght+1 with respect to θ.

3Alternatively, one could estimate it like Hansen and Huang (2016)

3.4. ESTIMATION 119

Lemma 4. Let θ1 = (β,τ′,α)′, θ2 = (ξ,φ,δ′)′, and θ3 = (ω,γ)′. Furthermore, define mt =(loght , zt , z2

t −1,ut

)′and nt =

(1, logσ2

t , zt , z2t −1

)′. The derivatives ∂ loght+1/∂θ are

given from the stochastic recursions

hµ,t+1 = ∂ loght+1

∂µ=C (zt )hµ,t +


) 1

σt, (3.37)


∂θ1=C (zt )hθ1,t +mt , (3.38)


∂θ2=C (zt )hθ2,t +αnt , (3.39)


∂θ3=C (zt )hθ3,t +D(zt )gθ3,t+1, (3.40)

where

gθ3,t+1 =∂ log g t

∂θ3= (

1, v ′t

)′ .


Finally, we present the score.

Theorem 2. Utilizing components from Lemma 3 and Lemma 4, the score with respect

to the parameters, θ =(µ,ω,β,τ′,α,ξ,φ,δ′,σ2

u ,γ′)′

, is given from

∂`t

∂µ= E(zt ,ut )hµ,t +

[zt −δ′ ut

σ2u

bzt

]1

σt, (3.41)

∂`t

∂θ1= E(zt ,ut )hθ1,t , (3.42)

∂`t

∂θ2= E(zt ,ut )hθ2,t +

ut

σ2u

nt , (3.43)

∂`t

∂σ2u

= 1

2

u2t −σ2

u

σ4u

, (3.44)

∂`t

∂θ3= E(zt ,ut )hθ3,t +E(zt ,ut )gθ3,t . (3.45)


3.4.3 Partial log-likelihood function

The conditional density of(rt , xt

)can be factorized as

f(rt , xt |Ft−1

)= f(rt |Ft−1

)f(xt |rt ,Ft−1

).

120 CHAPTER 3.

Implying that the Gaussian log-likelihood can be decomposed as

T∑t=1

`t(rt , xt , vt ;θ

)=−1

2

T∑t=1

[log2π+ logσ2

t + z2t

]− 1

2

T∑t=1

[log2π+ logσ2

u + u2t

σ2u

].

The first part is known as the partial log-likelihood function since it only measures

the goodness of fit of the return distribution. The partial log-likelihood is in particu-

lar relevant when comparing models utilizing realized measures with GARCH type

models.

3.4.4 Asymptotic properties of the estimators

To our knowledge, the asymptotic properties of even the standard REGARCH model

have not yet been established. Neither has the asymptotic properties of the EGARCH-

X model, but only the special case of an EGARCH(1,1) model (Wintenberger, 2013).

The asymptotic analysis of the REGARCH model and the extensions presented in this

paper is a complicated task and it is beyond the scope of this paper. However, we will

point the reader towards some of the contributions in the literature that may justify a

conjecture regarding consistency and asymptotic normality of the estimators. The

proof of consistency and asymptotic normality for stationary GARCH models can

be found in Francq and Zakoïan (2010) and references therein. Han and Kristensen

(2014) established the asymptotic theory for the QMLE in GARCH-X models with

stationary and non-stationary covariates. Han and Kristensen (2015) perform an

asymptotic analysis of a multiplicative GARCH-X model where a non-linear function

of exogenous covariates constitutes the ’long-run’ component.

Following Corollary 1 and the arguments in Hansen and Huang (2016), the score

function of both the additive and multiplicative decomposition is a martingale differ-

ence sequence provided that E[zt |Ft−1

]= 0, E[

z2t |Ft−1

]= 1, E

[ut |zt ,Ft−1

]= 0 and

E[

u2t |zt ,Ft−1

]=σ2

u . Based on this result, they conjecture that the distribution of the

QMLE is asymptotically normal:p

T(θ−θ0

)d→N

(0,T I−1JI−1

), (3.46)

where J is the limit of the outer-product of the scores and I is (minus) the limit of

the Hessian matrix for the log-likelihood functions. We make the same conjecture.

It can be noted that Vander Elst (2015) and Borup and Jakobsen (2017) use a para-

metric bootstrap to investigate the validity of this conjecture in their extension of the

REGARCH model. They found the asymptotic approximation to be reasonable with

sample sizes similar to the ones used in this paper.

3.5 Forecasting methodology

We seek to forecast the conditional variance k ≥ 1 days into the future. One-step ahead

forecasts are easily obtained directly from the GARCH equation (3.10) in the additive

3.6. FORECAST EVALUATION 121

case or by combining the GARCH equation (3.14) and the new component (3.16) in

the multiplicative case. However, some care must be taken when considering multi-

step ahead forecasting. First, we note that the introduction of exogenous variables

makes the dynamic structure incomplete unless we specify a dynamic model for

the exogenous variables. It is standard in the literature to circumvent this issue by

a random walk assumption such that vt+k|t = E[

vtk |Ft]= vt or by keeping g t fixed

(Engle et al., 2013; Dominicy and Vander Elst, 2015). We assume that the exogenous

variables are martingales. Secondly, we note that the dynamic structure for the models

are specified in terms of logσ2t (additive model) or loght and log g t (multiplicative

model). Jensen’s inequality implies that exp

(E[

logσ2t+k |Ft

])6= E

[exp

(logσ2

t+k

)|Ft

].

To obtain an unbiased forecast we therefore need to account for distributional aspects.

If M denotes the number of simulations, we obtain the conditional variance forecast

as the average

σ2t+k|t =

1

M

M∑m=1

exp(logσ2

t+k|t ,m

),

where logσ2t+k|t ,m is a simulated value of the conditional volatility at time t +k condi-

tional on the information available at time t . In the simulation procedure, we utilize

the empirical distributions of zt and ut .4 In the additive case, we note that recursive

substitution of the GARCH equation (3.14) implies

logσ2t+k =βk logσ2

t +k∑

i=1βi−1

(ω+τ(

zt+k−i)+αut+k−i +γ′vt+k−i

). (3.47)

In the multiplicative case, we have similarly

loght+k =βk loght +k∑

i=1βi−1

(τ(zt+k−i

)+αut+k−i

), (3.48)

and therefore,

logσ2t+k = loght+k + log g t+k =βk loght +

k∑i=1

βi−1(τ(zt+k−i

)+αut+k−i

)+ log g t+k .

(3.49)

3.6 Forecast evaluation

In order to evaluate forecasting performance in the empirical applications, we need a

proxy for latent volatility. As the RK only is a measure of open-to-close volatility, it will

not be an unbiased measure of the close-to-close volatility used in this paper. To deal

with this issue, we follow a strategy similar to Sharma and Vipul (2016) and Huang,

4Using the empirical distribution is standard in the literature, see e.g. Brownless and Gallo (2010).Alternatively, one can use the distribution applied in the MLE.

122 CHAPTER 3.

Liu, and Wang (2016), among others. Hence, as a proxy for the latent close-to-close

volatility, we use σ2t = ηRKt with

η=1T

∑Tt=1 r 2

t1T

∑Tt=1 RKt

. (3.50)

We have chosen to only use data for the out-of-sample period to calculate η due to

observed time-variation in the scaling factor.

A variety of evaluation criteria or loss functions has been suggested in the litera-

ture in order to ascertain the quality of forecasts. However, only a few are applicable

when the target is observed with error as in the case with latent volatility; see Hansen

and Lunde (2006). If the ranking of forecasting models based on a given criteria

depends on whether the proxy or true latent volatility are the target, the criteria is

said to be non-robust. Patton (2011) provides necessary and sufficient conditions on

the functional form of the loss functions ensuring consistency of the ordering when

using a proxy. Two robust measures are the Squared Forecasting Error (SFE) and the

Quasi-Likelihood (QLIKE) loss function defined as respectively

SFEt+k =(σ2

t+k −σ2t+k|t

)2,

QLIKEt+k = σ2t+k

σ2t+k|t

− logσ2

t+k

σ2t+k|t

−1,

where σ2t+k|t denotes the model based forecast. The main difference between the loss

functions is that the SFE solely depends on the forecast error σ2t+k −σ2

t+k|t while the

QLIKE solely depends on the ratio σ2t+k /σ2

t+k|t . Following the arguments in Patton

(2011), this implies that the average QLIKE loss will be less affected by the most

extreme observations while the MSFE will be sensitive to extreme observations and

the level of return volatility. Another important feature is that the SFE is symmetric

whereas the QLIKE penalizes underprediction more heavily than overprediction.

Patton and Sheppard (2009) show that QLIKE exhibits more statistical power in

differentiating between volatility forecasts.

3.6.1 Model Confidence Set

To statistically discriminate between competing forecasting models, we implement

the Model Confidence Set (MCS) of Hansen et al. (2011). The procedure starts with

the full set of candidate models M0 ={1, ...,m0

}. Hereafter, the MCS is obtained by

iteratively reducing the number of models in M0 to m < m0. The sequential testing

procedure is based on the loss differential between forecasts i and j , di j ,t+k , for

i > j constructed using the evaluation criteria listed above. At each step, the null

hypothesis of Equal Predictive Ability (EPA)

H0 : E(di j ,t+k

)= 0, ∀i > j ∈M

3.7. EMPIRICAL APPLICATION 123

is tested for a set of models M ∈M0 with M=M0 at the initial step. If H0 is rejected

at the significance level α, the worst performing model is removed and the process is

continued until no rejections occur. The surviving models constitute the MCS, M∗α. If

a fixed significance level α is used at each step, M∗α contains the best model from M0

with (1−α) confidence.

The test of EPA is based on the t-statistics

ti j =di j√

V(di j

) ,

where di j = 1H

∑Hh=1 di j ,h+k is the mean forecast differential over the H periods where

forecasts are available. The (m −1)m/2 unique t-statistics for the set M need to be

combined into one test statistic. Two possibilities are the range statistic,

TR = maxi> j∈M

|ti j | = maxi> j∈M

|di j |√V

(di j

) ,

and the semi-quadratic statistic,

TSQ = ∑i> j∈M,i< j

t 2i j =

∑i> j∈M,i< j

(di j

)2

√V

(di j

) .

Both test statistics indicate a rejection of the EPA hypothesis for large values. It is

necessary to obtain p-values of the test statistics using a bootstrap distribution as the

actual distribution is complicated and depends on the covariance structure between

the forecast models. In this paper, we showcase the results obtained using the semi-

quadratic statistic, but the results are qualitatively similar for the range statistic.

3.7 Empirical application

In this section, we investigate the empirical performance of the aforementioned

models on returns and realized measures for the exchange-traded index fund SPY,

which tracks the S&P 500 Index, and 20 individual stocks. The same series were also

investigated using the RGARCH model in Hansen et al. (2012) and the REGARCH

model in Hansen and Huang (2016) and Banulescu et al. (2014).

3.7.1 Data

The full dataset covers the period from January 2, 2002 to December 31, 2013 and

consists of daily close-to-close returns and daily RK of SPY and the 20 individual

stocks. In the computation of the RK, we restrict attention to the official trading

124 CHAPTER 3.

hours 9:30:00 and 16:00:00 New York time. For each stock, we remove short trading

days where trading spanned less than 20,000 seconds compared to typically 23,400

seconds for a full trading day.5

Table A.1 reports descriptive statistics for the daily returns and the RK. We com-

pute outlier-robust estimates of return skewness and kurtosis (Kim and White, 2004;

Teräsvirta and Zhao, 2011) along with their conventional estimates. The robust mea-

sures point to negligible skewness and quite mild kurtosis in the return series. This

stands in contrast to the moderately skewed, severely fat-tailed distributions sug-

gested by the conventional measures, corroborating the findings in Kim and White

(2004) that stylized facts of returns series change by the use of robust estimators.

We estimate the fractional integrated parameter d with the two-step exact local

Whittle estimator of Shimotsu (2010). Over the full sample all series have d > 0.5,

suggesting that log(RK) is highly persistent and non-stationary.6

Figure A.1 contains different financial time series related to the SPY: returns,

squared returns (annualized), log(RK) (annualized), and the autocorrelation of the

RK. The dotted line separates the estimation period from the out-of-sample period in

the forecasting exercise. As commonly observed in financial returns series and in line

with classical, stationary GARCH models, volatility clusters are observed throughout

the sample. However, the amplitude or baseline volatility seems to change over time.

In particular, there are a volatile period in the beginning of the sample, a tranquil

period in the middle of the sample, a volatile period around the Global Financial

Crisis (GFC), and finally a less volatile period at the end of the sample. The long-range

dependence in different measures of volatility is corroborated by the slowly decaying

autocorrelation of the log(RK) for the SPY.

Following the arguments in Han (2015) and Conrad and Kleen (2016), the long-

range dependence may motivate the inclusion of, probably quite persistent, exoge-

nous variables in the volatility dynamics. In the following, we present the exogenous

variables employed in this application.



Exogenous information

In this application, we consider three macro-finance variables observed at daily

frequency: VIX, EPU, and ADS. In the following, we describe the variables and some

previous applications.

Since Fleming, Ostdiek, and Whaley (1995), VIX has been widely used to forecast

financial market volatility. VIX is based on implied volatilities of the S&P 500 Index

5The data was kindly provided by Asger Lunde.6We estimated the parameters with m = bT q c for q =∈ {0.5,0.55, . . . ,0.8}, leading to no alterations of

the conclusions obtained for q = 0.65.


options and is therefore the risk-neutral expectation of the integrated variance of

the S&P 500 Index for the next 30 calender days.7 If options markets are efficient,

implied volatility should be an efficient forecast of future volatility (Christensen

and Prabhala, 1998). Hence, VIX is a natural predictor of future realized volatility

of the S&P 500 Index. Following Bekaert and Hoerova (2014), we however note that

Et

[RV22

t+1

]= VIX2

t −VPt , where VPt is (the negative of) the variance risk premium

and that the variance premium will be increasing in the risk aversion of the economy

and most likely time-varying with a mean different from zero. Due to its embedded

risk premium VIX will not be an unbiased predicator of future realized volatility.

Furthermore, VIX index is often viewed as an indicator of global market sentiment or

the investors’ fear gauge. Thus, it should also be relevant when forecasting volatility

of individual stocks or other indices than the S&P 500 Index. In the GARCH mod-

eling framework, implied volatility and VIX are often found to improve volatility

forecasts (Day and Lewis, 1992; Martens and Zein, 2004; Amado and Laakkonen,

2014; Kambouroudis and McMillan, 2016).8

The EPU of Baker et al. (2016) is a measure of uncertainty based on the appear-

ance of certain words in news articles. Extensive research has focused on linking

stock market return or volatility to economic uncertainty. Utilizing a GARCH-MIDAS

approach, Asgharian et al. (2015) find that macro economic uncertainty influences

long-run stock and bond volatility, but their out-of-sample results are fairly weak. In

a HAR framework, Liu and Zhang (2015) conclude that EPU contains significantly

predictive power of market volatility both in-sample as well as out-of-sample.

The ADS developed by Aruoba et al. (2009) and published by the Federal Re-

serve (Fed) Bank of Philadelphia proxies business conditions by combining six widely

followed macroeconomic indicators, namely weekly initial jobless claims, monthly

log-growth of non-agricultural payroll employment, industrial production, real man-

ufacturing and trade sales, real personal income minus transfers, and quarterly real

GDP. Dorion (2016) introduces a Macro-GARCH model by employing the ADS in a

GARCH-MIDAS model. The Macro-GARCH model is found to significantly reduce

option pricing errors.

Figure A.2 depicts the variables together with National Bureau of Economic Re-

search (NBER) recession periods and presents scatter plots of the relationship be-

tween the variables lagged one period and the realized variance measured by log(RK).

We have chosen to consider the following transformations: log(VIX2

t−1/250)

and

log(

EPUt−1100

). This is natural as we consider a log-volatility specification, but it also

7See Whaley (2000, 2009) and Mencía and Sentana (2013) for a more detailed description. See alsowww.cboe.com/VIX for a detailed index description.

8We note that there exists a large literature discussing the information content of implied volatilityand realized measures in predicting volatility in- and out-of-sample. A range of modeling approaches hasbeen employed on different data sets, resulting in very different conclusions. See e.g. Kruse et al. (2016),Han and Park (2013), Kambouroudis and McMillan (2016), and references therein. We do not investigatewhether VIX is an efficient forecast of future volatility, but only if it is a relevant covariate when taking themodeling framework as given.

www.cboe.com/VIX

126 CHAPTER 3.

makes the relationship between log(RK) and the exogenous variables more linear.

The intuition of the plots are clear. For the transformed version of VIX, we see, as

expected, an almost linear relationship with log(RK). It seems that EPU is a much

more noisy signal about future volatility. Economic uncertainty is positively corre-

lated with future volatility. Comparing the time series plot of EPU with the one for

RK in A.2 may indicate a structural change in the relationship around the GFC since

the EPU has not reverted to the same extent as volatility. Intuitively, ADS is negatively

correlated with future volatility. Clearly, volatility spikes and ADS plummets during

the GFC. The scatter plot may indicate that the relationship is somewhat non-linear.


3.7.2 Results for the S&P 500 Index

The results pertaining to the in-sample fit of the models are based on the full sample.

In the forecasting exercise, we estimate the models using the sample period until

December 31, 2010, while the remaining part of the sample is reserved for the out-of-

sample exercise.

Estimation results

Table A.2 contains the results from estimating EGARCH and REGARCH models. Natu-

rally, the results when not including exogenous variables are identical for the additive

and multiplicative decomposition except for the magnitude of ω. The REGARCH

models dominate the EGARCH models in terms of partial log-likelihood value with

a large increase of approximately 60 log-likelihood points. This illustrates the use-

fulness of including realized measures in a framework purely based on the financial

time series’ own history, i.e. the RK is a much stronger signal of volatility than squared

returns. The parameter estimates are in line with our intuition, e.g. β is close to one

indicating a high degree of persistence in the volatility dynamics and the leverage

functions show that negative information affects future volatility more than positive

information of the same magnitude.


We now include the exogenous covariates one by one and finally jointly in the

volatility models: the results are presented in Table A.3-A.6. First, we consider VIX. For

both the additive and the multiplicative version of the EGARCH model, the inclusion

of VIX leads to very large increase of approximately 60 points in the (partial) log-

likelihood value. The coefficient α is equal to zero for both EGARCH models which

shows that VIX dominates daily squared returns as a predictor of future volatility. For

both REGARCH models, VIX is also highly significant with an increase close to 100

points for the additive version and close to 200 points for the multiplicative version.


However, the partial log-likelihood values only increase marginally indicating that

the largest gains can be attributed to improved modeling of the realized measure. In

fact, the multiplicative EGARCH model now dominates the REGARCH models based

on the partial log-likelihood.

In Figure A.3, we depict the components of the multiplicative REGARCH models

for all covariates. For VIX, the story is quite clear. Since VIX is a measure of the

risk-neutral expectation of the integrated variance of the S&P 500 Index for the

next 30 calender days, its movements are less erratic than the conditional volatility.

The component g t can been seen as the forward looking measure that explains the

majority of conditional volatility, while ht , updated using the innovations to the

realized measure, helps explain the remaining part.


The second exogenous variable, EPU, is significant in the EGARCH models. Again,

with the largest improvement for the multiplicative model. The values of the coef-

ficient α are comparable in magnitude with the values in the benchmark EGARCH

models indicating that the variable not just subsumes the information in the squared

return. The picture differs for the REGARCH models. Here, the EPU seems to be most

relevant in the additive model. The positive coefficient on EPU is intuitive since it

is reasonable to believe that economic uncertainty drives financial market volatility.

From Figure A.3, it is clear that the majority of conditional volatility is explained by

ht in the multiplicative REGARCH model.

The ADS is significant for all considered models, but the log-likelihood gain is sub-

stantially larger for the additive models. As volatility is known to be countercyclical,

the sign of the coefficients is again intuitive. Furthermore, examining A.3, we see that

g t explains a small fraction of conditional volatility in the multiplicative REGARCH

model.

For all variables jointly, we observe highly significant log-likelihood improve-

ments, but EPU is generally insignificant. The value of the partial log-likelihood is

generally largest when including all exogenous variables. The log-likelihood improve-

ments are higher for the multiplicative models than the additive models primarily

driven by the inclusion of VIX.

To sum up, the inclusion of exogenous variables is found to be highly relevant for

modeling financial market volatility in-sample. This holds true also in a framework

employing realized measures in a dynamically complete fashion. One general point

to be made is that the relevance of including exogenous variables is dependent on

whether an additive or a multiplicative specification are considered.

¿ Insert Table A.3-A.6 about here À

128 CHAPTER 3.

Forecasting results

In this section, we present the results from a forecasting exercise. We reestimate the

models using data covering the period from January 2, 2002 to December 31, 2010

(2267 observations). The remaining part of the sample (754 observations) is used to

evaluate the forecasting performance of the different models.

We start by considering the four models one by one. Figure A.4 presents a compar-

ison of the forecasting performance when including exogenous variables relative to

the benchmark of no exogenous variables. For the EGARCH models, we find evidence

of statistically significant superior forecasting performance up to 10 days ahead when

including VIX or all variables by applying a Diebold-Mariano test (Diebold and Mari-

ano, 1995) using the QLIKE loss function.9 For the multiplicative REGARCH model,

we find superior one and two day ahead forecasting performance for VIX and all

variables jointly. For the additive REGARCH models, we find statistically significant

one and two day ahead outperformance when including ADS.


Now, we look at the overall picture by applying the MCS of Hansen et al. (2011) to

test for superior forecasting performance.10 Table A.7 contains mean QLIKE ratios rel-

ative to the EGARCH model and the MCS coverage based on the QLIKE loss function

for different forecasting horizons. Based on one day ahead forecasting only models

with VIX as an exogenous variable are included in the MCS. Although the mean QLIKE

is much smaller for the models including VIX for multi day ahead forecasting, it is

very hard to statically discriminate between the models.


The conclusion of this forecasting exercise is that the use of exogenous covari-

ates, in particular VIX, is beneficial for forecasting in both additive and multiplicative

EGARCH models. Although we realize short horizon gains, the benefits from including

VIX in the more sophisticated REGARCH model are dramatically smaller. A potential

explanation for this is that the REGARCH extensions with covariates are compet-

ing with benchmarks that in a dynamically complete way utilize realized measures

whereas the added covariates is assumed to follow a random walk. A possible way

to improve the multi day ahead forecasting results is to specify a more reasonable

dynamic structure for the covariates. We leave this for future research.

9The Diebold-Mariano test statistics, S = T 1/2d/p

V →d N(0,1), is based on the loss differentials, d ,defined in Subsection 3.6.1 of Section 3.6 and an HAC estimator of the long-run variance of loss differentials√

V . We note that the Diebold-Marino test formally cannot be used to compare nested models (see e.g.Clark and McCracken (2001)). Thus, the test can only be viewed as an approximation.

10We employ the code of Kevin Sheppard to calculate the MCS. The code is available at https://www.kevinsheppard.com/MFE_Toolbox.

https://www.kevinsheppard.com/MFE_Toolbox

https://www.kevinsheppard.com/MFE_Toolbox


3.7.3 Results for individual stocks

In this section, we present an empirical application to 20 stocks: AA (Alcoa), AIG

(American International Group), AXP (American Express), BA (Boing, CAT (Cater-

pillar), DD (DuPont), DIS (Walt Disney), GE (General Electric), IBM (International

Business Machines), INTC (Intel), JNJ (Johnson & Johnson), KO (Coca-Cola), MMM

(3M), MRK (Merck), MSFT (Microsoft), PG (Procter & Gamble), WHR (Whirlpool),

WMT (Walmart), and XOM (Exxon Mobil). The sample period is identical to the sam-

ple period in the previous example. Due to space considerations, we will not present

the results for each single stock but only in an aggregate form.

Figures A.5-A.8 present LR tests for the 20 stocks considered. The overall picture is

similar to the results for SPY. VIX is significant in most volatility specifications and the

gains are substantially larger in the multiplicative models. In fact, in a few cases VIX

is insignificant in the additive EGARCH and REGARCH specification. For all models

including VIX, the log-likelihood gains are smaller than for SPY. EPU is rarely or only

borderline significant in most models. ADS is significant in a large proportion of the

additive models, but rarely so in the multiplicative models. For almost all models, the

variables are jointly significant.


In Figure A.9, we depict g t for four stocks: AA, DD, GE and KO. Again, there is a

clear resemblance with g t for SPY depicted in A.3, but for the specifications including

VIX a smaller degree of the variation is now explained by the g t component. This is

naturally since we are using the implied volatility of the market and not the particular

assets. For the specifications with VIX, we can think of the g t component as capturing

the aggregate level of volatility while ht is modeling the idiosyncratic part of volatility.


For different forecast horizons, Table A.8 shows the percentage of times the dif-

ferent models is included in the 10% MCS. Again, multiplicative REGARCH models

including VIX seem to outperform at short horizons. At longer horizons, it is again

very hard to discriminate between models.


3.7.4 Discussion

The number of empirical studies on the ability of VIX, and implied volatility in general,

to forecast future volatility is vast. Some find that implied volatility dominates realized

volatility, while other in contrast find that implied volatility is a biased and inefficient

forecast of future volatility and contains little or no incremental information beyond

that in past realized volatility. The comparison between these studies is complicated

130 CHAPTER 3.

by differences in data, sampling frequency, forecasting horizon, econometric method-

ology, benchmark models, and testing procedures. A part of the literature is using

encompassing tests to compare the forecast obtained from GARCH models and other

volatility models with implied volatility. Fleming et al. (1995) find that VIX performs

well as a volatility forecast (one month ahead), but that it is not an unbiased forecast.

Also, Fleming (1998) finds that implied volatility is an upward biased forecast, but

also that it contains relevant information regarding future volatility. A linear model

using only the implied volatility appears to deliver a quality forecast of ex post volatil-

ity (one month ahead). Christensen and Prabhala (1998) find that implied volatility

works better than realized volatility on a monthly horizon. The studies of Szakmary,

Ors, Kim, and Davidson (2003), Corrado and Miller (2005), Carr and Wu (2006), Giot

and Laurent (2007), and Yu, Lui, and Wang (2010) also favor implied volatility and

some of the studies even find implied volatility or VIX to be close to unbiased. In

contrast, Becker, Clements, and White (2006) find that historical models may contain

additional information, when shorter forecasting horizons are considered. Martens

and Zein (2004) find that volatility forecasts based on historical intra day returns do

provide good volatility forecasts that can compete with and even outperform implied

volatility. The difference from previous studies is the use of an long-memory volatility

model with realized measures of volatility rather than plain vanilla GARCH models.

In GARCH type models with VIX added directly to the GARCH equation, Blair, Poon,

and Taylor (2001) find that VIX performs very well and subsumes the information in

squared returns. In an application only focusing on one step ahead volatility predic-

tions, Kambouroudis and McMillan (2016) find rather mixed results using a range of

different GARCH type specifications. In general, their forecasting models including

VIX performs marginally better. Our results confirm that the inclusion of VIX in a

plain vanilla volatility model using only squared return leads to both large in-sample

and out-of-sample gains. We also, corroborate the finding in Martens and Zein (2004)

that the use of dynamically complete volatility models employing realized measures

may alter results.

In the Macro-GARCH model Dorion (2016), ADS is significant in-sample and

found to significantly reduce option pricing errors in an out-of-sample exercise for

the S&P 500 Index. We obtain qualitatively similar in-sample results, but we do not

find out-of-sample gains. One issue with the analysis conducted in this paper is the

limited number of recession periods. The advantage of a model using macroeconomic

variables is to allow for different states in the form of expansions and recessions and

such a model is likely to only show its true potential in an out-of-sample exercise with

a change in states. In fact, Dorion (2016) also finds that the Macro-GARCH model

is particular useful in recessions. This may explain the difference in out-of-sample

performance. Therefore, the out-of-sample results of this paper should not discourage

the use of macroeconomic variables to improve GARCH type models. However, it

should be stressed that in many application with a limited span of data available, it is

3.8. CONCLUSION 131

likely unfruitful to try harvesting the explanatory power of macroeconomic variables

for volatility forecasting in GARCH type models.

EPU is the variable with the worst performance in our analysis. We find like

the existing literature, see Liu and Zhang (2015) and reference therein, that EPU

is statistically significant in-sample. However, we cannot obtain the encouraging

out-of-sample results of Liu and Zhang (2015).

3.8 Conclusion

In this paper, we present two different ways of including exogenous variables in the

Realized EGARCH framework. The first approach includes the exogenous variables

in the GARCH equation while the second approach disentangles the conditional

heteroskedasticity and the baseline volatility into two components.

The empirical application consider three exogenous variables: VIX, EPU, and ADS.

The exogenous variables seem to add information both in-sample and out-of-sample.

Especially, the multiplicative decompositions including VIX as a exogenous variable

perform very well. In line with the literature, we observe large forecasting gains for

the EGARCH model with VIX as a covariate. Superior forecasting performance is

restricted to short horizons in the REGARCH model. This may be explained by strong

performance of the benchmark REGARCH model and the likely unreasonable random

walk assumption for the exogenous variables. Proper forecasting may require explicit

modeling of the dynamic structure of the exogenous variables just as we explicitly

model the dynamics of the realized measure.

Interesting work remains to be undertaken. In the GARCH literature, we have

seen recent attempts to improve the information content from exogenous variables.

Examples include the GARCH-MIDAS model of Engle et al. (2013) and the Semi-

parametric Multiplicative GARCH-X model of Han and Kristensen (2015). Similar

approaches may be considered in a Realized EGARCH framework. Another avenue of

research is to allow for non-linear dynamics such as smooth transitions in a fashion

similar to Amado and Laakkonen (2014). The development of the asymptotic theory

and issues related to model specification, evaluation and estimation are also very

interesting research areas.

Acknowledgement

We thank Timo Teräsvirta, Asger Lunde, Peter Reinhard Hansen and Esther Ruiz Or-

tega for useful comments and suggestions. We also thank Asger Lunde for providing

cleaned high-frequency data. The authors acknowledge support from CREATES -

Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the

Danish National Research Foundation. Some of this research was carried out while

Johan S. Jakobsen was visiting the School of Economics and Finance, Queensland

132 CHAPTER 3.

University of Technology. He would like to acknowledge the generosity and hospi-

tality of the faculty at the department and in particular Stan Hurn and Annastiina

Silvennoinen.

3.9. REFERENCES 133

3.9 References

Alagidede, P., 2008. Day of the week seasonality in african stock markets. Applied

Financial Economics Letters 4 (2), 115–120.

Amado, C., Laakkonen, H., 2014. Modelling time-varying volatility in financial returns:

Evidence from the bond markets. Oxford University Press, 249–268.



Andersen, T. G., Bollerslev, T., Diebold, F. X., Ebens, H., 2001. The distribution of

realized stock return volatility. Journal of Financial Economics 61 (1), 43 – 76.


realized volatility. Econometrica 71, 579–625.



Asgharian, H., Christiansen, C., Hou, A. J., 2015. Effects of macroeconomic uncertainty

on the stock and bond markets. Finance Research Letters 13, 10–16.

Asgharian, H., Hou, A. J., Javed, F., 2013. The importance of the macroeconomic

variables in forecasting stock return variance: A GARCH-MIDAS approach. Journal

of Forecasting 32 (7), 600–612.

Baker, H. K., Rahman, A., Saadi, S., 2008. The day-of-the-week effect and conditional

volatility: Sensitivity of error distributional assumptions. Review of Financial Eco-

nomics 17 (4), 280–295.



Banulescu, G. D., Hansen, P. R., Huang, Z., Matei, M., 2014. Volatility during the finan-

cial crisis through the lens of high frequency data: A Realized EGARCH approach.



noise. Econometrica 76, 1481–1536.

Barndorff-Nielsen, O. E., Shephard, N., 2002. Econometric analysis of realized volatil-

ity and its use in estimating stochastic volatility models. Journal of the Royal Statis-

tical Society: Series B (Statistical Methodology) 64, 253–280.

Becker, R., Clements, A. E., White, S. I., 2006. On the informational efficiency of

S&P500 implied volatility. The North American Journal of Economics and Finance

17 (2), 139 – 153.

134 CHAPTER 3.

Bekaert, G., Hoerova, M., 2014. The VIX, the variance premium and stock market

volatility. Journal of Econometrics 183, 181–192.



Blair, B. J., Poon, S.-H., Taylor, S. J., 2001. Forecasting S&P 100 volatility: The incremen-

tal information content of implied volatilities and high-frequency index returns.

Journal of Econometrics 105 (1), 5 – 26.

Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity. Journal

of Econometrics 31, 207–327.

Bollerslev, T., Melvin, M., 1994. Bid-ask spreads and volatility in the foreign exchange

market: An empirical analysis. Journal of International Economics 36, 355–372.

Borup, D., Jakobsen, J. S., 2017. Volatility persistence in the Realized Exponential

GARCH model, working Paper.

Brenner, R. J., Harjes, R. H., Kroner, K. F., 1996. Another look at models of the short-

term interest rate. The Journal of Financial and Quantitative Analysis 31 (1), 85–107.

Brownless, C. T., Gallo, G. M., 2010. Comparison of volatility measures. A risk man-

agement perspective. Journal of Financial Econometrics 8, 29–56.

Carr, P., Wu, L., 2006. A tale of two indices. Journal of Derivatives 13 (3), 13–29.

Charles, A., 2010. The day-of-the-week effects on the volatility: The role of the asym-

metry. European Journal of Operational Research 202 (1), 143 –152.

Christensen, B., Prabhala, N., 1998. The relation between implied and realized volatil-

ity. Journal of Financial Economics 50 (2), 125 –150.

Christensen, K., Podolskij, M., 2007. Realized range-based estimation of integrated

variance. Journal of Econometrics 141, 323 –349.

Clark, T. E., McCracken, M., 2001. Tests of equal forecast accuracy and encompassing

for nested models. Journal of Econometrics 105 (1), 85–110.

Connolly, R. A., 1989. An examination of the robustness of the weekend effect. The

Journal of Financial and Quantitative Analysis 24 (2), 133–169.

Conrad, C., Kleen, O., 2016. On the statistical properties of multiplicative GARCH

models, working Paper.

Conrad, C., Loch, K., 2015. Anticipating long-term stock market volatility. Journal of

Applied Econometrics 30 (7), 1090–1114.

3.9. REFERENCES 135

Conrad, C., Schienle, M., 2015. Misspecification testing in GARCH-MIDAS models.

Tech. rep., University of Heidelberg, Department of Economics.

Corrado, C. J., Miller, Jr., T. W., 2005. The forecast quality of CBOE implied volatility

indexes. Journal of Futures Markets 25 (4), 339–373.

Day, T. E., Lewis, C. M., 1992. Stock market volatility and the information content of

stock index options. Journal of Econometrics 52 (1), 267 – 287.

Diebold, F. X., Mariano, R. S., 1995. Comparing predictive accuracy. Journal of Busi-

ness and Economic Statistics 13 (3), 253–263.

Dominguez, K., 1998. Central bank intervention and exchange rate volatility. Journal

of International Money and Finance 17 (1), 161–190.

Dominicy, Y., Vander Elst, H., 2015. Macro-driven VaR forecasts: From very high to

very low-frequency data, working Paper.

Dorion, C., 2016. Option valuation with macro-finance variables. Journal of Financial

and Quantitative Analysis 51 (4), 1359–1389.


425–446.

Engle, R., Ghysels, E., Sohn, B., 2013. Stock market volatility and macroeconomic

fundamentals. The Review of Economics and Statistics 95, 776–797.

Engle, R., Lee, H., 1999. Cointegration, causality, and forecasting: A festschrift in

honour of Clive W.J. Granger. Oxford University Press, Ch. A long-run and short-

run component model of stock return volatility, 475–497.

Engle, R. F., 1982. Autoregressive conditional heteroscedasticity with estimates of the

variance of United Kingdom inflation. Econometria 50, 987–1007.



Engle, R. F., Ng, V. K., December 1993. Measuring and testing the impact of news on

volatility. Journal of Finance 48 (5), 1749–78.


and its global macroeconomic causes. Review of Financial Studies 21, 1187–1222.

Fleming, J., 1998. The quality of market volatility forecasts implied by S&P 100 index

option prices. Journal of Empirical Finance 5 (4), 317 – 345.

Fleming, J., Kirby, C., Ostdiek, B., 2008. The specification of GARCH models with

stochastic covariates. Journal of Futures Markets 28 (10), 911–934.

136 CHAPTER 3.

Fleming, J., Ostdiek, B., Whaley, R. E., 1995. Predicting stock market volatility: A new

measure. Journal of Futures Markets 15 (3), 265–302.

Francq, C., Zakoïan, J.-M., 2010. GARCH models: Structure, statistical inference and

financial applications. John Wiley & Sons, Ltd, 311–340.

Gallo, G. M., Pacini, B., 2000. The effects of trading activity on market volatility. The

European Journal of Finance 6 (2), 163–175.

Gatheral, J., Oomen, R. C. A., 2010. Zero-intelligence realized variance estimation.

Finance and Stochastics 14, 249–283.

Giot, P., Laurent, S., 2007. The information content of implied volatility in light of the

jump/continuous decomposition of realized volatility. Journal of Futures Markets

27 (4), 337–359.

Girma, P. B., Mougoué, M., 2002. An empirical examination of the relation between

futures spreads volatility, volume, and open interest. Journal of Futures Markets

22 (11), 1083–1102.

Glosten, L. R., Jagannathan, R., Runkle, D. E., 1993. On the relation between the

expected value and the volatility of the nominal excess return on stocks. The

Journal of Finance 48 (5), 1779–1801.

Gray, S. F., 1996. Modeling the conditional distribution of interest rates as a regime-

switching process. Journal of Financial Economics 42 (1), 27–62.

Hagiwara, M., Herce, M. A., 1999. Endogenous exchange rate volatility, trading vol-

ume and interest rate differentials in a model of portfolio selection. Review of

International Economics 7 (2), 202–218.

Han, H., 2015. Asymptotic properties of GARCH-X processes. Journal of Financial

Econometrics 13 (1), 188–221.

Han, H., Kristensen, D., 2014. Asymptotic theory for the QMLE in GARCH-X models

with stationary and nonstationary covariates. Journal of Business & Economic

Statistics 32 (3), 416–429.

Han, H., Kristensen, D., 2015. Semparametric multiplicative GARCH-X model: Adapt-

ing economic variables to explain volatility. Tech. rep., University College London.

Han, H., Park, M. D., 2013. Comparison of realized measure and implied volatility in

forecasting volatility. Journal of Forecasting 32 (6), 522–533.


of volatility. Journal of Business and Economic Statistics 34, 269–287.

3.9. REFERENCES 137

Hansen, P. R., Huang, Z., Shek, H. H., 2012. Realized GARCH: A joint model for returns

and realized measures of volatility. Journal of Applied Econometrics 27, 877–906.

Hansen, P. R., Large, J., Lunde, A., 2008. Moving average-based estimators of integrated

variance. Econometric Reviews 27 (1-3), 79–111.




79, 453–497.


Econometrics 129, 121–138.

Hodrick, R. J., 1989. Risk, uncertainty, and exchange rates. Journal of Monetary Eco-

nomics 23 (3), 433 – 459.

Huang, Z., Liu, H., Wang, T., 2016. Modeling long memory volatility using realized

measures of volatility: A realized HAR GARCH model. Economic Modelling 52,

812–821.

Kambouroudis, D. S., McMillan, D. G., 2016. Does VIX or volume improve GARCH

volatility forecasts? Applied Economics 48 (13), 1210–1228.

Kim, T.-H., White, H., 2004. On more robust estimation of skewness and kurtosis.

Finance Research Letters 1, 56–73.

Kruse, R., Leschinski, C., Will, C., 2016. Comparing predictive ability under long

memory - with an application to volatility forecasting, CREATES Working Paper.

Lamoureux, C. G., Lastrapes, W. D., 1990. Heteroskedasticity in stock return data:

Volume versus GARCH effects. The Journal of Finance 45 (1), 221–229.

Laursen, B., Jakobsen, J. S., 2017. Realized EGARCH models with time-varying uncon-

ditional variance, working Paper.

Liu, L., Zhang, T., 2015. Economic policy uncertainty and stock market volatility.

Finance Research Letters 15, 99 – 105.


Business 36, 394–394.

Martens, M., Zein, J., 2004. Predicting financial volatility: High-frequency time-series

forecasts vis-a-vis implied volatility. Journal of Futures Markets 24 (11), 1005–1028.

Mencía, J., Sentana, E., 2013. Valuation of VIX derivatives. Journal of Financial Eco-

nomics 108 (2), 367 – 391.

138 CHAPTER 3.

Mikosch, T., Starica, C., 2004. Nonstationarities in financial time series, the long-

range dependence, and the IGARCH effects. Review of Economics and Statistics 86,

378–390.

Morana, C., 2002. IGARCH effects: An interpretation. Applied Economics Letters 9,

745–748.

Nelson, D. B., 1991. Conditional heteroskedasticity in asset returns: A new approach.

Econometrica 59 (2), 347–370.


Journal of Econometrics 160, 246 – 256.

Patton, A. J., Sheppard, K., 2009. Handbook of financial time series. Springer Berlin

Heidelberg, Berlin, Heidelberg, Ch. Evaluating volatility and correlation forecasts,

801–838.

Paye, B. S., 2012. Déja vol: Predictive regressions for aggregate stock market volatility

using macroeconomic variables. Journal of Financial Economics 106 (3), 527–546.

Sharma, P., Vipul, 2016. Forecasting stock market volatility using Realized Garch


59, 222 – 230.

Shephard, N., Sheppard, K., 2010. Realising the future: Forecasting with high-


197–231.

Shimotsu, K., 2010. Exact local whittle estimation of fractional integration with un-

known mean and time trend. Econometric Theory 26 (2), 501–540.

Szakmary, A., Ors, E., Kim, J. K., Davidson, W. N., 2003. The predictive power of

implied volatility: Evidence from 35 futures markets. Journal of Banking & Finance

27 (11), 2151 – 2175.

Teräsvirta, T., Zhao, Z., 2011. Stylized facts of return series, robust estimates and three

popular models of volatility. Applied Financal Economics 21 (1-2), 67–94.

Vander Elst, H., 2015. FloGARCH: Realizing long memory and asymmetries in returns

volatility, working paper.

Wagner, N., Marsh, T. A., 2005. Surprise volume and heteroskedasticity in equity

market returns. Quantitative Finance 5 (2), 153–168.

Whaley, R. E., 2000. The investor fear gauge. Journal of Portfolio Management 26 (3),

12–17.

3.9. REFERENCES 139

Whaley, R. E., 2009. Understanding the VIX. Journal of Portfolio Management 35 (3),

98–105.

Wintenberger, O., 2013. Continuous invertibility and stable QML estimation of the


Yu, W. W., Lui, E. C., Wang, J. W., 2010. The predictive power of the implied volatility of

options traded otc and on exchanges. Journal of Banking & Finance 34 (1), 1 – 11.

Zhang, L., 2006. Efficient estimation of stochastic volatility using noisy observations:

A multi-scale approach. Bernoulli 12, 1019–1043.

Zhang, L., Mykland, P. A., Aït-Sahalia, Y., 2005. A tale of two time scales: Determin-

ing integrated volatility with noisy high-frequency data. Journal of the American

Statistical Association 100, 1394–1411.

140 CHAPTER 3.

Appendix

A.1 Derivation of Scores


First, consider A(zt ) = ∂ logσ2t+1/∂ logσ2

t . Using zt = rt−µσt

, it can easily be shown that

∂zt

∂ logσ2t

=−1

2zt . (A.1)

For ut = log xt −φ logσ2t −δ

(zt

), we find that

∂ut

∂ logσ2t

=− ∂δ(zt

)∂ logσ2

t

−φ=−δ′bzt

∂zt

∂ logσ2t

−φ, (A.2)

Similarly, we have

∂τ(zt

)∂ logσ2

t

= τ′azt

∂zt

∂ logσ2t

= τ′azt zzt

∂zt

∂ logσ2t

, (A.3)

Inserting the above components in the following expression for A(zt ),

A(zt ) =β+ ∂τ(zt

)∂ logσ2

t

+α ∂ut

∂ logσ2t

, (A.4)

yields

A(zt ) = (β−αφ)+ 1

2


)zt . (A.5)

Next, we turn to B(zt ,ut ) = ∂`t /∂ logσ2t . The term logσ2

t enters the log-likelihood

contribution at time t directly due to the logσ2t term and indirectly through z2

t and

u2t . Thus, we have

B(zt ,ut ) =−1

2

[1+ ∂z2

t

∂ logσ2t

+ 1

σ2u

2ut∂ut

∂ logσ2t

]. (A.6)

where∂z2

t

∂ logσ2t

=−z2t . (A.7)

Combing the different expressions yields,

B(zt ,ut ) =−1

2

[(1− z2

t

)+ ut

σ2u

(δ′bzt zt −2φ

)]. (A.8)

�



For hµ,t+1 = ∂ logσ2t+1/∂µ, we have

hµ,t+1 =β∂ logσ2

t+1

∂σt

∂ logσ2t

∂µ+ ∂τ

(zt

)∂µ

+α∂ut

∂µ, (A.9)

with

∂τ(zt

)∂µ

= ∂τ(zt

)∂zt

∂zt

∂µ

= τ′ ∂a(zt

)∂zt

[−1

2zt∂ logσ2

t

logµ− 1

σt

]

= τ′azt

[−1

2zt hµ,t − 1

σt

], (A.10)

and

∂ut

∂µ= ∂ut

∂µ

=[−φ logσ2

t

∂µ− ∂δ

(zt

)∂zt

∂zt

∂µ

]

=[−φhµ,t −δ′bzt

[−1

2zt hµ,t − 1

σt

]]. (A.11)

Inserting (A.10) and (A.11) into (A.9) and manipulating yields

hµ,t+1 = A(zt )hµ,t +(αδ′bzt −τ′azt

) 1

σt.

For hθ1,t+1 = ∂ logσ2t+1/∂θ1, we have

hθ1,t+1 =β∂ logσ2

t

∂θ1+ ∂τ

(zt

)∂θ1

+α∂ut

∂θ1+

(1, logσ2

t , zt , z2t −1,ut

)′. (A.12)

However, we remember that τ(zt

)and ut only depends on θ1 through logσ2

t such

that the three first terms can be collapsed to one. Thus, we can instead write

hθ1,t+1 = A(zt )hθ1,t +mt . (A.13)

Exactly, what we wanted to show. For hθ2,t = ∂ logσ2t+1/∂θ2, we have

hθ2,t+1 =β∂ logσ2

t

∂θ2+ ∂τ

(zt

)∂θ2

+α∂ut

∂ut

(1, log

(σ2

t

), zt , z2

t −1

)′. (A.14)

Using the same arguments as before, we obtain

hθ2,t+1 = A(zt )hθ2,t +αnt . (A.15)

142 CHAPTER 3.

Finally, for hγ,t+1 = ∂ logσ2t+1/∂γ, we have

hγ,t+1 = A(zt )hγ,t +vt .

�

A.1.3 Proof of Theorem 1

The parameter µ enters the log-likelihood contribution at time t through logσ2t , zt ,

and u2t such that

∂`t

∂µ=−1

2

∂ logσ2t

∂µ− 1

2

∂z2t

∂µ− 1

2

1

σ2u

∂u2t

∂µ

= ∂`t

∂ logσ2t

∂ logσ2t

∂µ−

[zt −δ′ ut

σ2u

∂b(zt

)∂zt

]1

σt

= B(zt ,ut )hµ,t −[

zt +δ′ ut

σ2u

bzt

]1

σt.

As θ1,θ2, θ3, and γ only enter the log-likelihood contribution at time t indirectly

through logσ2t , an application of the chain-rule yields

∂`t

∂θ1= B(zt ,ut )hθ1,t , (A.16)

∂`t

∂θ3= B(zt ,ut )hθ3,t , (A.17)

∂`t

∂γ= B(zt ,ut )hγ,t , (A.18)

and as(ξ,φ,δ′

)′ enters through logσ2t and u2

t , an application of the chain-rule yields

∂`t


ut

σ2u

nt . (A.19)

σ2u only enters directly into the log-likelihood contribution such that we obtain

∂`t

∂σ2u= 1

2

u2t −σ2

u

σ4u

. (A.20)

�

A.1.4 Proof of Lemma 3, Lemma 4, and Theorem 2

The proofs are almost identical to the additive case, so we do not present the deriva-

tions. See Lemma 1 and 2, and Theorem 1.

A.2. FIGURES 143

A.2 Figures

2002 2005 2008 2010 2013-0.15

-0.1

-0.05

0

0.05

0.1

0.15

2002 2005 2008 2010 20130

1

2

3

4

5

6

2002 2005 2008 2010 20130

0.5

1

1.5

2

50 100 150 2000

0.2

0.4

0.6

0.8

1

Figure A.1: Data plotsThe upper-left panel: SPY daily (close-to-close) return. The lower-left panel: SPY daily RK(annualized). The upper-right panel: SPY daily squared (close-to-close) returns (annualized).The lower-right panel: SPY daily log(RK) autocorrelation function.

144 CHAPTER 3.

2002 2005 2008 2010 20130

0.5

1

-12 -10 -8 -6 -4

-12

-10

-8

-6

-4

2002 2005 2008 2010 2013-4

-2

0

2

4

-4 -2 0 2

-12

-10

-8

-6

-4

2002 2005 2008 2010 2013

-4

-2

0

2

-4 -2 0 2

-12

-10

-8

-6

-4

Figure A.2: Plots of exogenous variablesThe upper-left panel: VIX. The middle-left panel: log(EPU/100). The lower-left panel: ADS.

The upper-right panel: logRKt plotted against log(VIX2

t−1/250). The middle-left panel: logRKt

plotted against log(

EPUt−1100

). The lower-right panel: logRKt plotted against ADSt−1.

A.2. FIGURES 145

2002 2005 2008 2010 20130

5

10

15

20

25

30

35

40

45

50

2002 2005 2008 2010 20130

5

10

15

20

25

30

35

40

2002 2005 2008 2010 20130

5

10

15

20

25

30

35

40

45

2002 2005 2008 2010 20130

5

10

15

20

25

30

35

40

45

50

Figure A.3: The components of the multiplicative REGARCH modelIllustrates ht (red), gt (blue) and σ2

t of the multiplicative REGARCH models with exogenouscovariates. The upper-left panel: VIX. The upper-right panel: ADS. The lower-left panel: EPU.The lower-right panel: all exogenous variables.

146 CHAPTER 3.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.70

0.80

0.90

1.00

1.10

1.20

1.30

Additive EGARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.70

0.80

0.90

1.00

1.10

1.20

1.30

Multiplicative EGARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.70

0.80

0.90

1.00

1.10

1.20

1.30

Additive REGARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200.70

0.80

0.90

1.00

1.10

1.20

1.30

Multiplicative REGARCH

Figure A.4: Forecast evaluation with Diebold-Mariano testOut-of-sample ratio of mean QLIKE for the four presented models with different exogenousvariables relative to their benchmark case without exogenous variables. For each model we

consider log(VIX2

t−1/250)

(blue), log(EPU/100) (yellow), ADSt−1 (red), and all (purple). A

filled marker indicates rejection of equal forecast at a 10% significance level using the Diebold-Mariano test (one-sided alternative).

A.2. FIGURES 147

AA AIG AXP BA CAT DD DIS GE IBM INTC JNJ KO MMM MRK MSFT PG VZ WHR WMT XOM0

50

100

150

200

250

Additve EGARCH


Additive REGARCH


Figure A.5: LR-test statistics for including VIXLR statistics for including VIX. The dashed line indicates the 5% critical value.


50

100

150

200

250

Additve EGARCH


Additive REGARCH


Figure A.6: LR-test statistics for including EPULR statistics for including EPU. The dashed line indicates the 5% critical value.

148 CHAPTER 3.


50

100

150

200

250

Additve EGARCH


Additive REGARCH


Figure A.7: LR-test statistics for including ADSLR statistics for including ADS. The dashed line indicates the 5% critical value.


50

100

150

200

250

Additve EGARCH


Additive REGARCH


Figure A.8: LR-test statistics for including exogenous variablesLR statistics for including all exogenous variables. The dashed line indicates the 5% criticalvalue.

A.2. FIGURES 149

2002 2005 2008 2010 20130

50

100

150

Cond. Vol. BenchmarkVIXADSEPUVIX, ADS, EPU

2002 2005 2008 2010 20130

10

20

30

40

50

60

70


2002 2005 2008 2010 20130

20

40

60

80

100

120

140


2002 2005 2008 2010 20130

5

10

15

20

25

30

35


Figure A.9: Exogenous components of the multiplicative REGARCH modelDepicts gt from the extension with exogenous covariates and σ2

t from the multiplicativeREGARCH model. The upper-left panel: AA. The upper-right panel: DD. The lower-left panel:GE. The lower-right panel: KO.

150 CHAPTER 3.

A.3 Tables

Table A.1: Descriptive statistics for daily returns and RKThis table reports descriptive statistics for the daily returns and the realized kernel. Dailyreturns and the RK are in percentages. Robust skewness and kurtosis are from Kim and White(2004). The fractional integrated parameter d is estimated using the two-step exact localWhittle estimator of Shimotsu (2010) and bandwith of m = [T 0.65].

Return Log(RK)

No. ofobs.

MeanStd.Dev.

SkewRobustSkew

Ex. Kurt.Robust

Ex. Kurt.Median Mean

Std.Dev.

Median d

SP500 3020 0.02 1.32 0.07 -0.08 11.67 1.03 0.08 -0.35 1.00 -0.50 0.66

AA 3004 0.00 2.73 0.23 0.02 8.92 0.99 0.00 1.13 0.86 0.98 0.64AIG 2999 -0.00 4.55 1.42 0.01 54.58 1.17 -0.03 1.07 1.30 0.88 0.64AXP 2994 0.07 2.44 0.55 0.04 11.12 1.07 0.02 0.60 1.18 0.38 0.70BA 2996 0.07 1.89 0.23 0.01 4.07 0.84 0.07 0.60 0.82 0.47 0.64CAT 2998 0.07 2.09 0.11 0.03 5.06 0.92 0.06 0.73 0.82 0.59 0.67DD 2995 0.04 1.78 -0.04 0.01 5.68 0.88 0.04 0.51 0.85 0.38 0.63DIS 2997 0.07 1.91 0.51 -0.02 6.76 0.88 0.06 0.55 0.88 0.38 0.66GE 3005 0.02 1.99 0.38 0.03 10.30 1.06 0.00 0.40 1.05 0.22 0.68IBM 2996 0.03 1.53 0.20 0.01 6.87 0.87 0.02 0.10 0.83 -0.05 0.65INTC 3016 0.03 2.20 -0.22 -0.00 6.09 0.90 0.04 0.85 0.80 0.74 0.63JNJ 2997 0.03 1.16 -0.28 0.03 20.06 0.95 0.02 -0.28 0.86 -0.43 0.68KO 2999 0.04 1.24 0.32 -0.02 11.96 0.92 0.04 -0.10 0.81 -0.22 0.63MMM 2992 0.05 1.45 -0.06 0.02 5.54 0.97 0.06 0.13 0.80 0.03 0.64MRK 2994 0.03 1.80 -1.21 0.04 24.18 0.87 0.03 0.38 0.85 0.26 0.61MSFT 3016 0.03 1.81 0.37 0.02 8.94 0.96 0.00 0.46 0.81 0.32 0.63PG 2998 0.04 1.14 -0.02 0.01 6.74 0.92 0.03 -0.18 0.77 -0.30 0.61VZ 2995 0.03 1.57 0.34 -0.03 7.37 0.90 0.05 0.31 0.89 0.14 0.67WHR 2992 0.07 2.52 0.40 0.03 5.14 0.96 -0.01 1.01 0.86 0.92 0.58WMT 3001 0.03 1.31 0.30 -0.03 5.57 0.88 0.02 0.05 0.80 -0.09 0.65XOM 3001 0.05 1.60 0.34 -0.01 13.37 0.81 0.07 0.24 0.86 0.14 0.68

Table A.2: Estimation results for S&P 500 Index for benchmark modelsMaximum likelihood parameter estimates for the four different models. Standard errors aregiven in parenthesis. p denotes the total number of parameters in the given model.

EGARCH EGARCH Realized EGARCH Realized EGARCH(additive) (multiplicative) (additive) (multiplicative)

µ 0.021(0.0144) 0.021(0.0144) 0.013(0.0127) 0.013(0.0127)ω 0.001(0.0030) 0.059(0.1521) −0.002(0.0035) −0.069(0.1237)β 0.980(0.0034) 0.980(0.0034) 0.972(0.0036) 0.972(0.0037)α 0.111(0.0163) 0.111(0.0163) 0.319(0.0199) 0.319(0.0199)ξ −0.281(0.0328) −0.281(0.0328)σ2

u 0.154(0.0055) 0.154(0.0055)τ1 −0.140(0.0119) −0.140(0.0119) −0.146(0.0076) −0.146(0.0076)τ2 0.029(0.0047) 0.029(0.0047)δ1 −0.115(0.0084) −0.115(0.0084)δ2 0.044(0.0052) 0.044(0.0052)φ 1.004(0.0330) 1.004(0.0330)

p 5 5 11 11logL -4,242.04 -4,242.04 -5,640.07 -5,640.07logLp -4,242.04 -4,242.04 -4,181.19 -4,181.19

A.3. TABLES 151

Table A.3: Estimation results for S&P 500 Index with VIXMaximum likelihood parameter estimates for the four different models when including

log(VIX2

t−1/250)

as an exogenous variable. Standard errors are given in parenthesis. p de-

notes the total number of parameters in the given model.


µ 0.008(0.0133) 0.009(0.0107) 0.012(0.0147) 0.011(0.0091)ω −0.069(0.0061) −0.488(0.0000) −0.091(0.0098) 9.149(6.9455)β 0.860(0.0057) 0.923(0.0104) 0.814(0.0047) 0.894(0.0221)α 0.005(0.0244) 0.000(0.0000) 0.312(0.0194) 0.242(0.0022)ξ −0.280(0.0329) −0.282(2.2690)σ2

u 0.145(0.0054) 0.140(0.1373)τ1 −0.237(0.0181) −0.135(0.0125) −0.150(0.0084) −0.068(0.0379)τ2 0.028(0.0060) 0.022(0.0578)δ1 −0.121(0.0083) −0.120(0.3865)δ2 0.041(0.0048) 0.042(0.7918)φ 0.994(0.0199) 0.978(0.0247)γV I X 0.149(0.0056) 1.030(0.0000) 0.195(0.0057) 1.047(0.2693)

p 6 6 12 12logL -4,183.71 -4,175.25 -5,547.82 -5,497.26logLp -4,183.71 -4,175.25 -4,177.91 -4,179.33LR 116.66∗∗∗ 133.58∗∗∗ 184.49∗∗∗ 285.61∗∗∗

152 CHAPTER 3.

Table A.4: Estimation results for S&P 500 Index with EPUMaximum likelihood parameter estimates for the four different models when including

log EPUt−1100 as an exogenous variable. Standard errors are given in parenthesis. p denotes the

total number of parameters in the given model.


µ 0.018(0.0141) 0.019(0.0148) 0.015(0.0132) 0.013(0.0126)ω 0.003(0.0032) 0.080(0.1415) −0.000(0.0038) −0.068(0.1219)β 0.974(0.0036) 0.978(0.0034) 0.964(0.0037) 0.972(0.0037)α 0.106(0.0150) 0.112(0.0159) 0.319(0.0198) 0.318(0.0198)ξ −0.281(0.0329) −0.280(0.0321)σ2

u 0.153(0.0055) 0.154(0.0055)τ1 −0.152(0.0121) −0.139(0.0122) −0.149(0.0077) −0.146(0.0075)τ2 0.029(0.0047) 0.030(0.0046)δ1 −0.116(0.0085) −0.115(0.0084)δ2 0.044(0.0051) 0.044(0.0050)φ 1.004(0.0323) 0.999(0.0319)γEPU 0.015(0.0051) 0.210(0.0762) 0.022(0.0054) 0.037(0.0172)

p 6 6 12 12logL -4,236.56 -4,233.45 -5,633.07 -5,636.90logLp -4,236.56 -4,233.45 -4,180.50 -4,178.28LR 10.95∗∗∗ 17.17∗∗∗ 13.99∗∗∗ 6.34∗∗

A.3. TABLES 153

Table A.5: Estimation results for S&P 500 Index with ADSMaximum likelihood parameter estimates for the four different models when including ADSt−1as an exogenous variable. Standard errors are given in parenthesis. p denotes the total numberof parameters in the given model.


µ 0.019(0.0142) 0.016(0.0113) 0.010(0.0135) 0.009(0.0134)ω −0.005(0.0038) −0.033(0.1191) −0.010(0.0045) −0.164(0.1132)β 0.970(0.0033) 0.973(0.0044) 0.959(0.0036) 0.962(0.0044)α 0.109(0.0171) 0.116(0.0170) 0.316(0.0199) 0.325(0.0194)ξ −0.281(0.0324) −0.281(0.0326)σ2

u 0.152(0.0055) 0.153(0.0055)τ1 −0.152(0.0125) −0.147(0.0126) −0.147(0.0077) −0.148(0.0078)τ2 0.030(0.0048) 0.031(0.0049)δ1 −0.115(0.0084) −0.116(0.0084)δ2 0.044(0.0052) 0.044(0.0052)φ 1.001(0.0309) 1.004(0.0317)γADS −0.017(0.0029) −0.281(0.0981) −0.023(0.0039) −0.311(0.0839)

p 6 6 12 12logL -4,228.36 -4,238.29 -5,624.99 -5,635.31logLp -4,228.36 -4,238.29 -4,180.65 -4,182.06LR 27.35∗∗∗ 7.50∗∗∗ 30.15∗∗∗ 9.51∗∗∗

154 CHAPTER 3.

Table A.6: Estimation Results for S&P 500 Index with all exogenous variablesMaximum likelihood parameter estimates for the four different models when including

log(VIX2

t−1/250),(log EPUt−1

100

)2, and ADSt−1 as exogenous variables. Standard errors are given

in parenthesis. p denotes the total number of parameters in the given model.


µ 0.009(0.0140) 0.009(0.0138) 0.011(0.0147) 0.012(0.0145)ω −0.074(0.0091) −0.480(0.0644) −0.100(0.0104) −0.507(0.0522)β 0.860(0.0055) 0.918(0.0104) 0.805(0.0047) 0.883(0.0129)α 0.003(0.0227) 0.000(0.0162) 0.306(0.0199) 0.243(0.0186)ξ −0.280(0.0326) −0.281(0.0309)σ2

u 0.144(0.0054) 0.139(0.0053)τ1 −0.232(0.0183) −0.141(0.0154) −0.149(0.0084) −0.075(0.0080)τ2 0.029(0.0061) 0.023(0.0048)δ1 −0.121(0.0083) −0.120(0.0081)δ2 0.041(0.0048) 0.042(0.0045)φ 0.992(0.0192) 0.975(0.0196)γV I X 0.141(0.0104) 0.921(0.0742) 0.193(0.0077) 0.973(0.0389)γEPU −0.008(0.0112) 0.095(0.0623) −0.006(0.0081) 0.022(0.0165)γADS −0.020(0.0079) −0.124(0.0481) −0.025(0.0060) −0.134(0.0350)

p 8 8 14 14logL -4,177.35 -4,167.78 -5,537.53 -5,487.50logLp -4,177.35 -4,167.78 -4,177.50 -4,177.10LR 129.38∗∗∗ 148.52∗∗∗ 205.07∗∗∗ 305.14∗∗∗

A.3. TABLES 155

Table A.7: MCS coverage for S&P 500 Index (QLIKE)This table reports the ratio of out-of-sample QLIKE relative to the EGARCH model and theMCS coverage.

Specification of volatility dynamicsEGARCH EGARCH REGARCH REGARCH(additive) (multiplicative) (additive) (Multiplicative)

1 day aheadBenchmark 1.00 1.00 0.70 0.70VIX 0.82 0.79 0.70 0.63EPU 1.12 1.04 0.70 0.70ADS 1.04 0.99 0.70 0.70VIX, EPU, ADS 0.83 0.82 0.70 0.63





MCS Coverage 99% 95% 90%

156 CHAPTER 3.

Table A.8: Out-of-sample evaluation for individual stocksThis table contains the percentage of times each model is included in the MCS at a 10% levelusing the QLIKE as the loss function, when looking at the 20 individual stocks.

Specification of volatility dynamicsEGARCH EGARCH REGARCH REGARCH

(additive) (multiplicative) (additive) (multiplicative)

1 day aheadBenchmark 0.00% 0.00% 10.00% 5.00%VIX 0.00% 5.00% 15.00% 75.00%EPU 0.00% 0.00% 10.00% 10.00%ADS 0.00% 0.00% 5.00% 5.00%VIX, EPU, ADS 5.00% 0.00% 25.00% 85.00%





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ISBN: 9788793195684

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Modeling Financial Market Volatility: A Component Model ... · GARCH is a state-of-the-art volatility model that uses realized measures of volatility for predicting daily latent volatility