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Time-Varying Loadings in Factor Models:

Theory and Applications

2016-22

Jakob Guldbæk Mikkelsen

PhD Dissertation

DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS

AARHUS BSS AARHUS UNIVERSITY DENMARK

TIME-VARYING LOADINGS IN FACTOR MODELS:THEORY AND APPLICATIONS


A PhD dissertation 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

August 2016

CREATESCenter for Research in Econometric Analysis of Time Series

This version: November 20, 2016 © Jakob Guldbæk Mikkelsen

PREFACE

This dissertation represents the tangible results of my PhD studies at the Department

of Economics and Business Economics at Aarhus University from September 2013

to August 2016. I am grateful to the department and to the Center for Research

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

Research Foundation for providing an excellent research environment and funding

for numerous courses and conferences. Furthermore, financial support from Aage og

Ylva Nimbs Fond, Augustinus Fonden, Oticon Fonden, and Knud Højgaards Fond is

gratefully acknowledged.

Several people deserve my gratitude for their contributions to this dissertation.

First and foremost, I would like to thank my supervisor Eric Hillebrand for our many

fruitful discussions, his encouragement and many helpful comments and guidance. I

have enjoyed working together on our projects, and I am grateful for his support. I

would also like to thank Solveig N. Sørensen for always being available for help, for

proof-reading the dissertation chapters, and for being the glue that keeps everything

together at CREATES.

In the fall of 2015, I had the pleasure of visiting Professor Giovanni Urga at Cass

Business School, City University of London to work on a joint research project. His

many comments and suggestions have been very helpful, and I appreciate all his

advice. I have enjoyed working together with him, both during and after my visit at

Cass Business School. I am grateful to both him and the school for the hospitality

during my stay.

During my studies, I have had the privilege to be surrounded by many great

colleagues and friends. I am grateful to all of them for providing a fantastic academic

and social environment. In particular, I would like to thank all my fellow PhD students.

First and foremost, thanks to Carsten for being a great office mate. Special thanks also

go to Alexander, Anders, Bo, Heidi, Johan, Jonas, Kasper, Mikkel, Niels, Silvana, and

Simon. You have all made it enjoyable to come to work each day. Our countless coffee

breaks and social activities have truly made my studies a memorable experience.

Finally, I would also like to thank my friends and family for their support and patience

i

ii

they have shown me over the last three years.


Aarhus, August 2016

UPDATED PREFACE

The pre-defence meeting was held on October 11, 2016, in Aarhus. I am grateful to

the members of the assessment committee consisting of Professor Catherine Doz,

Senior Professor Marco Lippi, and Professor Martin Møller Andreasen for their careful

reading of the dissertation and their many insightful comments and suggestions.

Some of the suggestions have been incorporated into the present version of the

dissertation while others remain for future research.


Copenhagen, November 2015

iii

CONTENTS

Summary vii

Danish summary xi

1 Maximum Likelihood Estimation of Time-Varying Loadings in High-DimensionalFactor Models 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Model and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Asymptotic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Monte carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 An empirical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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

2 Testing for Time-Varying Loadings in Factor Models 432.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2 Testing for time-varying loadings . . . . . . . . . . . . . . . . . . . . . 45

2.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4 Small sample properties . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3 Foreign Exchange Rates and Macroeconomic Factors: Evidence from Time-Varying Loadings 873.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.2 Model and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

v

vi CONTENTS

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

SUMMARY

Factor models have become an important tool for analysing and forecasting large

datasets. Factors models are useful in summarizing the information content in large

datasets and have proven to be useful for both prediction and structural analysis.

A few factors are often found to explain a large fraction of the variance in macroe-

conomic and financial series. When the data spans over long time horizons, often

several decades, the factor loadings are likely to exhibit some instability. This disser-

tation comprises three chapters that concern time-varying factor loadings in factor

models. The first chapter provides a maximum likelihood estimator of time-varying

loadings in factor models. The second chapter develops a test for time-varying factor

loadings. The third chapter examines the relationship between changes in exchange

rates and macroeconomic fundamentals through a factor model with time-varying

loadings.

The first chapter "Maximum likelihood estimation of time-varying loadings in

high-dimensional factor models" (joint work with Eric Hillebrand and Giovanni

Urga) develops a maximum likelihood estimator for time-varying factor loadings.

The loadings evolve as stationary autoregressions, and we show that the parameters

of the loadings can be consistently estimated by a two-step estimation procedure.

In the first step, the common factors are estimated by the principal components

estimator where the loadings are treated as constants. Bates, Plagborg-Møller, Stock,

and Watson (2013) show that the principal components estimator gives consistent

factor estimates when there is temporal instability in the loadings, and our result

builds on their work. In the second step, we estimate the parameters of the loadings

by a set of regression models with time-varying coefficients. The estimated factors act

as the regressors, and the loadings are the time-varying coefficients. Our result shows

that the estimation error from the factor estimates does not affect the likelihood

function, and the factors can therefore be treated as observed regressors for the

purpose of estimating the parameters of the time-varying factor loadings. The finite

sample properties of the estimator are investigated through an extensive simulation

study.

The second chapter "Testing for time-varying loadings in factor models" develops

a test for the presence of time-varying factor loadings. The test can detect stationary

variations in factor loadings. The test is based on the observation that the squared

vii

viii SUMMARY

residuals and squared factors in a factor model are correlated when there is temporal

variation in the loadings. The principal components estimator is used to obtain esti-

mates of the factors and residuals. The test is constructed as the squared correlation

times the sample size from a regression of the squared residuals on the squared

factors. Under the null hypothesis of constant factor loadings, the test statistic has a

limiting χ2 distribution with degrees of freedom equal to the number of factors. The

test procedure is applied to two datasets. The first application uses a large macroe-

conomic dataset for the US economy. There is found to be evidence of time-varying

factor loadings in over half of the series, irrespective of the number of factors that

are included in the analysis. In the second application, we consider return portfolios

sorted on size and book-to-market. The results show that over 80% of the portfolios

are associated with time-varying factor loadings, which indicate that the portfolios

have time-varying exposures to the risk embedded in the factors.

The third chapter "Foreign exchange rates and macroeconomic factors: evidence

from time-varying loadings" (joint work with Eric Hillebrand and Giovanni Urga)

examines the relationship between changes in exchange rates and macroeconomic

fundamentals. We specify a model for the exchange rate using the class of models

of Engel and West (2005). The changes in the exchange rate are determined by a

linear combination of macroeconomic fundamentals. The parameters in the linear

combination are subject to unobservable shocks that affect the weights on the funda-

mentals. This leads to a time-varying relationship between variations in exchange

rates and fundamentals. We extract a set of macroeconomic factors from a large

macroeconomic dataset to represent the information available in the fundamentals.

The factors can be interpreted as indicators of real activity, inflation, and the housing

market. The relationship between the exchange rate and the factors is estimated as

time-varying factor loadings. The empirical results show that the weights on the fac-

tors are highly unstable and vary considerably over the sample period. In particular,

the factor loadings on the real activity indicator exhibit large increases during the

financial crisis for all exchange rates in the analysis. The weights on the inflation and

housing indicators are also highly unstable and add substantial explanatory power

to the model. The model explains a large fraction of the variation in exchange rates

and the directional changes. We benchmark our results against a factor model with

constant factor loadings. The model with time-varying factor loadings is found to be

superior in explaining the variations and directional changes in exchange rates, and

the results clearly show that the relationship between changes in exchange rates and

fundamentals is unstable.

ix

References

Bates, B., Plagborg-Møller, M., Stock, J., Watson, M., 2013. Consistent factor estima-

tion in dynamic factor models with structural instability. Journal of Econometrics

177, 289–304.

Engel, C., West, K. D., 2005. Exchange rates and fundamentals. Journal of Political

Economy 113 (3).

DANISH SUMMARY

Faktormodeller er et vigtigt redskab til at analysere og fremskrive store datasæt. Fak-

tormodeller kan anvendes til at beskrive de fælles variationer i store datasæt ved

hjælp af et lille antal faktorer. På denne måde kan informationen i store dataset anven-

des til at analysere og fremskrive økonomiske variable ved hjælp af få faktorer fremfor

de flere hundrede variable, som er til rådighed. Man finder ofte, at en stor del af varia-

tionen i økonomiske variable kan beskrives med bare et få antal faktorer. Økonomiske

analyser beskæftiger sig ofte med datasæt, der strækker sig over lange perioder, ofte

flere årtier. Når dette er tilfældet vil faktorernes påvirkning på de enkelte variable, de

såkaldte faktorloadings, ofte varierer over tid. Denne afhandling beskæftiger sig med

faktormodeller, hvor de enkelte faktorloadings er tidsvarierende. Det første kapitel

omhandler en metode til at estimere parametrene for disse tidsvarierende loadings

i faktormodeller. I det andet kapitel udvikles en metode til at teste, hvorvidt de en-

kelte faktorloadings er tidsvarierende eller konstante. I det tredje kapitel analyseres

sammenhængen mellem ændringer i valutakurser og makroøkonomiske faktorer ved

hjælp af en faktormodel med tidsvarierende loadings.

Det første kapitel "Maximum likelihood estimation of time-varying loadings

in high-dimensional factor models” (skrevet i samarbejde med Eric Hillebrand og

Giovanni Urga) handler om estimation af tidsvarierende loadings i faktormodeller. I

modellen følger de enkelte loadings stationære autoregressioner, og vi viser at parame-

trene for disse kan estimeres konsistent ved hjælp af maximum likelihood-estimation.

Estimationsproceduren består af to dele. Først estimeres de latente faktorer med

principal components-estimatoren. Denne del bygger på resultatet af Bates et al.

(2013), som viser, at denne estimator giver konsistente faktorestimater, når de tilhø-

rende faktorloadings er ustabile over tid. I den anden del af estimationsproceduren

specificerer vi regressionsmodeller med tidsvarierende koefficienter. I disse regres-

sionsmodeller fungerer de estimerede faktorer som regressorer, og de tidsvarierende

koefficienter fungerer som de tidsvarierende faktorloadings. Vores resultat viser, at

likelihoodfunktionen ikke påvirkes af estimationsfejlen, der følger fra estimationen

af de latente faktorer. Faktorerne kan således betragtes som observerbare regressorer,

når faktorloadingsparametrene skal estimeres. Estimatorens egenskaber i endelige

stikprøvestørrelser undersøges i et omfattende simulationsstudie.

I det andet kapitel "Testing for time-varying loadings in factor models” udvikles

xi

xii DANISH SUMMARY

en test for tidsvarierende faktorloadings. Testen kan opfange stationære variationer i

disse loadings. Testen konstrueres ud fra de estimerede faktorer og residualer, som

estimeres med principal components-estimatoren. Hvis de tilhørende faktorloadings

varierer over tid, vil de kvadrerede residualer og faktorer være korrelerede. Hypote-

sen om konstante faktorloadings kan således testes ved at regressere de kvadrerede

residualer på de kvadrerede faktorer. Den kvadrerede korrelationskoefficient ska-

leret med stikprøvestørrelsen fra denne regression vil følge en χ2-fordeling under

nulhypotesen. Testen anvendes på to forskellige datasæt. Det første datasæt består

af makroøkonomiske variable for den amerikanske økonomi. Der findes evidens

for tidsvarierende faktorloadings for over halvdelen af variablene uanset antallet af

faktorer, der medtages i analysen. Det andet datasæt, hvorpå testen anvendes, består

af afkast på aktieporteføljer. Testen viser, at over 80% af porteføljerne har tidsvarie-

rende faktorloadings. Dette indikerer, at disse porteføljers eksponering overfor de

underliggende risikofaktorer varierer over tid.

Det tredje kapitel "Foreign exchange rates and macroeconomic factors: evidence

from time-varying loadings"(skrevet i samarbejde med Eric Hillebrand og Giovanni

Urga) omhandler sammenhængen mellem valutakurser og makroøkonomiske fakto-

rer. Sammenhængen analyseres ved brug af en model, der tager sin inspiration fra

klassen af modeller af Engel og West (2005). Ændringerne i valutakursen bestemmes

af en lineær kombination af makroøkonomiske faktorer. Vægtene i denne lineære

kombination påvirkes af uobserverbare chok, således at vægtene kan variere over

tid. Dette medfører, at sammenhængen mellem valutakursen og de makroøkono-

miske faktorer er ustabil. I den empiriske analyse anvender vi faktorer fra et stort

makroøkonomisk datasæt. Disse faktorer kan tolkes som indeks for henholdsvis realø-

konomien, samt inflation og boligmarkedet. Sammenhængen mellem valutakursen

og disse faktorer estimeres ved hjælp af tidsvarierende faktorloadings. Vores resul-

tater viser, at vægtene på de makroøkonomiske faktorer er ustabile og udviser stor

variation over tid for alle valutakurser, som vi betragter i analysen. I særdeleshed

stiger vægtene på indekset for realøkonomien under den finansielle krise. Vægtene

på inflations- og boligmarkedsindikatorerne udviser også stor variabilitet og tilføjer

betragtelig forklaringsgrad til modellen. Modellen forklarer en stor del af variation

og retningsændringerne i valutakurserne. Vi sammenholder vores resultater med en

model, hvor vægtene på faktorerne ikke varierer over tid. I denne model forklarer

faktorerne kun en lille del af variationen i valutakurserne, hvorimod modellen med

tidsvarierende vægte forklarer en langt større del af variationerne og retningsændrin-

gerne. Vi finder således stærk evidens for, at sammenhængen mellem valutakurser

og makroøkonomiske faktorer er ustabil.

xiii

Litteratur



177, 289–304.


Economy 113 (3).

C H A P T E R 1MAXIMUM LIKELIHOOD ESTIMATION OF

TIME-VARYING LOADINGS IN

HIGH-DIMENSIONAL FACTOR MODELS

Jakob Guldbæk MikkelsenAarhus University and CREATES

Eric HillebrandAarhus University and CREATES

Giovanni UrgaCass Business School

Abstract

In this paper, we develop a maximum likelihood estimator of time-varying loadings

in high-dimensional factor models. We specify the loadings to evolve as stationary

vector autoregressions (VAR) and show that consistent estimates of the loadings

parameters can be obtained by a two-step maximum likelihood estimation procedure.

In the first step, principal components are extracted from the data to form factor

estimates. In the second step, the parameters of the loadings VARs are estimated as a

set of univariate regression models with time-varying coefficients. We document the

finite-sample properties of the maximum likelihood estimator through an extensive

simulation study and illustrate the empirical relevance of the time-varying loadings

structure using a large quarterly dataset for the US economy.

1

2

CHAPTER 1. MAXIMUM LIKELIHOOD ESTIMATION OF TIME-VARYING LOADINGS IN


1.1 Introduction

In this paper, we develop a consistent maximum likelihood estimator of time-varying

loadings in high-dimensional factor models where factors are estimated with princi-

pal components.

The problem of time-varying loadings in factor models is important because the

assumption of constant loadings has been found to be implausible in a number of

studies considering structural instability in factor models. In a large macroeconomic

dataset for the U.S., Stock and Watson (2009) find considerable instability in factor

loadings around 1984, and they improve factor-based forecast regressions of individ-

ual variables by allowing factor coefficients to change after the break point. Breitung

and Eickmeier (2011) develop Chow-type tests for structural breaks in factor load-

ings and find similar evidence of structural instability around 1984. They also find

evidence of structural breaks in the Euro area around 1992 and 1999. Del Negro and

Otrok (2008), Liu, Mumtaz, and Theophilopoulou (2011), and Eickmeier, Lemke, and

Marcellino (2015) estimate factor models in which the factor loadings are modelled

as random walks using large panels of data, but theoretical results for models with

time-varying parameters in a high-dimensional setting are scant.

The econometric theory on factor models explicitly addresses the high dimen-

sionality of these datasets by developing results in a large N and large T framework.

The central results in the literature on consistent estimation of the factor space by

principal components as N ,T →∞ have been developed in Stock and Watson (1998,

2002), and Bai and Ng (2002). Forni, Hallin, Lippi, and Reichlin (2000) consider esti-

mation in the frequency domain. Principal components have the advantage of being

easy to compute and feasible even when the cross-sectional dimension N is larger

than the sample size T . Bates et al. (2013) characterize the types and magnitudes

of structural instability in factor loadings under which the principal components

estimator of the factor space is consistent. Another strand of literature is concerned

with estimation by maximum likelihood. Bai and Li (2012, 2016) consider maximum

likelihood estimation of factor loadings and idiosyncratic variances, while Doz, Gian-

none, and Reichlin (2012) study functions of maximum likelihood estimators, also in

a large N ,T setting. Their analyses applies to factor models with constant loadings.

We consider a factor model of the form Xi t = λ′i t Ft + ei t for i = 1, ...N and t =

1, ...,T , where the data Xi t depend on a small number r ¿ N of unobserved common

factors Ft . The r ×1 vector of factor loadings λi t evolves over time. We model λi t for

each i as a stationary vector autoregression, and our main contribution is to show

that the parameters of these time-varying loadings can be consistently estimated by

maximum likelihood. Our estimation procedure consists of two steps. In the first step,

the common factors are estimated by principal components, and in the second step

we estimate the loadings parameters by maximum likelihood, treating the principal

components as observed data.

The principal components estimator is robust to stationary variations in the load-

1.2. MODEL AND ESTIMATION 3

ings. By averaging over the cross-section, the temporal instabilities in the loadings

are smoothed out and the factor space is consistently estimated. Average consistency

in t of the factor space is shown by Bates et al. (2013), and we extend the result to

uniform consistency in t to analyse the maximum likelihood estimator.

In the second step, we estimate a panel of regression models with time-varying

coefficients where the principal components are treated as the observed regressors,

and the loadings are the time-varying coefficients. We allow for heteroskedasticity

and serial correlation in the idiosyncratic errors ei t , but restrict our attention to

cross-sectionally uncorrelated errors to avoid proliferation of parameters. Condi-

tional on the factor estimates, the variables in the panel of regressions are therefore

uncorrelated, and the loadings parameters can be estimated as a set of N univariate

regression models with time-varying coefficients. Under the condition that TN 2 → 0,

the maximum likelihood estimator of the time-varying loadings is consistent as

N ,T →∞, and estimation error from the principal components can be ignored.

We point out that the computation of the maximum likelihood estimator is rel-

atively simple. Principal components are simple to compute, and the set of N uni-

variate regression models with time-varying parameters can be readily estimated by

Kalman-filter procedures.

The rest of the paper is organized as follows. Section 1.2 introduces the model

and the two-step estimation procedure. Section 1.3.1 states the assumptions and

consistency results for the principal components estimator, and Section 1.3.2 dis-

cusses identification of the loadings parameters. Our main result on consistency

of the maximum likelihood estimator of the time-varying loadings and the associ-

ated assumptions are stated in Section 1.3.3. In Section 1.4 we report the results of a

Monte Carlo study, and in Section 1.5 we provide an empirical illustration. Section

1.6 concludes.

1.2 Model and estimation

We consider the following model:

X t =Λt Ft +et , (1.1)

where X t = (X1t , ..., XN t )′ is the N -dimensional vector of observed data at time t . The

observations are generated by a small number r ¿ N of unobserved common factors

Ft = (F1t , ...,Fr t )′, time-varying factor loadingsΛt = (λ1t , ...,λN t )′, and idiosyncratic

errors et = (e1t , ...,eN t ) with covariance matrix E (et e ′t ) =Ψ0. The N×r loadings matrix

Λt = (λ1t , ...,λN t )′ is time-varying and each λi t ∈Rr×1 evolves as an r -dimensional

vector autoregression:

B 0i (L)(λi t −λ0

i ) = ηi t , (1.2)

where λ0i = E(λi t ) is the unconditional mean, and B 0

i (L) = I −B 0i ,1L − ...−B 0

i ,p Lp is

a p th-order lag polynomial where the roots of |B 0i (L)| are outside the unit circle.

4



The autoregressive order p can be allowed to vary over i such that pi differs over

i . We suppress the subscript for notational convenience. The innovations ηi t have

covariance matrix E(ηi tη′i t ) =Q0

i .

Our goal is to estimate the parameters of each of the loadings processes (1.2)

and the idiosyncratic variance matrix Ψ0. To achieve a sufficiently parsimonious

parametrization of the N × N matrix Ψ0, we specify it to be diagonal, such that

the idiosyncratic errors are cross-sectionally uncorrelated, Ψ0 = di ag (ψ01, ...,ψ0

N ).

Conditional on the factors, Xi is therefore uncorrelated over i , and the model can be

written as:

Xi = FΛi +ei , (1.3)

where Xi = (Xi 1, ..., Xi T )′, ei = (ei 1, ...,ei T )′, F = di agF ′

t

t=1,...,T is a T × r T block-

diagonal matrix, and Λi = (λ′i 1, ...,λ′

i T )′. The mean and variance of Xi are E(Xi ) =(F ′

1λ0i , ...,F ′

Tλ0i )′ and Σi := V ar (Xi ) = FΦi F′+ψi IT where Φi = V ar (Λi ) is of dimen-

sion r T × r T . We can thus specify a Gaussian likelihood function for Xi conditional

on the factors F = (F1, ...,FT )′ as:

LT (Xi |F ;θi ) =−1

2log(2π)− 1

2Tlog|Σi |− 1

2T(Xi −E(Xi ))′Σ−1

i (Xi −E(Xi )), (1.4)

with parameter vector θi =Bi (L),λi ,Qi ,ψi

. Equations (1.2) and (1.3) can be written

as a linear state-space model and the likelihood can therefore be calculated with the

Kalman filter.

It is not feasible to estimate θi with (1.4), however, as the likelihood depends on

the unobservable factors F . We therefore replace the unobservable factors F in (1.4)

with an estimate F to form the feasible likelihood function LT (Xi |F ;θi ). This gives us

a set of N likelihood functions to estimate the parameters θi for each i . Define the

estimator θi which maximizes the feasible likelihood function as:

θi = argmaxθ

LT (Xi |F ;θi ). (1.5)

This is our object of interest and we show that the estimator θip→ θ0

i for each i , where

θ0i =

B 0

i (L),λ0i ,Q0

i ,ψ0i

is the true value of the parameters.

We use the principal components estimator to estimate the factors. The principal

components estimator treats the loadings as being constant over time,Λt ≡Λ, and

solves the minimization problem:

(F ,Λ) = minF,Λ

(N T )−1N∑

i=1

T∑t=1

(Xi t −λ′i Ft )2, (1.6)

where F is T × r and Λ is N × r . To uniquely define the minimizers, it is necessary to

impose identifying restrictions on the estimators, as only Xi t is observed. By concen-

trating outΛ and using the normalization F ′F /T = Ir , the problem is equivalent to

maximizing tr (F ′(X X ′)F ), where X = (X1, ..., XT )′ is the T ×N matrix of observations.

1.3. ASYMPTOTIC THEORY 5

The resulting estimator F is given byp

T times the eigenvectors corresponding to

the r largest eigenvalues of the T ×T matrix X X ′. The solution is not unique: any

orthogonal rotation of F is also a solution. Bai and Ng (2008b) give an extensive treat-

ment of the principal components estimator. We use F to form the feasible likelihood

function LT (Xi |F ;θi ).

The estimation procedure thus consists of two steps. In the first step, we extract

principal components from the observable data to estimate the factors Ft , under

the assumption of constant loadings. In the second step, we use the factor estimates

together with the observable data to maximize the likelihood function and estimate

the parameters θi of the time-varying loadings. Our main result in Section 1.3.3 shows

that this yields a consistent estimator for the parameters of the time-varying loadings.

1.3 Asymptotic theory

In this section, we present the asymptotic theory for the two-step estimation method

discussed in Section 1.2. The main result is Theorem 1 on consistent estimation of

the loadings parameters by maximum likelihood; it is given in Section 1.3.3. Our

result builds on the work by Bates et al. (2013), who show average consistency of the

principal components estimator when loadings are subject to structural instability.

We use a different rotation of the principal components estimator, and in Section

1.3.1 we therefore restate their result in Lemma 1. Furthermore, we provide a result

on uniform consistency in t of the principal components estimator in Proposition

1. Section 1.3.2 discusses identification of the factors and loadings parameters. All

results are for N ,T →∞, and the factor rank r is assumed to be known.

We introduce the following notation. ‖A‖ = [tr(A′A)]1/2 denotes the Frobenius

norm of the matrix A. The subscripts i , j are cross-sectional indices, t , s are time

indices, and p, q are factor indices. The constant M ∈ (0,∞) is a constant common to

all the assumptions below. Finally, define CN T = minp

N ,p

T .

1.3.1 Principal Components Estimation

Let ξi t :=λi t −λ0i = B 0

i (L)−1ηi t be the loadings innovations and write (1.1) as:

X t =Λ0Ft +ξt Ft +et ,

whereΛ0 = (λ01, ...,λ0

N )′ and ξt = (ξ1t , ...,ξN t )′ are the N×r matrices of loadings means

and innovations, respectively. The vector ξi t is the moving average representation of

the loadings. Assumptions A-C are standard for factor models and are the same as

Assumptions A-C in Bai and Ng (2002):

6



Assumption A (Factors). E‖Ft‖k ≤ M <∞ for some k ≥ 4, and T −1 ∑Tt=1 Ft F ′

tp→

ΣF for some r × r positive definite matrix ΣF .

Assumption B (Loadings). ‖λ0i ‖ ≤ M < ∞, and ‖Λ0′Λ0/N −ΣΛ‖ → 0 for some

positive definite matrix ΣΛ.

Assumption C (Idiosyncratic Errors). There exists a positive constant M <∞ such

that for all N and T :

1. E(ei t ) = 0, E |ei t |8 ≤ M .

2. E(e ′s et /N ) = E(N−1 ∑Ni=1 ei s ei t ) = γN (s, t ), |γN (s, s)| ≤ M for all s, and

T −1 ∑Ts,t=1 |γN (s, t )| ≤ M .

3. E(ei t e j t ) = τi j ,t with |τi j ,t | ≤ |τi j | for some τi j and for all t . In addition

N−1 ∑Ni , j=1 |τi j | ≤ M .

4. E(ei t e j s ) = τi j ,t s , and (N T )−1 ∑Ni , j=1

∑Tt ,s=1 |τi j ,t s | ≤ M .

5. For every (s, t ), E |N−1/2 ∑Ni=1[ei s ei t −E(ei s ei t )]|4 ≤ M .

We leave the moment condition on the factors in Assumption A unspecified, as the

result in Proposition 1 depends on k. Assumption B requires the columns of Λ0

to be linearly independent, such that the matrix ΣΛ is non-singular. Assumptions

A and B together imply the existence of r common factors. Assumption C allows

for heteroskedasticity and limited time-series and cross-section dependence in the

idiosyncratic errors. It will later be strengthened to ei t being independent over i and

t . Note that if ei t is independent for all i and t , Assumptions C.2-C.5 follow from C.1.

We impose the following assumption on the factor loadings innovations and the

factors:

Assumption D (Factor Loadings Innovations). The following conditions hold for

all N ,T and factor indices p1, q1, p2, q2 = 1, ...,r :

1. sups,t

∑Ni , j=1 |E(ξi sp1ξ j t q1 Fsp1 Ft q1 )| =O(N ).

2.∑T

s,t=1∑N

i , j=1 |E(ξi sp1ξ j sq1 Fsp1 Fsq1 Ft p2 Ft q2 )| =O(N T 2).

3. supt

∑Ts=1

∑Ni , j=1 |E(ξi sp1ξ j sq1ξi t p2ξ j t q2 Fsp1 Fsq1 Ft p2 Ft q2 )| =O(N 2)+O(N T ).

Assumption D is identical to Bates et al. (2013) except for D.3, which is stronger

than their corresponding assumption. Assumption D.3 is needed for uniform consis-

tency of the principal components and is still reasonable. We do not require indepen-

dence between the factors and loadings, as the effect of the factors on the observable


variables might reasonably be expected to change when the factors differ substan-

tially from their mean levels. However, if the factors and loadings are assumed to be

independent, and the loadings evolve as stationary vector autoregressions that are

independent over i , Assumptions D.1-D.3 can easily be shown to hold: For simplicity,

take r = 1. By Assumption A and cross-sectional independence of the loadings, the

supremum in D.1 can be bounded by:

sups,t

|E(Fs Ft )|N∑

i , j=1|E(ξi sξ j t )| ≤ Msup

s,t

N∑i , j=1

|E(ξi sξ j t )| = MN∑

i=1sup

s,t|E(ξi sξi t )|.

The terms E(ξi sξi t ) are the autocovariances of the moving average representation of

the loadings. As the loadings are stationary, these autocovariances are bounded, and

the rate O(N ) follows. The rate O(N T 2) in D.2 follows from D.1 when the factors and

the loadings are independent. The sum in D.3 can be bounded by:

Msupt

T∑s=1

N∑i , j=1

|E(ξi sξ j sξi tξ j t )| = Msupt

T∑s=1

N∑i=1

|E(ξ2i sξ

2i t )|

+Msupt

T∑s=1

N∑i 6= j

|E(ξi sξi t )E(ξ j sξ j t )| ≤ MT∑

s=1

N∑i=1

supt|E(ξ4

i s )|

+Msupt

N∑i 6= j

(T∑

s=1|E(ξi sξi t )|2

)1/2 (T∑

s=1E |(ξ j sξ j t )|2

)1/2

.

The first term is O(N T ) if E (ξ4i s ) <∞, and the second term is O(N 2) if the autocovari-

ances E (ξi sξi t ) are square-summable. Assumption D.3 is therefore satisfied when the

loadings and the factors are independent. We assume the same rates to hold without

imposing independence between the factors and the loadings.

Finally, we impose independence between the idiosyncratic errors and the factors

and loadings innovations.

Assumption E (Independence). For all (i , j , s, t ), ei t is independent of (Fs ,ξ j s ).

Assumptions A-E are sufficient to consistently estimate the space spanned by the

factors. For this purpose, we use the result of Lemma 1 below, which is a modified

version of Theorem 1 in Bates et al. (2013).1 We use a rescaled estimator that is more

convenient for the rest of the analysis and therefore restate their result:

Lemma 1. Under Assumptions A-E there exists an r × r matrix H such that

T −1T∑

t=1‖Ft −H ′Ft‖2 =Op (C−2

N T )

1Bates et al. (2013) use the estimator F = FVN T , where VN T is the diagonal matrix of the r largesteigenvalues of (N T )−1 X X ′.

8



as N ,T →∞.

Proof. See the Appendix.

Lemma 1 shows that the mean-squared deviation between the principal com-

ponents and the common factors disappears as the sample size T and the cross-

sectional dimension N tend to infinity.2 The convergence rate CN T is the same as

in Bai and Ng (2002), and the principal components estimator is thus robust to sta-

tionary deviations in the loadings around a constant mean. Note that the common

factors are only identified up to a rotation, so the principal components converge to

a rotation of the common factors.

Lemma 1 does not imply uniform convergence in t , but only average consistency

of the principal components. In order to analyse the properties of the feasible likeli-

hood function LT (Xi , F |θi ), we need uniform consistency of the estimated factors, in

addition to the average consistency of Lemma 1. To establish uniform convergence,

we make additional assumptions, as in Bai and Ng (2006, 2008a):

Assumption F There exists a positive constant M <∞ such that for all N and T :

1.∑T

s=1 |γN (s, t )| ≤ M for all t .

2. E‖(N T )−1/2 ∑Ts=1

∑Nk=1 Fs [eks ekt −E(eks ekt )]‖2 ≤ M for all t .

3. E‖N−1/2 ∑Ni=1λ

0i ei t‖8 ≤ M for all t .

Assumption F.1 is stronger than C.2, but still reasonable: If ei t is assumed to be

stationary with absolutely summable autocovariances, Assumption F.1 holds. As-

sumptions F.2 and F.3 are reasonable as they involve zero-mean random variables.

We can now present the uniform consistency result for the estimated factors.

Proposition 1. Under Assumptions A-F and additionally if maxt

‖Ft‖ =Op (αT ), and

T /N 2 → 0,

maxt

‖Ft −H ′Ft‖ =Op

(T 1/8

N 1/2

)+Op (αT N−1/2)+Op (αT T −1)+Op (C−1

N T ).


2Lemma 1 also holds when the factor rank is unknown. By setting the number of estimated factors toany fixed k ≥ 1, the Lemma can be stated as T−1 ∑T

t=1 ‖F kt −Hk′Ft ‖2 =Op (C−2

N T ), where F kt is k ×1 and

Hk is a r ×k matrix, and F kt consistently estimates the space spanned by k of the true factors


Proposition 1 shows that the maximum deviation between the factors and the

principal components depends on αT . The convergence rate thus depends on the

assumption imposed on maxt

‖Ft‖. The factors can be modelled as a dynamic process

with arbitrary dynamics to determineαT . However, if the parameters governing these

dynamics are not of direct interest, nothing is lost by assuming the factors to be a

sequence of fixed and bounded constants. Thus maxt

‖Ft‖ ≤ M .3 We can take Op (αT )

to be O(1) in our results, and maxt

‖Ft −H ′Ft‖ = op (1). However, Proposition 1 is of

independent interest, e.g. for deriving the limiting distribution for the maximum

likelihood estimator, so we state Proposition 1 in its more general form. Bai (2003) and

Bai and Ng (2008a) derive a similar result for factor models with constant loadings.

Uniform convergence when loadings undergo small variations is also considered by

Stock and Watson (1998), who obtain a much slower convergence rate and require

T = o(N 1/2). Thus, Proposition 1 extends the uniform consistency result to the case

of time-varying loadings.

1.3.2 Identification

It is well known that without identifying restrictions, factors and loadings are not

separately identified in (1.1). The common component Ct =Λt Ft is identified, but

normalizations are needed to separate factor and loadings from the common com-

ponent. This has implications for the identification of the loadings parameters as

well, which we now illustrate. The model defined by (1.1) and (1.2) is observationally

equivalent to:

X t =Λt H ′−1H ′Ft +et ,

Bi (L)H−1(λi t −λi ) = H−1ηi t , f or i = 1, ..., N .

Lemma 1 states that the principal components estimator Ft is a consistent estimate

of a rotation of the true factors, H ′Ft . The two-step estimation procedure fixes the

rotational indeterminacy by imposing the normalization in the principal compo-

nents step. By replacing the unobserved factors Ft with Ft for maximum likelihood

estimation, we are thus estimating the parameters of λ∗i t = H−1λi t .

To clarify the issue, consider the following example. Using the same notation as

previously, the elements of the r ×1 vector λi t = (λi t ,1...,λi t ,r )′ refer to the loadings

of variable i at time t on each of the r factors, and λi = E(λi t ) = (λi ,1, ...,λi ,r )′ are

the corresponding unconditional expectations of the factor loadings. Assume that

the matrices ΣF and ΣΛ are diagonal. In this case it is not hard to show that the

rotation matrix H converges to Σ−1/2F . Let the number of factors r = 2 with variance

3Bai and Li (2012, 2016) treat the factors as a sequence of fixed constants when providing inferentialtheory for maximum likelihood estimation of factor models with constant loadings.

10



ΣF = di ag (σ21,σ2

2) and let the data-generating parameters of the loadings be

λ0i =

(λi ,1

λi ,2

), Q0

i =(

qi ,1 0

0 qi ,2

), B 0

i (L) = I2 −(

bi ,11 0

0 bi ,22

).

We can now make precise what θ is estimating. With the normalization F ′F /T = I2,

the principal components will be close to Σ−1/2F Ft in large samples. Using the princi-

pal components in place of the unobserved factors means that we are estimating the

following model:

X t =Λ∗t Ft +et ,

λ∗i t −λ∗

i = B∗i (λ∗

i ,t−1 −λ∗i )+ vi t , f or i = 1, ..., N ,

where λ∗i t = Σ1/2

F λi t =(σ1λi t ,1

σ2λi t ,2

)and vi t = Σ1/2

F ηi t . The loadings λi t are scaled by

the standard deviations of the unobserved factors, and it is the parameters of the

rotated loadings λ∗i t that can be estimated. In large samples the estimate of the

loadings mean λ∗i will be therefore close to

Σ1/2F λ0

i =(σ1λi ,1

σ2λi ,2

),

and the variance estimate V ar (vi t ) =V ar (Σ1/2F ηi t ) will be close to

Σ1/2F Q0

i Σ1/2F =

(σ2

1qi ,1 0

0 σ22qi ,2

).

The mean and variance parameters are thus scaled by the standard deviation of the

factors. The matrices Bi (L) and Q0i of the data-generating model are diagonal in this

example, so the diagonal elements of B∗i are the autocorrelations of λ∗

i t ,1 and λ∗i t ,2. In

large samples the first diagonal element of B∗i will therefore be close to

b∗i ,11 =

Cov(λ∗i t ,1,λ∗

i ,t−1,1)

V ar (λ∗i t ,1)

= Cov(σ1λi t ,1,σ1λi ,t−1,1)

V ar (σ1λi t ,1)

= σ21Cov(λi t ,1,λi ,t−1,1)

σ21V ar (λi t ,1)

= Cov(λi t ,1,λi ,t−1,1)

V ar (λi t ,1)= bi ,11,

and similarly for b∗i ,22. The estimates of the autoregressive matrix B∗

i are therefore

unaffected by the normalization imposed on the principal components, and the esti-

mate of B∗i is consistent for the autoregressive parameters Bi of the data-generating

process λi t .

The arguments of this example apply to the general setting as well. The maximum

likelihood estimator (1.5) of the loadings parameters is estimating Bi (L), Hλi , and

HQi H ′. The mean and variance parameters of (1.2) are identified up to the unknown


rotation matrix H , while the dynamic parameters Bi (L) are not subject to any rotation.

The rotation is determined by the restriction used to identify the principal compo-

nents. Using another normalization in the first step will thus change the estimates

of λi and Qi , while the estimate of Bi (L) is unaffected, except for small numerical

differences owing to numerical optimization of the likelihood. The dynamic proper-

ties of the loadings are therefore uniquely identified. In the following, we assume for

simplicity that H = Ir . This is just a normalization and can be achieved by imposing

further assumptions on the matrices ΣF and ΣΛ.

1.3.3 Maximum Likelihood Estimation

Our method of proof relies on showing that the likelihood function (1.4) with principal

components is asymptotically equivalent to the likelihood function with unobserved

factors. To establish our result, we impose distributional assumptions on the loadings

and idiosyncratic errors that enable maximum likelihood estimation of the parame-

ters θi =Bi (L),λi ,Qi ,ψi

. We make the following assumptions:

Assumption G (Distributions) For all i = 1, ..., N , the following statements hold:

1. The loadings λi t follow a finite-order Gaussian VAR:

Bi (L)(λi t −λi ) = ηi t ,

with the r × r filter Bi (L) = I − Bi ,1L − ... − Bi ,p Lp having roots outside the

unit circle, and ηi t is an r -dimensional Gaussian white noise process, ηi t ∼i.i.d. N(0,Qi ), where Qi is positive definite with all elements bounded.

2. The idiosyncratic errors et are cross-sectionally independent Gaussian white

noise, et ∼ N(0,Ψ), where Ψ is a diagonal matrix with elements ψi > 0 and

bounded for all i .

G.1 assumes the loadings to evolve as stationary vector autoregressions. We rule

out the possibility of I (1) loadings as this would be in violation of Assumption D. With

non-stationary loadings the principal components estimator cannot consistently

estimate the factor space.4 G.2 assumes the idiosyncratic errors to be i.i.d. over both

t and i . This assumption can be relaxed to allow for serial correlation. The key part

of Assumption G.2 is the cross-sectional independence. This enables us to analyse

the likelihood separately for each i . This is a set of N independent univariate regres-

sions with time-varying parameters. With observed regressors, consistency is known

to hold, see e.g. Pagan (1980). Rather than proving consistency for the maximum

4Bates et al. (2013) consider random walk loadings of the form λi t =λi ,t−1 +T−3/4ζi t and show thatAssumption D is satisfied with this specification. However, the scaling of the loadings innovations bythe factor T−3/4 is crucial for Lemma 1 to hold. With a pure random walk of the form λi t =λi ,t−1 +ζi t ,principal components cannot estimate the factor space consistently.

12



likelihood estimator with observed factors we therefore assume consistency in the

following assumption.

Assumption H (MLE with observed factors) For each i , the function LT (Xi |F ;θi )

satisfies:

1. There exists a function L0(Xi |F ;θi ) that is uniquely maximized at θ0.

2. θ0i is in the interior of a convex setΘi , and LT (Xi |F ;θi ) is concave.

3. LT (Xi |F ;θi )p→L0(Xi |F ;θi ) for all θi ∈Θi .

Under Assumptions G and H, the maximum likelihood estimator with observed

factors θi = argmaxθ

LT (Xi |F ;θi ) exists with probability approaching 1 and is con-

sistent for each i : θip→ θ0

i . This follows from standard arguments as in Newey and

McFadden (1994).

Replacing the unobserved factors with the principal components estimates yields

the feasible likelihood function LT (Xi |F ;θi ) and the maximum likelihood estimator

defined in (1.5). We now state our main result.

Theorem 1. Let Assumption A-H hold. For each i , the estimator θi defined in (1.5)

exists with probability approaching 1 and

θip→ θ0

i .


Theorem 1 states that using the principal component estimates instead of the

unobserved factors does not affect the consistency of the maximum likelihood es-

timator. The main argument in proving Theorem 1 is that the feasible likelihood

function converges uniformly to the infeasible likelihood function. Asymptotically,

the feasible likelihood function therefore has the same properties as the infeasible

likelihood function, for which consistency is known to hold. Assumption H thus holds

for LT (Xi |F ;θi ) and consistency follows. In the proof of Theorem 1 we use the follow-

ing normalization that is convenient for the calculations: If F ′F /T = Ir andΛ0′Λ0 is a

diagonal matrix with distinct elements, we show in the Appendix that the rotation

matrix H converges to the identity Ir . Lemma 1 and Proposition 1 then holds with H

replaced by the identity matrix, and θi can be estimated asymptotically without rota-

tion. Such normalizations are inconsequential for the results as H is asymptotically

bounded, and they are only imposed to avoid unnecessary complications. Without

such normalizations the feasible likelihood converges to LT (Xi |F H ;θi ) and θi is

consistent for the parameters of the process λ∗i t = Hλi t as discussed in Section 1.3.2.

1.4. MONTE CARLO SIMULATIONS 13

We have assumed that the factors are estimated by the method of principal com-

ponents. Note, however, that the proof of Theorem 1 does not rely on the principal

components estimator. Theorem 1 holds for all estimators F that satisfy the condi-

tions for Lemma 1 and Proposition 1.

Our analysis does not make any formal statements about the limiting distribution

of θi . Simulation evidence in Section 1.4 does, however, suggest that an asymptotic

normality result holds for θi as well. The simulations further indicate that the limiting

distribution of θi is unaffected by the estimation error of Ft . Two-step estimators typ-

ically require adjusting the limiting distribution to account for estimation error from

first-step estimation as in Newey (1984) and Pagan (1986), but that does not seem

to be the case here. Bai and Ng (2006) show that the estimated-regressor problem

can be ignored when using principal components in place of the unobserved factors

in factor-augmented VARs. We expect that a similar result holds for our model, but

leave a formal proof for future research.

In Assumption G.2 and the proof of Theorem 1 we assume that the model has an

exact factor structure in the sense that idiosyncratic errors have no cross-sectional or

temporal dependence. It is straightforward to relax the assumption of no temporal

dependence. We could model the idiosyncratic errors as cross-sectionally uncor-

related autoregressions and estimate the parameters by including ei t in the state

equation of the state space representation of the model and compute the likelihood

with the Kalman filter. The proof of Theorem 1 applies with very minor changes. The

assumption of no temporal dependence in ei t is thus only for expositional simplicity.

Relaxing the assumption of no cross-sectional correlation to allow for an approxi-

mate factor structure requires substantially more work. The essential contribution of

Assumption G.2 is that it enables the likelihood of the full panel of data X = (X1, ..., XN )

to be analysed separately for each Xi . When the idiosyncratic elements of the model

are cross-sectionally correlated, E(et e ′t ) =Ψ0 is non-diagonal, and we cannot con-

dition on the factors to make Xi independent over i . Analysis of the model with

cross-sectionally correlated errors requires a different method of proof and is beyond

the scope of this paper.5 However, in the next section we provide simulation evidence

showing that our results are robust to the assumption of no cross-sectional correla-

tion in the idiosyncratic elements.

1.4 Monte carlo simulations

In this section, we conduct a simulation study to assess the finite-sample perfor-

mance of the two-step estimator. We provide results for both principal components

5Bai and Li (2016) analyse factor models with constant loadings in a maximum likelihood settingunder weak cross-sectional correlation in ei t . They show that the estimates of ψ0

i are consistent for the

diagonal elements of T−1 ∑Tt=1 E(et e′t ). However, they rely on a method of proof that differs substantially

from ours and their results do not readily apply to our model.

14



and maximum likelihood estimates. Section 1.4.1 describes the simulation design,

and Section 1.4.2 reports and discusses the results.

1.4.1 Design

The simulation design broadly follows that of Stock and Watson (2002):

Xi t =λ′i t Ft +ei t ,

(Ir −Bi L)(λi t −λi ) = ηi t ,

Ft p = ρFt−1,p +ut p ,

(1−αL)ei t = vi t ,

ηi t ∼ i.i.d. N(0,Qi ),

ut p ∼ i.i.d. N(0,1−ρ2),

vt ∼ i.i.d. N(0,Ω),

where i = 1, ..., N , t = 1, ...,T , p = 1, ...,r . The processes ηi t ,ut p , and vt are mutu-

ally independent. The autoregressive matrix Bi determines the degree of persistence

of the loadings and has eigenvalues inside the unit circle in all simulations. The

unconditional mean of the loadings is λi = (λi 1, ...,λi r )′ and λi p ∼ i.i.d. N(0,1) in all

simulations. The matrix Qi is the covariance matrix of the loadings innovations. The

model allows for cross-sectional and temporal dependence in the errors ei t . The

parameter α determines the degree of serial correlation in the idiosyncratic errors,

and cross-sectional correlation is modelled by specifying the variance matrix of vt

asΩ=(β|i− j |√ψiψ j

)i j

for i , j = 1, ..., N . The matrix is thus a Toeplitz matrix and the

cross-sectional correlation between the idiosyncratic elements is therefore limited

and determined by the coefficient β. If β= 0, α= 0, ei t is independent across i and t ,

and the model is an exact factor model and correctly specified according to Assump-

tion G. We allow for factor persistence through the coefficient ρ. Furthermore, we

consider the case where the loadings are weakly dependent across i . We model the

correlation such that Cor r (ηi pt ,η j pt ) = θ|i− j | for i , j = 1, ..., N . Finally, we introduce

correlation between factors and loadings through ut and ηi t . For each i , we simulate

the variables

(u∗

t p

η∗i t p

)∼ AN(0, Ir ), where A is the lower triangular cholesky decompo-

sition matrix such that A A′ =(

1 γ

γ 1

). The variables u∗

t p and η∗i t p are then rescaled

to get the innovations ut p = u∗t p (1−ρ2)1/2 and ηi t p = η∗i t p q1/2

i , respectively.

We generate the model 2000 times for each of the different combinations of T

and N . To avoid any dependence on initial values of the simulated processes we have

a ’burn-in’ period of 200 observations for each simulation. The principal components

are calculated with the estimator Ft defined in (1.6). The data Xi t are standardized

to have mean zero and variance equal to one prior to extracting principal compo-

nents. The principal components are identified only up to an orthogonal rotation. In

order to directly compare the maximum likelihood estimates with data-generating

parameters, we therefore rotate the principal components to resemble the simulated


factors. More specifically, we solve for the orthogonal r × r matrix A∗ that maximizes

tr[corr(F, F A)].6 The estimates are then rescaled to have the same standard deviation

as the true simulated factors:

F∗p = σ(Fp )

σ(Fp )Fp , p = 1, ...,r

where Fp is the p th column of the rotated principal components matrix F A∗. Such

rotations are innocuous and allow us to directly compare the estimated parameter

values with the data-generating parameters. The principal components are treated as

data, and we maximize the likelihood LT (Xi |F∗;θi ) to estimate θi .

The performance of the principal component estimator F is measured by the

trace statistic:

R2F ,F

= E [tr(F ′F (F ′F )−1F ′F )]

E [tr(F ′F )],

where E denotes the average over the Monte Carlo simulations. The trace statistic

R2F ,F

is a multivariate R2 from a regression of the true data-generating factors on the

principal components. It is smaller than 1 and tends to 1 as the canonical correlation

between the factors and the principal components tends to 1.

For the maximum likelihood estimates θi we compute the mean estimates over

the Monte Carlo repetitions for each parameter.7 However, for the mean parameter

λi we report the bias of the estimates λi as the true value of λi changes for each

combination of N ,T . Furthermore, we calculate the root-mean-squared error of the

estimates θi and also of the infeasible estimates θi where the true data-generating

factors are used in the maximum likelihood estimation. We report the relative root-

mean-squared error between the estimates θi and θi . This gives us a measure of the

estimation error in θi that is due to estimation error from the principal components

estimates.

The parameters are identically chosen across the cross-section.8 The properties of

the estimated parameters θi are thus the same for all i and we only report the results

for a single cross-section index.9 In the baseline case, we set Bi = diagbi p p=1,...,r ,

6The solution to this is A∗ =V U ′ where V and U are the orthogonal matrices of the singular valuedecomposition corr(F, F ) =U SV ′. When the number of principal components k is not equal to the truenumber of factors r , we only rotate the first l = mink,r principal components. Eickmeier et al. (2015) usethe same rotation.

7Convergence is generally very good, with all 1-factor calibrations having over 99% convergence rate,and most calibrations with 2 and 3 factors have over 98% convergence rate. Exceptions are sample sizes ofT = 50 for the 2- and 3-factor models where the lowest convergence rate is 92%. However, this is expectedas we are estimating up to 10 parameters in a highly non-linear model with 50 observations. Convergencestatistics using the true factors are similar, but with somewhat better convergence rates for calibrationswith 2 and 3 factors and T = 50.

8The mean parameters λi are not the same for all i . This is necessary for Assumption B to be satisfied.With λi identical over i , the matrixΛ0 does not have full rank andΛ0′Λ0/N will not converge to a positivedefinite matrix.

9Simulations with loadings parameters calibrated with heterogeneous values across i show similarresults as in Table 1.1. The results are available upon request.

16



Qi = diagqi p p=1,...,r , and choose the loadings persistence and variance parameters

to be bi p = 0.9 and qi p = 0.2. The idiosyncratic errors are cross-sectionally and

temporally uncorrelated, i.e. α = 0, β = 0, and the variance is set at ψi = 1. The

loadings are cross-sectionally independent, θ = 0, and also independent of the factors,

γ= 0. Finally, we set ρ = 0 such that the factors are white noise.

We introduce serial correlation and cross-sectional dependence separately in the

idiosyncratic errors. We setα= 0.5 and estimate this parameter by including ei t in the

state equation. To consider the effect of cross-sectional correlation, i.e. misspecifying

the model, we set β = 0.5. We also report results with persistent factors with the

factor persistence set at both 0.9 and 0.5. Results with cross-sectionally correlated

loadings are reported for θ = 0.3, and the correlation parameter between loadings

and factors is set to γ = 0.3. Finally, we consider the consequences of estimating

the wrong number of factors, i.e. extracting one fewer or one additional principal

component than the true number of factors.

1.4.2 Results

Table 1.1 reports the results for one factor, r = 1. Panel I shows the results for the

baseline model with no serial, no cross-sectional dependence in errors, and no factor

dependence. The R2F ,F

statistics show that the factor estimates are close to the true

factors even for small sample sizes. For the autoregressive parameter bi , the estimates

improve as the sample size T increases. Increasing the cross-sectional dimension N

only gives minor improvements for fixed T . This is unsurprising as a larger N can only

improve the parameter estimates through better factor estimates which are already

quite good even for N = 50. The estimate of the loadings innovation variance qi is

closely related to the estimate of bi . As bi gets closer to its true value, so does qi , and

vice versa. For T ≥ 200 the estimates are close to the true values. The small-sample

bias of bi is not a consequence of estimation error from principal components. Using

the true factors instead of principal components to estimate the parameters of the

latent process λi t also shows that T ≥ 200 is needed for the bias of bi and qi to be

less than 10% of the true value. The loadings mean λi and the error variance ψi are

very precisely estimated for all sample sizes.

In Panel II, the idiosyncratic errors are serially correlated, and the autoregressive

parameters for the errors are estimated along with the other parameters. The R2F ,F

statistic is hardly affected by serially correlated errors. The results are very close to

the corresponding values in the first panel. The results for the loadings parameters

are also very similar and are not markedly affected. The autoregressive parameter for

the errors α and the variance parameter ψ are very close to their true value for all

sample sizes. The model with serially correlated errors can thus be estimated equally

well as the model with i.i.d. errors.

Next, in Panel III we consider the effect of cross-sectional correlation in the errors.

The factor estimates are again not affected. The misspecification of the variance


matrix for the idiosyncratic errors deteriorates the estimates of bi and qi for the

smaller sample sizes. Compared to the results for i.i.d. errors, estimates are worse,

but do converge for the larger sample sizes. The mean parameter λi is not affected.

One can show that the information matrix of the likelihood is block diagonal be-

tween λi and the other parameters, so misspecifying the variance does not lead to

estimation error in the mean. Cross-sectionally correlated errors inflate the estimate

of the idiosyncratic error variance ψi , however, but the loadings parameters remain

consistent. This indicates that our result in Theorem 1 is robust to the assumption of

no cross-sectional correlation in ei t .

High factor persistence has a larger impact on the R2F ,F

statistic. Panel IV shows

much lower values of these statistics for all but the largest sample sizes. However, this

estimation error does not seem to influence the estimate of the loadings parameters.

The estimates for bi , and accordingly qi , are similar to the case of white noise factors.

The most notable impact of the lower R2F ,F

is in the estimate of ψi . The increase

in factor estimation error seems to inflate the error variance, which is larger for all

sample sizes, but the results do show convergence for the largest sample size. Results

for more moderate levels of factor persistence are shown in Panel V. The drop in the

R2F ,F

is less severe in this case and the estimate of ψi thus less biased.

In Table 1.2, the relative root-mean-squared errors of the estimates using princi-

pal components and the true simulated factors are reported. Values close to 1 indicate

that the asymptotic variance of the parameter estimates is unaffected by the estima-

tion error from principal components estimation of the factors. In Panels I-III, all the

statistics are close to 1 even for the smallest sample sizes. In Panel IV, the statistics for

the loadings parameters are somewhat higher for the smaller sample sizes, but close

to one for large sample sizes. The statistics for the idiosyncratic variances are much

larger than 1. This is partly due to the bias of these estimates evident in Panel IV of

Table 1.1, but also reflects higher variability of the estimates. High factor persistence

thus mainly affects the idiosyncratic variance parameters. Unreported results show

that the estimates improve for larger sample sizes. In Panel V, the factor persistence is

more moderate and the relative root-mean-squared errors are much closer to 1.

Panel I in Table 1.3 reports results for the case where the loadings are cross-

sectionally correlated. The results are very similar to Panel I in Table 1.1. The R2F ,F

statistic and the parameter estimates are not influenced by cross-sectional depen-

dence in the loadings. Unreported results show that stronger cross-sectional depen-

dence has only very minor effects on the results. The R2F ,F

statistics generally falls by

a single percentage point, but the loadings parameters are not affected.

In Panel II, the loadings and factors are correlated. Correlation between factors

and loadings leads to a minor inflation of the estimates of qi for the largest sample

sizes. This is not simply sampling variation. Unreported results for larger sample

sizes show that the estimates of qi do not converge to 0.2. When factors and loadings

are correlated, the data exhibits some variance that is not captured by a parameter

18



in the model. The variance in the data that is due to Cov(ξi t ,Ft ) shows up in the

estimate of the loadings variance. Stronger correlation between factors and loadings

inflates the estimate of qi further. However, this is not a source of inconsistency for

the estimator. Using the simulated factors instead of the principal components to

estimate the model leads to similar parameter estimates. This is evident from Table

1.4, Panel II. Here we report the relative root-mean-squared errors of the estimates

using principal components and the true simulated factors. The results are all close

to 1. The estimates using principal components and simulated factors are therefore

consistent for the same parameter. If they were not, the relative root-mean-squared

errors would not be close to 1. Correlation between factors and loadings do not affect

the estimates of the other parameters. They are similar to the results in Panel I, Table

1.1.

Table 1.5 displays the simulations results for the model with 2 and 3 factors

with i.i.d. errors and white noise factors. Compared to the 1-factor model, the R2F ,F

statistics are lower, reflecting the increasing difficulties in extracting additional factors.

In Panel I, the estimates for the second set of loadings parameters are worse than

for the first set and the same pattern is evident for the 3-factor model (Panel II). The

results for the third set of loadings parameters are worse than for the second, which

are worse than for the first. However, all the estimates are converging to their true

values. Compared to the 1-factor model, larger sample sizes are generally needed

to get precise estimates due to the increased number of parameters. Introducing

serial and cross-sectional correlation in the errors, correlated factors and loadings,

or persistence in factors does not reveal any additional insights compared to the

1-factor model. The results generalize and are therefore omitted. Table 1.6 shows the

relative root-mean-squared errors for the 2- and 3-factor model. The statistics are

somewhat larger than 1 for the smaller sample sizes, but get increasingly closer to

one as the sample sizes grow. This indicates that the estimation error of the principal

components does not affect the asymptotic variance of the estimates.

Table 1.7 shows the results of estimating the wrong number of factors. For these

simulations, we report two convergence statistics for the principal components. The

first is the R2 from a regression of the principal components on the true factors,

R2(1)F ,F

= E [F ′F (F ′F )−1F ′F ]E [F ′F ]

and the second is the R2 from a regression of the true factors

on the principal components. In Panel I, the simulated data have two factors, but only

1 principal component is extracted. The first statistic R2(1)F ,F

is close to 1 for all sample

sizes. Hence, the two factors explain all the variation in the single principal compo-

nent. The second statistic R2(2)F ,F

does not tend to 1, as a single principal component

cannot span the two-dimensional factor space. The results show that the loadings

parameters for the first factor can still be estimated consistently. The consequence of

excluding a factor is that the estimate of the error variance ψi gets larger, reflecting

the variability in the data from the excluded factor and its loadings. Panel II displays

results for the 1-factor model with two principal components extracted from the data.

1.5. AN EMPIRICAL ILLUSTRATION 19

R2(2)F ,F

tends to 1, and the two principal components thus explain all the variation in

the single factor. The other measure R2(1)F ,F

tends to 0.5 as the single factor can only

span half of the two-dimensional space of the principal components. The loadings

on the first factor are estimated consistently. For the second factor, the mean and

variance of the loadings are being estimated as zero.10 The estimated parameters thus

show that the data do not load on the second factor and therefore correctly dismiss

the second factor. The results are thus very encouraging even with the number of

principal components different from the true number of factors.

The results can be summarized as follows:

• The loadings and idiosyncratic variance parameters are estimated consistently.

The sample size T needs to be sufficiently large (≥ 200) for the bias in the

autoregressive parameters to be less than 10%.

• The results are robust to the assumption of cross-sectionally uncorrelated

errors. The loadings parameters are not affected by this misspecification, only

the estimate of the error variance.

• Loadings parameters are consistently estimated even when an incorrect num-

ber of principal components are extracted. Too few principal components

increase the error variance estimate, and loadings means and variances are cor-

rectly estimated as zero for principal components in excess of the true number

of factors.

• The relative root-mean-squared errors indicate that the asymptotic variance is

unaffected by replacing the factors with the principal components estimates

and that the estimates have the same limiting distribution as if the factors are

observed.

1.5 An empirical illustration

We provide an empirical illustration of the model using the data set of Stock and Wat-

son (2009), who analyse a balanced panel of 144 quarterly time series for the United

States, focusing on structural instability in factor loadings and its consequences for

forecast regression. The data set consist of 144 quarterly time series for the United

States, spanning 1959:I-2006:IV. The data series are transformed to be stationary,

and the first two quarters are thus excluded because of differencing, resulting in

T = 190 observations used for estimation. We exclude a number of series that are

higher-level aggregates of the included series, which brings the number of series

used for estimation to N = 109. For a complete data description and details on data

transformations, see the appendix of Stock and Watson (2009).

10The results for bi 2 are not indicative of any convergence. Histograms of the estimated values showthat the parameter is not identified as the values are randomly estimated anywhere between -1 and 1.

20



Stock and Watson (2009) argue for 4 factors in the sample, and perform robust-

ness checks of their results using different numbers of factors. We therefore extract

4 principal components from the standardized data and estimate the loadings pa-

rameters and the idiosyncratic variances for each of the 109 variables. The system

matrices Bi and Qi are specified to be diagonal, i.e., the loadings are estimated as

univariate autoregressions uncorrelated over the factor indices. The lag polynomials

Bi (L) are of order one for all i .

Stock and Watson (2009) test for breaks in the factor model and find evidence of

structural instability in a large number of the factor loadings. Using Chow statistics

to test for breaks in loadings in 1984:I, they reject the null of no instability for 41% of

the variables. We provide similar evidence of structural instability in the loadings. For

each variable Xi , we test the null hypothesis of constant loadings by likelihood ratio

statistics. In the restricted model, we thus set the diagonal elements of the matrix Qi

equal to zero, in which case the maximum likelihood estimator of the restricted model

can be computed by ordinary least squares. The last column of Table 1.8 reports the

rejection frequencies grouped by variable category as well as the rejection frequency

for the entire panel. For 85% of the series, the likelihood ratio statistics rejects at

the 5% significance level, and 76% of the series are rejected at the 1% level. When

comparing the rejection frequencies across categories, no obvious pattern emerges:

The rejection frequencies are high for all categories. The category with the lowest

rejection frequency is consumption variables, where the null of constant loadings

cannot be rejected for a third of the variables.

We also test the null hypothesis of constant loadings on each individual factor.

We reestimate the model for each variable 4 times and let the loadings on a single

factor be constant each time, while the rest of the loadings follows first-order au-

toregressions. Columns F1 −F4 of Table 1.8 report the rejection frequencies for these

likelihood ratio statistics. For the first two factors, approximately two thirds of the

series show evidence of time-varying loadings. For the last two factors, the rejec-

tion frequencies are smaller, but time-variation in the loadings is still evident for a

non-trivial number of the series. Comparing the rejection frequencies across variable

categories does not reveal any general patterns: The rejection frequencies vary a lot

across categories and factors. Closer inspection of the individual series indeed reveals

that the way in which the data load on the factors is very heterogeneous across series.

Comparing our results to those of Stock and Watson (2009), we find that a larger

share of the variables have time-varying loadings. However, as our model considers

stationary variations around a constant mean, and Stock and Watson (2009) consider

a one-time break at a fixed point in time, it is possible for the likelihood ratio statistics

to reject the null of constant loadings, even when the Chow tests of Stock and Watson

(2009) cannot reject the null of no breaks.

Given the large number of series for which we reject the null of constant load-

ings, we can obtain a better in-sample fit of the common component by letting the

1.6. CONCLUSION 21

loadings vary over the time dimension. Figure 1.1 displays the estimated loadings

paths for the one-year/three-month Treasury term spread. This series exhibits very

strong evidence of structural instability in the loadings. The p-values of the likelihood

ratio statistics for joint and single restrictions are all zero to at last 4 decimal places.

The loadings show a considerable amount of variation over the sample period with

quite persistent dynamics. Figure 1.2 displays the standardized Treasury term spread

and two estimates of its common component, computed with time-varying loadings

and constant loadings, respectively. The common component based on time-varying

loadings tracks the data much closer than the one with constant loadings. The corre-

lations of the two estimates of the common component with the data are 0.99 and

0.19, respectively. This clearly illustrates the improvement in the in-sample fit by

modelling the loadings as autoregressive processes.

1.6 Conclusion

We proposed a two-step maximum likelihood estimator for time-varying loadings in

high-dimensional factor models. The loadings parameters are estimated by a set of N

univariate regression models with time-varying coefficients, where the unobserved re-

gressors are estimated by principal components. Replacing the unobservable factors

with principal components gives a feasible likelihood function that is asymptotically

equivalent to the infeasible one with unobservable factors and therefore gives consis-

tent estimates of the loadings parameters as N ,T →∞. The finite-sample properties

of our estimator were assessed via an extensive simulation study. The results showed

that the loadings means and idiosyncratic error variances are estimated precisely

even for small sample sizes. A somewhat larger sample size is needed to get precise

estimates of the loadings variance and dynamic parameters. Furthermore, the simu-

lations showed very satisfactory results when the number of principal components

is different from the number of factors in the data. We illustrated the empirical rel-

evance of the time-varying loadings structure using the large quarterly dataset of

Stock and Watson (2009) for the US economy. For the majority of the variables we

found evidence of time-varying loadings, and we showed that a large increase in the

in-sample fit of the common component can be obtained by modelling the loadings

as time-varying.

Acknowledgements

Jakob Guldbæk Mikkelsen and Eric Hillebrand acknowledge support from The Danish

Council for Independent Research (DFF 4003-00022) and CREATES - Center for

Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish

National Research Foundation. Part of the research was conducted while the first

22



named author was visiting the Centre for Econometric Analysis at Cass Business

School in the Fall of 2015.

1.6. CONCLUSION 23

Table 1.1: Simulation results for 1-factor model.

T N R2F ,F

bi λi qi ψi α

Panel True values 0.9 0 0.2 1

α= 0, β= 0, ρ = 0, θ = 0, γ= 0

I

50 50 0.943 0.603 -0.022 0.302 0.970 -100 50 0.941 0.785 -0.005 0.258 1.005 -

50 100 0.960 0.607 0.023 0.307 0.978 -100 100 0.969 0.792 -0.043 0.260 1.010 -100 200 0.976 0.797 -0.011 0.256 0.993 -200 200 0.984 0.864 0.003 0.224 1.003 -400 200 0.986 0.884 -0.019 0.213 1.009 -600 300 0.991 0.890 0.012 0.208 0.997 -

α= 0.5, β= 0, ρ = 0, θ = 0, γ= 0

II

50 50 0.939 0.617 -0.005 0.296 0.934 0.500100 50 0.934 0.805 -0.009 0.246 0.992 0.494

50 100 0.956 0.614 0.033 0.290 0.929 0.494100 100 0.966 0.805 -0.060 0.246 0.978 0.509100 200 0.974 0.812 0.015 0.242 0.969 0.493200 200 0.982 0.870 0.001 0.223 0.990 0.497400 200 0.985 0.885 -0.027 0.212 0.994 0.500600 300 0.989 0.890 0.011 0.206 0.995 0.499

α= 0, β= 0.5, ρ = 0, θ = 0, γ= 0

III

50 50 0.944 0.540 -0.042 0.334 1.279 -100 50 0.942 0.762 -0.002 0.276 1.326 -

50 100 0.959 0.556 0.029 0.335 1.295 -100 100 0.968 0.777 -0.029 0.275 1.324 -100 200 0.976 0.775 -0.004 0.273 1.328 -200 200 0.983 0.857 -0.005 0.235 1.326 -400 200 0.986 0.881 -0.019 0.215 1.353 -600 300 0.990 0.889 0.003 0.210 1.321 -

α= 0, β= 0, ρ = 0.9, θ = 0, γ= 0

IV

50 50 0.652 0.587 0.000 0.355 1.177 -100 50 0.785 0.784 0.019 0.255 1.134 -

50 100 0.651 0.606 -0.031 0.392 1.257 -100 100 0.809 0.828 0.010 0.291 1.385 -100 200 0.806 0.781 0.009 0.269 1.116 -200 200 0.896 0.865 -0.024 0.237 1.065 -400 200 0.941 0.894 -0.050 0.221 1.122 -600 300 0.961 0.895 0.026 0.211 1.040 -

α= 0, β= 0, ρ = 0.5, θ = 0, γ= 0

V

50 50 0.905 0.625 -0.033 0.306 1.017 -100 50 0.920 0.803 -0.010 0.249 1.023 -

50 100 0.916 0.606 0.044 0.307 1.045 -100 100 0.950 0.803 -0.060 0.254 1.076 -100 200 0.955 0.803 0.019 0.254 1.015 -200 200 0.973 0.865 -0.010 0.226 1.012 -400 200 0.981 0.885 -0.022 0.214 1.026 -600 300 0.987 0.889 0.012 0.209 1.006 -

NOTE: The columns T and N report the sample sizes. The column R2F ,F

reports the convergence statistic for the principal components estimator.The remaining columns report the mean of the parameter estimates overthe Monte Carlo simulations. For the parameter λi , the bias is reported.

24



Table 1.2: Relative root-mean-squared error for 1-factor model.

T N R2F ,F

bi λi qi ψi α


α= 0, β= 0, ρ = 0, θ = 0, γ= 0

I

50 50 0.943 1.043 1.056 0.983 1.024 -100 50 0.941 1.083 1.062 0.986 1.024 -

50 100 0.960 1.032 1.042 1.039 1.030 -100 100 0.969 1.065 1.033 1.017 1.076 -100 200 0.976 1.009 1.033 1.032 1.020 -200 200 0.984 1.029 1.019 0.992 1.005 -400 200 0.986 0.995 1.013 1.008 1.039 -600 300 0.991 0.979 1.011 0.989 0.997 -

α= 0.5, β= 0, ρ = 0, θ = 0, γ= 0

II

50 50 0.939 1.040 1.065 1.063 0.997 1.006100 50 0.934 1.022 1.072 0.985 1.025 1.014

50 100 0.956 1.021 1.054 0.989 0.988 1.001100 100 0.966 1.105 1.051 1.032 1.015 1.017100 200 0.974 0.999 1.043 1.020 0.998 1.002200 200 0.982 0.977 1.021 1.011 0.998 0.999400 200 0.985 0.985 1.019 1.018 1.015 0.999600 300 0.989 0.980 1.012 1.007 1.000 1.007

α= 0, β= 0.5, ρ = 0, θ = 0, γ= 0

III

50 50 0.944 1.013 1.055 0.998 1.053 -100 50 0.942 1.081 1.066 1.043 1.095 -

50 100 0.959 1.016 1.041 0.989 1.068 -100 100 0.968 0.970 1.021 0.984 1.133 -100 200 0.976 1.022 1.033 1.011 1.049 -200 200 0.983 0.962 1.019 0.996 1.035 -400 200 0.986 1.001 1.013 0.995 1.076 -600 300 0.990 0.981 1.006 0.984 0.982 -

α= 0, β= 0, ρ = 0.9, θ = 0, γ= 0

IV

50 50 0.652 1.060 1.147 1.339 1.926 -100 50 0.785 1.106 1.127 1.060 1.778 -

50 100 0.651 1.038 1.147 1.451 2.415 -100 100 0.809 0.854 1.166 1.468 4.506 -100 200 0.806 1.047 1.085 1.083 1.665 -200 200 0.896 0.947 1.085 1.101 1.455 -400 200 0.941 0.851 1.099 1.130 2.743 -600 300 0.961 0.921 1.044 1.021 1.432 -

α= 0, β= 0, ρ = 0.5, θ = 0, γ= 0

V

50 50 0.905 1.061 1.064 0.936 1.086 -100 50 0.920 1.043 1.067 1.049 1.039 -

50 100 0.916 1.024 1.046 1.030 1.241 -100 100 0.950 0.956 1.043 1.060 1.415 -100 200 0.955 1.090 1.040 1.070 1.057 -200 200 0.973 1.012 1.028 1.023 1.038 -400 200 0.981 0.984 1.014 1.013 1.169 -600 300 0.987 0.971 1.011 0.990 1.040 -

NOTE: The columns T and N report the sample sizes. The column R2F ,F

reports the convergence statistic for the principal components estimator.The remaining columns report the relative root-mean-squared errorof the parameter estimates using principal components and the truesimulated factors.

1.6. CONCLUSION 25

Table 1.3: Simulation results for 1-factor model.

T N R2F ,F

bi λi qi ψi


α= 0, β= 0, ρ = 0, θ = 0.3, γ= 0

I

50 50 0.944 0.584 -0.016 0.311 0.970100 50 0.941 0.796 -0.003 0.253 1.008

50 100 0.959 0.591 0.032 0.311 0.975100 100 0.969 0.794 -0.038 0.248 1.024100 200 0.976 0.804 -0.023 0.256 0.993200 200 0.983 0.864 -0.008 0.225 1.001400 200 0.986 0.885 -0.025 0.212 1.010600 300 0.990 0.889 0.003 0.205 1.000

α= 0, β= 0, ρ = 0, θ = 0, γ= 0.3

II

50 50 0.941 0.604 -0.020 0.303 0.986100 50 0.939 0.797 0.004 0.258 1.012

50 100 0.957 0.572 0.008 0.316 0.985100 100 0.967 0.788 -0.045 0.259 1.028100 200 0.973 0.793 -0.000 0.264 0.997200 200 0.983 0.861 -0.011 0.239 1.001400 200 0.985 0.883 -0.026 0.213 1.014600 300 0.990 0.888 0.007 0.214 1.001

NOTE: The columns T and N report the sample sizes. Thecolumn R2

F ,Freports the convergence statistic for the princi-

pal components estimator. The remaining columns report themean of the parameter estimates over the Monte Carlo simula-tions. For the parameter λi , the bias is reported.

26



Table 1.4: Relative root-mean-squared error for 1-factor model.

T N R2F ,F

bi λi qi ψi


α= 0, β= 0, ρ = 0, θ = 0.3, γ= 0

I

50 50 0.944 1.059 1.057 0.989 1.019100 50 0.941 1.036 1.062 1.026 1.035

50 100 0.959 1.066 1.047 1.047 1.014100 100 0.969 1.069 1.031 1.014 1.096100 200 0.976 1.019 1.040 1.017 1.016200 200 0.983 1.047 1.016 0.998 1.010400 200 0.986 0.978 1.017 1.004 1.033600 300 0.990 0.991 1.008 0.996 1.005

α= 0, β= 0, ρ = 0, θ = 0, γ= 0.3

II

50 50 0.941 1.057 1.052 1.009 1.025100 50 0.939 1.061 1.057 1.039 1.053

50 100 0.957 1.088 1.038 0.979 1.042100 100 0.967 1.074 1.029 0.991 1.180100 200 0.973 1.040 1.043 1.017 1.019200 200 0.983 0.998 1.019 1.008 1.016400 200 0.985 0.976 1.011 0.967 1.101600 300 0.990 0.982 1.004 0.982 1.018

NOTE: The columns T and N report the sample sizes. The col-umn R2

F ,Freports the convergence statistic for the principal

components estimator. The remaining columns report the rela-tive root-mean-squared error of the parameter estimates usingprincipal components and the true simulated factors.

1.6. CONCLUSION 27

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0.73

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033

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100

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629

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271

0.49

0-0

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0.36

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--

0.97

110

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953

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90.

047

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40.

735

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287

--

-1.

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100

200

0.97

30.

813

-0.0

560.

240

0.73

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0.29

1-

--

1.08

820

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978

0.87

2-0

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0.21

60.

854

0.00

20.

239

--

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400

200

0.98

20.

888

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205

0.88

30.

036

0.21

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--

1.02

960

030

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987

0.89

20.

004

0.20

40.

891

0.02

10.

210

--

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013

Th

ree

fact

ors

–α=

0,β=

0,ρ=

0,θ=

0,γ=

0

II

5050

0.88

30.

584

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000.

244

0.43

4-0

.143

0.33

40.

340

0.00

60.

516

1.39

610

050

0.87

50.

813

0.02

90.

196

0.75

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0.24

50.

587

-0.1

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372

1.35

350

100

0.90

60.

605

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240

0.47

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0.33

30.

351

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536

1.13

410

010

00.

938

0.83

1-0

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60.

770

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030.

262

0.63

90.

051

0.36

11.

132

100

200

0.96

00.

838

0.02

30.

219

0.77

40.

034

0.26

20.

654

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365

1.04

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020

00.

974

0.87

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0.21

00.

862

0.01

40.

226

0.82

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062

0.27

61.

015

400

200

0.97

50.

889

0.02

20.

207

0.88

60.

043

0.21

40.

880

0.05

10.

231

1.09

960

030

00.

985

0.89

3-0

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40.

890

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210

0.88

7-0

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61.

025

NO

TE

:Th

eco

lum

ns

Tan

dN

rep

ort

the

sam

ple

size

s.T

he

colu

mn

R2 F

,Fre

po

rts

the

con

verg

ence

stat

isti

cfo

rth

ep

rin

cip

al

com

pon

ents

esti

mat

or.T

he

rem

ain

ing

colu

mn

sre

por

tth

em

ean

ofth

ep

aram

eter

esti

mat

esov

erth

eM

onte

Car

losi

mu

lati

ons.

For

the

par

amet

erλ

i,th

eb

ias

isre

po

rted

.

28



Table

1.6:Relative

roo

tmean

squ

arederro

rsfo

r2-

and

3-factor

mo

del.

TN

R2F

,Fb

i1λ

i1q

i1b

i2λ

i2q

i2b

i3λ

i3q

i3ψ

i

Panel

True

values

0.90

0.20.9

00.2

0.90

0.21

Two

factors

—α=

0,β=

0,ρ=

0,θ=

0,γ=

0

I

5050

0.9201.210

1.1051.151

1.0611.167

1.446-

--

1.462100

500.934

0.9891.063

1.0791.048

1.0901.027

--

-1.040

50100

0.9531.087

1.0591.049

1.0541.086

1.110-

--

1.069100

1000.953

1.2501.055

1.0301.097

1.0931.062

--

-1.118

100200

0.9731.059

1.0591.059

1.0531.112

1.082-

--

1.388200

2000.978

0.9781.027

1.0070.929

1.0391.019

--

-1.027

400200

0.9820.988

1.0271.010

0.9861.069

1.012-

--

1.082600

3000.987

0.9631.037

1.0000.970

1.0221.002

--

-1.042

Th

reefacto

rs—α=

0,β=

0,ρ=

0,θ=

0,γ=

0

II

5050

0.8831.409

1.1131.221

1.1871.140

1.3721.030

1.5661.557

1.788100

500.875

1.6041.082

0.9991.236

1.1191.164

1.1991.877

1.4661.821

50100

0.9061.359

1.0911.178

1.1141.112

1.3871.016

1.5371.727

1.403100

1000.938

1.2041.079

1.0211.150

1.1041.121

1.0161.312

1.2861.285

100200

0.9601.235

1.0511.051

1.0871.082

1.0811.029

1.2171.186

1.109200

2000.974

1.0731.030

1.0241.062

1.0531.045

1.0291.100

1.1561.047

400200

0.9750.967

1.0361.041

1.0031.090

1.0911.063

1.1951.109

1.320600

3000.985

0.9701.014

1.0210.987

1.0211.037

0.8931.033

1.0611.051

NO

TE:T

he

colum

ns

Tan

dN

reportth

esam

ple

sizes.Th

ecolu

mn

R2F

,Frep

ortsth

econ

vergence

statisticfor

the

prin

cipal

comp

onen

tsestim

ator.Th

erem

ainin

gcolu

mn

srep

ortsth

erelative

root-mean

-squ

arederror

ofthe

param

eterestim

atesu

sing

prin

cipalco

mp

on

ents

and

the

true

simu

latedfacto

rs.

1.6. CONCLUSION 29

Tab

le1.

7:Si

mu

lati

on

resu

lts

for

inco

rrec

tnu

mb

ero

fpri

nci

pal

com

po

nen

ts.

TN

R2(

1)F

,FR

2(2)

F,F

bi1

λi1

qi1

bi2

λi2

qi2

ψi

Pan

elTr

ue

valu

es0.

90

0.2

0.9

00.

21

1p

rin

cip

alco

mp

on

ent,

two

fact

ors

—α=

0,β=

0,ρ=

0,θ=

0,γ=

0

I

5050

0.95

40.

797

0.49

70.

011

0.38

7-

--

6.70

310

050

0.95

60.

789

0.83

50.

056

0.23

0-

--

2.35

450

100

0.96

70.

807

0.64

9-0

.070

0.31

3-

--

3.68

110

010

00.

971

0.80

80.

831

0.04

00.

238

--

-2.

587

100

200

0.98

10.

819

0.81

7-0

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0.24

8-

--

3.56

620

020

00.

985

0.82

00.

873

-0.0

310.

218

--

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078

400

200

0.98

90.

823

0.88

50.

023

0.21

5-

--

4.88

560

030

00.

992

0.82

60.

889

0.02

30.

212

--

-6.

630

2p

rin

cip

alco

mp

on

ents

,1fa

cto

r—α=

0,β=

0,ρ=

0,θ=

0,γ=

0

II

5050

0.47

50.

951

0.31

5-0

.022

0.19

80.

128

-0.0

080.

100

0.88

610

050

0.47

40.

948

0.64

2-0

.003

0.19

60.

241

0.01

60.

075

0.92

850

100

0.48

20.

964

0.31

70.

005

0.21

00.

126

0.00

60.

106

0.90

110

010

00.

486

0.97

20.

605

-0.0

100.

208

0.24

8-0

.003

0.07

90.

961

100

200

0.48

90.

978

0.58

30.

008

0.19

90.

268

0.01

30.

074

0.95

320

020

00.

492

0.98

40.

826

-0.0

070.

201

0.24

10.

002

0.04

70.

967

400

200

0.49

30.

986

0.88

4-0

.018

0.19

50.

236

-0.0

070.

025

0.99

460

030

00.

495

0.99

10.

889

0.01

40.

196

0.22

7-0

.004

0.01

70.

991

NO

TE

:Th

eco

lum

ns

Tan

dN

rep

ort

the

sam

ple

size

s.T

he

colu

mn

sR

2(1)

F,F

and

R2(

1)F

,Far

eth

etw

oco

nve

r-

gen

cest

atis

tics

for

the

pri

nci

pal

com

po

nen

tses

tim

ato

r.T

he

rem

ain

ing

colu

mn

sre

po

rtth

em

ean

oft

he

par

amet

eres

tim

ates

over

the

Mo

nte

Car

losi

mu

lati

on

s.Fo

rth

ep

aram

eterλ

i,th

eb

ias

isre

po

rted

.

30



Table 1.8: Rejection frequencies of likelihood ratio statistics.

Category F1 F2 F3 F4 All Factors

Output 0.63 0.75 0.38 0.25 0.88Consumption 0.33 0.33 0.00 0.67 0.67

Labour market 0.70 0.61 0.30 0.17 0.78Housing 1.00 0.60 0.60 0.20 1.00

Investment 0.63 0.88 0.50 0.38 1.00Prices & Wages 0.50 0.68 0.21 0.29 0.79

Financial variables 0.68 0.74 0.68 0.53 0.95Money & Credit 0.50 0.38 0.50 0.50 0.88

Other 0.71 0.71 0.29 0.00 0.86All 0.62 0.66 0.39 0.31 0.85

NOTE: Column F1 reports the rejection frequencies across variablegroups of the likelihood ratio statistics for testing the null hypothesisof constant loadings on the first factor and similarly for columnsF2−F4. The last column reports the rejection frequencies for the nullhypothesis of constant loadings on all factors. All tests are evaluatedat the 5% significance level.

1.6. CONCLUSION 31

Figure 1.1: Factor loadings for the one-year/three-month Treasury term spread.

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

-1.5

-1

-0.5

0

0.5

1

1.5

Loading 1

Loading 2

Loading 3

Loading 4

Figure 1.2: One-year/three-month Treasury term spread and two estimates of its commoncomponent (CC).

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

-2

-1

0

1

2

3

4

Term spread

CC: Time-varying loadings

CC: Constant loadings

32



1.7 References

Bai, J., 2003. Inferential theory for factor models of large dimensions. Econometrica

71, 135–171.

Bai, J., Li, K., 2012. Statistical analysis of factor models of high dimension. The Annals

of Statistics 40 (1), 436–465.

Bai, J., Li, K., 2016. Maximum likelihood estimation and inference for approximate

factor models of high dimension. Review of Economics and Statistics 98 (2), 298–

309.

Bai, J., Ng, S., 2002. Determining the number of factors in approximate factor models.

Econometrica 70 (1), 191–221.

Bai, J., Ng, S., 2006. Confidence intervals for diffusion index forecasts and inference

for factor-augmented regressions. Econometrica 74, 1133–1150.

Bai, J., Ng, S., 2008a. Extremum estimation when the predictors are estimated from

large panels. Annals of Economics and Finance 9 (2), 201–222.

Bai, J., Ng, S., 2008b. Large dimensional factor analysis. Foundations and Trends in

Econometrics 3, 8–163.

Bai, J., Ng, S., 2013. Principal components estimation and identification of static

factors. Journal of Econometrics 176 (1), 18–29.



177, 289–304.

Breitung, J., Eickmeier, S., 2011. Testing for structural breaks in dynamic factor models.

Journal of Econometrics 163 (1), 71–84.

Del Negro, M., Otrok, C., 2008. Dynamic factor models with time-varying parame-

ters: measuring changes in international business cycles. Staff report 326, Federal

Reserve Bank of New York, New York.

Doz, C., Giannone, D., Reichlin, L., 2012. A quasi–maximum likelihood approach for

large, approximate dynamic factor models. Review of Economics and Statistics

94 (4), 1014–1024.

Eickmeier, S., Lemke, W., Marcellino, M., 2015. Classical time varying factor-

augmented vector auto-regressive models – estimation, forecasting and structural

analysis. Journal of the Royal Statistical Society: Series A (Statistics in Society)

178 (3), 493–533.

1.7. REFERENCES 33

Forni, M., Hallin, M., Lippi, M., Reichlin, L., 2000. The generalized dynamic-factor

model: identification and estimation. Review of Economics and Statistics 82, 540–

554.

Liu, P., Mumtaz, H., Theophilopoulou, A., 2011. International transmission of shocks:

A time-varying factor-augmented VAR approach to the open economy. Working

paper 425, Bank of England, London.

Newey, W. K., 1984. A method of moments interpretation of sequential estimators.

Economics Letters 14 (2), 201–206.

Newey, W. K., McFadden, D., 1994. Large sample estimation and hypothesis testing.

Handbook of Econometrics 4, 2111–2245.

Pagan, A., 1980. Some identification and estimation results for regression models

with stochastically varying coefficients. Journal of Econometrics 13 (3), 341–363.

Pagan, A., 1986. Two stage and related estimators and their applications. The Review

of Economic Studies 53 (4), 517–538.

Stock, J., Watson, M., 1998. Diffusion indexes. Working paper, National Bureau of

Economic Research.

Stock, J. H., Watson, M., 2009. Forecasting in dynamic factor models subject to struc-

tural instability. In: Hendry, D.F., Castle, J., Shephard, N. (Eds.), The Methodology

and Practice of Econometrics. A Festschrift in Honour of David F. Hendry. Oxford

University Press, Oxford, 173–205.

Stock, J. H., Watson, M. W., 2002. Forecasting using principal components from a

large number of predictors. Journal of the American Statistical Association 97 (460),

1167–1179.

34



A.1 Appendix

Let X = (X1, ..., XT )′ be the T ×N matrix of observations, and let VN T be the r × r

diagonal matrix of the r largest eigenvalues of (N T )−1X X ′ in decreasing order. By

the definition of eigenvalues and eigenvectors, we have (N T )−1X X ′F = FVN T or

(N T )−1X X ′FV −1N T = F , where F ′F /T = Ir . Let H = (Λ0′Λ0/N )(F ′F /T )V −1

N T be the r ×r

rotation matrix. Assumption A and B together with Lemma A.1 below implies that

‖H‖ =Op (1). Let wt = ξt Ft . We can write (1.1) as:

X t =Λ0Ft +ξt Ft +et =Λ0Ft +wt +et .

Define e = (e1, ...,eT )′ and w = (w1, ..., wT )′. We use the following expression from

Bates et al. (2013):

X X ′ = FΛ0′Λ0F ′+FΛ0′(e +w)′+ (e +w)Λ0F ′+ (e +w)(e +w)′. (A.1)

Let vt denote a conforming unit vector with zeros in all entries except the t th . We

then have:

X X ′v = FΛ0′Λ0Ft +FΛ0′(et +wt )+ (e +w)Λ0Ft + (e +w)(et +wt ).

Using the definition of Ft and H , we can then write:

Ft −H ′Ft =V −1N T (N T )−1F ′X X ′v −V −1

N T (F ′F /T )(Λ0′Λ0/N )Ft

=V −1N T (N T )−1

F ′FΛ0′et + F ′eΛ0Ft + F ′eet

+F ′FΛ0′wt + F ′wΛ0Ft + F ′w wt + F ′ewt + F ′wet

.

Denote each term on the right-hand as A1t , ..., A8t , respectively. We get:

Ft −H ′Ft =V −1N T

8∑n=1

Ant . (A.2)

The following is a generalization of Lemma A.3 in Bai (2003). They consider constant

loadings; we generalize the proof to autoregressive loadings.

Lemma A.1. Under Assumptions A-E, as N,T →∞:

(i)∥∥VN T − F ′F

TΛ0′Λ0

NF ′F

T

∥∥2 =Op (C−2N T ),

(ii) F ′FT

Λ0′Λ0

NF ′F

T

p→V ,

where V is the diagonal matrix consisting of the eigenvalues of ΣΛΣF .

A.1. APPENDIX 35

Proof. From VN T = T −1F ′(N T )−1X X ′F we get using (A.1):

VN T − F ′FT

Λ0′Λ0

N

F ′FT

= T −1F ′(N T )−1

FΛ0′(e +w)′ +(e +w)Λ0F ′+ (e +w)(e +w)′)

F

= T −1T∑

t=1Ft

8∑n=1

A′nt .

Hence,

‖T −1T∑

t=1Ft

8∑n=1

A′nt‖2 ≤

(T −1

T∑t=1

‖Ft‖2

)(T −1

T∑t=1

∥∥∥∥ 8∑n=1

Ant

∥∥∥∥2)

≤ 8r T −1T∑

t=1

8∑n=1

‖Ant‖2,

where the last inequality uses tr (F′F /T ) = tr (Ir ) = r and Loève’s inequality. The right-

hand side is Op (C−2N T ) by Theorem 1 of Bates et al. (2013).11 Statement (i) follows.

Statement (ii) is implicitly proven by Stock and Watson (1998). It should be

noted that their paper considers the model X t =Λ0Ft + et , i.e. a factor model with

constant loadings. However, their proof only uses the asymptotic representation

VN T = F ′FT

Λ0′Λ0

NF ′F

T + op (1) and the normalization F ′F /T = Ir . Their proof is thus

applicable for our model as well.

Proof of Lemma 1. From (A.2) we have:

T −1T∑

t=1‖Ft −H ′Ft‖2 ≤ ‖V −1

N T ‖28T −1T∑

t=1

8∑n=1

‖Ant‖2.

Since VN T converges to a positive definite matrix, it follows that ‖V −1N T ‖2 =Op (1). The

right-hand side is thus Op (C−2N T ) by Theorem 1 in Bates et al. (2013).

Proof of Proposition 1. Using (A.2) we have:

maxt

‖Ft −H ′Ft‖ = maxt

‖V −1N T

8∑n=1

Ant‖ ≤ ‖V −1N T ‖

8∑n=1

maxt

‖Ant‖.

Lemma A.1 implies that ‖V −1N T ‖ =Op (1). We can write A1t as:12

A1t = (N T )−1T∑

s=1(Fs −H ′Fs )F ′

sΛ0′et + (N T )−1

T∑s=1

H ′Fs F ′sΛ

0′et .

11Our Assumption D.3 differs from the corresponding assumption in Bates et al. (2013). They as-sume that

∑Ts,t=1

∑Ni , j=1 |E (ξi sp1ξ j sq1ξi t p2ξ j t q2 Fsp1 Fsq1 Ft p2 Ft q2 )| is bounded by the envelope function

Q3(N ,T ), with C 2N T Q3(N ,T ) = O(N 2T 2). This is implied by Assumption D.3, and their Theorem 1 thus

holds with Assumption A-E.12The terms A1t , A2t , A3t have been shown to be Op (αT T−1)+Op (T 1/8)N−1/2 by Bai and Ng (2008a).

They do, however, rely on intermediate results, which we have not proved for the model with time-varyingloadings. We therefore provide an alternative proof for A1t , A2t , A3t .

36



The first term is less than:(T −1

T∑s=1

‖Fs −H ′Fs‖2

)1/2 (N−2T −1

T∑s=1

‖F ′sΛ

0′et‖2

)1/2

.

We have:

N−2T −1T∑

s=1‖F ′

sΛ0′et‖2 ≤ N−1‖N−1/2Λ0′et‖2T −1

T∑s=1

‖Fs‖2.

By Assumption F.3, the maximum of ‖N−1/2Λ0′et‖2 over t is Op (T 1/4), and Assump-

tion A implies T −1 ∑Ts=1 ‖Fs‖2 =Op (1). By Lemma 1, we have T −1 ∑T

s=1 ‖Fs −H ′Fs‖2 =Op (C−2

N T ). Taking the square root then gives that the first term is Op

(C−1

N T

)Op

(T 1/8

N 1/2

).

For the second term, we have:

(N T )−1‖T∑

s=1H ′Fs F ′

sΛ0′et‖ ≤ N−1/2‖H‖‖N−1/2Λ0′et‖T −1

T∑s=1

‖Fs‖2,

where ‖H‖ = Op (1) and T −1 ∑Ts=1 ‖Fs‖2 = Op (1) by Assumption A. The maximum

of ‖N−1/2Λ0′et‖ over t is Op (T 1/8). The second term is thus equal to Op

(T 1/8

N 1/2

)and

dominates the first.

Consider A2t , which can be written as:

(N T )−1T∑

s=1(Fs −H ′Fs )e ′sΛ

0Ft + (N T )−1T∑

s=1H ′Fs e ′sΛ

0Ft .

The first term is bounded by(T −1

T∑s=1

‖F −H ′Fs‖2

)1/2 (N−2T −1

T∑s=1

‖e ′sΛ0Ft‖2

)1/2

.

Now,

N−2T −1T∑

s=1‖e ′sΛ

0Ft‖2 ≤ maxt

‖Ft‖2N−1T −1T∑

s=1‖N−1/2e ′sΛ

0‖2 =Op (α2T )N−1

by Assumption F.3. The first term is thus equal to Op (C−1N TαT N−1/2). The second term

is equal to:

(N T )−1T∑

s=1

N∑i=1

H ′Fs ei sλ0′i Ft ,

which is bounded by:

N−1/2Mmaxt

‖Ft‖‖H‖(

T −1T∑

s=1‖Fs‖2

)1/2(N T )−1

T∑s=1

N∑i , j=1

ei s e j s

1/2

.

This is equal to Op (αT )N−1/2 by Assumption C.3 and dominates the first term.

A.1. APPENDIX 37

We can write A3t as:

(N T )−1T∑

s=1(Fs −H ′Fs )[e ′s et −E(e ′s et )]+ (N T )−1

T∑s=1

H ′Fs [e ′s et −E(e ′s et )]

+ (N T )−1T∑

s=1(Fs −H ′Fs )E(e ′s et )+ (N T )−1

T∑s=1

H ′Fs E(e ′s et ).

The first term is bounded by:

(T −1

T∑s=1


)1/2N−1T −1

T∑s=1

∣∣∣∣∣N−1/2N∑

i=1[e ′i s ei t −E(e ′i s ei t )]

∣∣∣∣∣21/2

.

By Assumption C.5, maxt

∣∣∣N−1/2 ∑Ni=1[e ′i s ei t −E(e ′i s ei t )]

∣∣∣2 =Op (p

T ), so the first term

is equal to Op (C−1N T )Op ( T 1/4

N 1/2 ). The second term is bounded by:

(N T )−1/2‖H‖‖(N T )−1/2T∑

s=1

N∑i=1

Fs [e ′i s ei t −E(e ′i s ei t )]‖.

By Assumption F.2, the maximum of this expression over t is Op (N−1/2). The third

term is bounded by:

T −1/2

(T −1

T∑s=1


)1/2 (T∑

s=1γN (s, t )2

)1/2

.

By Assumption F.1 and Lemma 1, this is equal to T −1/2Op (C−1N T ). The fourth term is

bounded by:

T −1maxt

‖Ft‖‖H‖T∑

s=1|γN (s, t )|,

which is Op (αT T −1).

For A4t , we have:

(N T )−1‖F ′FΛ0′wt‖ ≤ ‖T −1/2F‖‖T −1/2F‖‖N−1Λ0′wt‖.

The first two terms are both Op (1). The last term can be bounded in expectation:

E

∥∥∥∥Λ0′wt

N

∥∥∥∥2

≤ N−2N∑

i , j=1|E(wi t w j t )λ0′

i λ0j | ≤ M 2N−2

N∑i , j=1

|E(ξi t Ftξ j t Ft )|

≤ M 2r 2N−2supp.q

N∑i , j=1

|E(ξi t pξ j t q Ft p Ft q )| =Op (N−1)

uniformly in t by Assumption D.1, so the maximum of the last term over t is Op (N−1/2).

38



Consider A5t :

(N T )−1‖T∑

s=1Fs w ′

sΛ0Ft‖ ≤ max

t‖Ft‖

(T −1

T∑s=1

‖Fs‖2

)1/2 (T −1

T∑s=1

‖N−1w ′sΛ

0‖2

)1/2

.

By Assumption D.1, this is equal to Op (αT N−1/2).

For A6t , we have:

(N T )−1‖T∑

s=1Fs w ′

s wt‖ ≤ N−1T −1/2

(T −1

T∑s=1

‖Fs‖2

)1/2 (T∑

s=1‖w ′

s wt‖2

)1/2

.

By Assumption D.3, we have:

T∑s=1

E(w ′s wt )2 =

T∑s=1

N∑i , j

E(wi s wi t w j s w j t )

≤ r 4 supp1,p2,q1,q2

T∑s=1

N∑i , j

|E(ξi sp1ξ j sq1ξi t p2ξ j t q2 Fsp1 Fsq1 Ft p2 Ft q2 )| =O(N 2)+O(N T )

uniformly in t . We therefore have that maxt

‖A6t‖ = N−1T −1/2[Op (N )+Op (N 1/2T 1/2)] =Op (C−1

N T ).

The seventh term is bounded by:

(T −1

T∑s=1

‖Fs‖2

)1/2 (N−2T −1

T∑s=1

‖e ′s wt‖2

)1/2

.

The first term is O(1). We can bound the second term in expectation:

N−2T −1T∑

s=1E‖e ′s wt‖2 = N−2T −1

T∑s=1

N∑i , j=1

E(ei s e j s )E(wi t w j t )

≤ N−2T −1T∑

s=1

N∑i , j=1

E(e2i s )1/2E(e2

j s )1/2|E(wi t w j t )|

≤ Mr 2N−2T −1supp,q

T∑s=1

N∑i , j=1

|E(ξi ptξ j qt Fpt Ft q )|,

which is Op (N−1) uniformly in t by Assumption D.1. Taking the square root then

gives Op (N−1/2).

Finally, A8t is bounded by:

(N T )−1‖F ′wet‖ =(

T −1T∑

s=1‖Fs‖2

)1/2 (N−2T −1

T∑s=1

‖w ′s et‖2

)1/2

.

A.1. APPENDIX 39

The first term is again O(1), and the last term can be bounded in expectation:

N−2T −1T∑

s=1E‖w ′

s et‖2 = N−2T −1T∑

s=1

N∑i , j=1

E(ei t e j t )E(wi s w j s )

≤ N−2T −1T∑

s=1

N∑i , j=1

E(e2i t )1/2E(e2

j t )1/2|E(wi s w j s )|

≤ Mr 2N−2T −1supp,q

T∑s=1

N∑i , j=1

|E(ξi psξ j qs Fps Fsq )|,

which is Op (N−1) uniformly in t by Assumption D.1. The last term is thus Op (N−1/2).

All terms are dominated by Op ( T 1/8

N 1/2 )+Op (αT N−1/2)+Op (αT T −1)+Op (C−1

N T ), and

Proposition 1 follows.

Lemma A.2. Let Assumption A-E hold. If F ′F /T = Ir and Λ0′Λ0 is a diagonal

matrix with distinct entries,

H = Ir +Op (C−2N T ).

Proof. First we need to show that (F − F H)′F /T and (F − F H)′F /T are both

Op (C−2N T ). We have:

‖(F −F H)′F /T ‖2 = ‖T −1T∑

t=1(Ft −H ′Ft )F ′

t‖2

≤(

T −1T∑

t=1‖Ft −H ′Ft‖2

)(T −1

T∑t=1

‖Ft‖2

)=Op (C−2

N T ),

where the last equality follows from Lemma 1 and Assumption A. By similar argu-

ments (F −F H)′F /T = Op (C−2N T ). The rest of the proof is identical to the proof of

equation (2) in Bai and Ng (2013).

Lemma A.2 shows that if the imposed normalization holds for the process gen-

erating the data, the factors can be estimated without rotation. This implies that

θi can be estimated without rotation as well. In the proof of Theorem 1 below, we

assume that H = Ir , and note that in general, the feasible likelihood converges to

LT (Xi |F H ;θi ), and θi is consistent for a rotation of θ0i as discussed in Section 1.3.2.

Proof of Theorem 1. It suffices to show that the feasible likelihood function

LT (Xi |F ;θi ) converges uniformly to the infeasible one LT (Xi |F ;θi ).13 This will imply

13Pointwise convergence would suffice as Assumption H(iii) requires convergence for all θi ∈ Θi .However, there are no additional difficulties in showing uniform convergence, and we therefore proveconvergence uniformly inΘi .

40



that LT (Xi |F ;θi ) satisfies the conditions of Assumption H and θip→ θ0

i . We thus need:

supθ∈Θ

∣∣∣LT (Xi |F ;θi )−LT (Xi |F ;θi )∣∣∣ p→ 0.

By the mean value expansion, we can write:

LT (Xi |F ;θi ) =LT (Xi |F ;θi )+T∑

t=1∇Ft LT (Xi |F∗;θi )(Ft −Ft ),

where ∇Ft LT (Xi |F∗;θi ) = ∂LT (Xi |F ;θi )∂Ft

∣∣∣F=F∗ , and F∗ is between F and F . For uniform

convergence the last term needs to be op (1) uniformly in Θ, when F∗t is in a neigh-

bourhood of Ft , such that maxt

‖F∗t −Ft‖ = op (1).

Let λmax(A) and λmin(A) denote the largest and smallest eigenvalue of a matrix A,

and let (A)(s,t ) denote entry (s, t ) of a T ×T matrix A. Furthermore, let φi be the r × r

block matrix on the diagonal ofΦi , i.e. φi =V ar (λi t ). The derivative of LT (Xi |F ;θi )

takes the form:14,15

∇Ft LT (Xi |F ;θi )′ =−T −1φi FtΣ−1i ,(t ,t ) +T −1λi

T∑s=1

(Xi s −F ′sλi )Σ−1

i ,(s,t )

+T −1φi Ft

(Σ−1

i (Xi −E(Xi ))(Xi −E(Xi ))′Σ−1i

)t ,t

,

where Ft is to be evaluated at F∗t . Denote the three terms above by Bnt , for n = 1, ...,3.

We can then write:

supθ∈Θ

∣∣∣LT (Xi |F ;θi )−LT (Xi |F ;θi )∣∣∣= sup

θ∈Θ

∣∣∣∣∣ T∑t=1

3∑n=1

(Ft −Ft )′Bnt

∣∣∣∣∣≤ sup

θ∈Θ

∣∣∣∣∣ T∑t=1

(Ft −Ft )′B1t

∣∣∣∣∣+ supθ∈Θ

∣∣∣∣∣ T∑t=1

(Ft −Ft )′B2t

∣∣∣∣∣+ supθ∈Θ

∣∣∣∣∣ T∑t=1

(Ft −Ft )′B3t

∣∣∣∣∣ .

(A.3)

For the term involving B1t , we have:∣∣∣∣∣T −1T∑

t=1(Ft −Ft )′φi F∗

t Σ−1i ,(t ,t )

∣∣∣∣∣≤λmax(Σ−1)T −1T∑

t=1‖Ft −Ft‖‖φi‖‖F∗

t ‖, (A.4)

since each entry in Σ−1i is bounded by the largest eigenvalue. For the largest eigen-

value ofΣ−1i , we have λmax(Σ−1

i ) = [λmin(Σi )]−1, and it therefore follows from the Weyl

inequality that λmax(Σ−1i ) ≤ M as:16

λmin(Σi ) ≥λmin(FΦi F′)+λmin(ψi IT ) ≥ψi > 0

14The calculations of the derivative are omitted for brevity. They are available upon request.15With autocorrelated errors, the derivative takes the same form, but the variance matrix is Σi =

FΦi F ′+Ψi , whereΨi = E(ei e′i ) is non-diagonal.16This also holds with Ψi = E(ei e′i ) non-diagonal, as we can bound the smallest eigenvalue of Σi

uniformly iΘi .

A.1. APPENDIX 41

uniformly inΘi . The term ‖φi‖ is also uniformly bounded, as the parameters of Bi (L)

are in the stationary region, and the elements of Qi are bounded. We can therefore

bound (A.4) by:

O(1)T −1T∑

t=1‖Ft −Ft‖‖F∗

t −Ft‖+O(1)T −1T∑

t=1‖Ft −Ft‖‖Ft‖.

Since F∗t is between Ft and Ft , the first term is less than T −1 ∑

t ‖Ft −Ft‖2 and is

Op (C−2N T ) by Lemma 1. Note that T −1 ∑

t ‖Ft −Ft‖2 does not depend on θi , and the

result is thus uniform inΘi . For the second term, we can write:

T −1T∑

t=1‖Ft −Ft‖‖Ft‖ ≤

(T −1

T∑t=1

‖Ft −Ft‖2

)1/2 (T −1

T∑t=1

‖Ft‖2

)1/2

,

which is Op (C−1N T ) by Lemma 1 and Assumption A, also uniformly inΘi .

For the term involving B3t in (A.3), we can write:∣∣∣∣∣T −1T∑

t=1(Ft −Ft )′φi F∗

t

(Σ−1


)t ,t

∣∣∣∣∣≤max

t

∣∣∣(Ft −Ft )′φi F∗t

∣∣∣T −1

∣∣∣∣∣ T∑t=1

(Σ−1


)t ,t

∣∣∣∣∣ .

For the term outside the sum, we have:

maxt

∣∣∣(Ft −Ft )′φi F∗t

∣∣∣≤ ‖φi‖maxt

‖Ft −Ft‖‖F∗t ‖

≤O(1)maxt

‖Ft −Ft‖2 +O(1)maxt

‖Ft −Ft‖‖Ft‖.

If we take Ft to be a sequence of fixed and bounded constants, maxt

‖Ft‖ ≤ M , and

the second term is then op (1) by Proposition 1, which is uniform inΘi as the proof of

Proposition 1 does not depend on θi . The first term is bounded by the second.

The term involving the sum can be written as

T −1

∣∣∣∣∣ T∑t=1

(Σ−1


)t ,t

∣∣∣∣∣= T −1

∣∣∣∣tr(Σ−1


)∣∣∣∣ ,

(A.5)

which is bounded by

λmax(Σ−2)T −1|tr(Xi −E(Xi ))(Xi −E(Xi ))′| ≤ M 2T −1T∑

t=1‖Xi t −F∗′

t λi‖2

≤ 4M 2T −1T∑

t=1

(‖F ′

tλ0i ‖2 +‖F ′

t (λi t −λ0i )‖2 +‖ei t‖2 +‖F∗′

t λi‖2)

.

42



The first term in the sum is bounded by T −1M 2 ∑Tt=1 ‖Ft‖2 =Op (1). For the second

term in the sum, we can write:

T −1T∑

t=1‖F ′

t (λi ,t −λ0i )‖2 ≤

(T −1

T∑t=1

‖Ft‖4

)1/2 (T −1

T∑t=1

‖λi ,t −λ0i ‖4

)1/2

.

This is Op (1) by Assumption A and G. By Assumption C we have T −1 ∑Tt=1 e2

i t =Op (1),

and for the last term, we can write:

T −1T∑

t=1‖F∗′

t λi‖2 ≤ M 2T −1T∑

t=1‖F∗′

t −Ft‖2 +M 2T −1T∑

t=1‖Ft‖2 =Op (C−2

N T )+Op (1),

as λi is estimated in a bounded parameter space. The second term in (A.3) is thus

maxt

‖Ft −Ft‖Op (1) = op (1) uniformly inΘi .

For the term involving B2t in (A.3), we can write:∣∣∣∣∣T −1T∑

t=1(Ft −Ft )′λi

T∑s=1

(Xi s −F∗′s λi )Σ−1

i ,(s,t )

∣∣∣∣∣≤

(T −1

T∑t=1

|(Ft −Ft )′λi |2)1/2

T −1T∑

t=1

∣∣∣∣∣ T∑s=1

(Xi s −F∗′s λi )Σ−1

s,t

∣∣∣∣∣21/2

.

The first term in parentheses is less than M 2T −1 ∑Tt=1 ‖Ft −Ft‖2 =Op (C−2

N T ) uniformly

inΘi . The second term in parentheses is equal to

T −1∣∣∣∣tr

(Σ−1


)∣∣∣∣ ,

which is Op (1) uniformly in Θi from the arguments above, see (A.5). By taking the

square root, the second term is thus Op (C−1N T ) and dominated by the third. Collecting

the results gives:

supθ∈Θ

∣∣∣LT (Xi |F ;θi )−LT (Xi |F ;θi )∣∣∣=Op

(max

t‖Ft −Ft‖

)= op (1).

C H A P T E R 2TESTING FOR TIME-VARYING LOADINGS IN

FACTOR MODELS


Abstract

In this paper we develop a test for time-varying factor loadings in factor models. The

test is simple to compute and is constructed from estimated factors and residuals

using the principal components estimator. The hypothesis is tested by regressing the

squared residuals on the squared factors. The squared correlation coefficient times

the sample size has a limiting χ2 distribution. The test can be made robust to serial

correlation in the idiosyncratic errors. We find evidence for factor loadings variance

in over half of the variables in a dataset for the US economy, while there is evidence

of time-varying loadings on the risk factors underlying portfolio returns for around

80% of the portfolios.

43

44 CHAPTER 2. TESTING FOR TIME-VARYING LOADINGS IN FACTOR MODELS

2.1 Introduction

Large factor models have become an important tool for analysing and forecasting

large economic datasets spanning several hundred of variables. Factors extracted

from large panels of disaggregated macroeconomic variables explain a large part

of the comovement in the series. Estimated factors are successful in summarizing

the predictive content in large datasets when used in predictive regression, see e.g.

Stock and Watson (2002). Examples of structural analysis using factor models are

Bernanke, Boivin, and Eliasz (2005) and Giannone, Reichlin, and Sala (2006). In

financial applications, an underlying factor structure is often assumed, and factors

can be extracted from portfolios of asset returns to form risk factors that separate

the returns into to systematic and non-systematic risk. When the data spans over

long periods, often several decades, the factor loadings are likely to be exhibit some

instability. We propose a simple test for time-varying loadings in factor models.

The test is constructed using principal components estimates of the common

factors and residuals. From a regression of the squared residuals on the squared

factors we obtain the test statistics as the squared correlation coefficient R2 times the

sample size T . Under the null hypothesis of constant factor loadings, the test statistic

has a limiting χ2 distribution with degrees of freedom equal to the number of factors.

The result is based on the observation that a factor model with time-varying factor

loadings can be written as xi t =λ′i t Ft +ei t =λ′

i Ft +ξ′i t Ft +ei t . The observed data for

variable i at time t is xi t , Ft is the vector of common factors, λi t is the time-varying

factor loadings, and ei t is the idiosyncratic errors. The variable ξi t = λi t −λi is the

stationary variations in the factor loadings around the constant λi . The principal

components estimator estimates the factors Ft and the constant λi , and the residuals

are thus an estimate of ξ′i t Ft +ei t , which we denote ui t . In the single factor case, the

second moment of ui t is E(u2i t ) = E(ξ2

i t )E(F 2t )+E(e2

i t ). A regression of the squared

residuals on the squared factors will thus give an estimate of the variance of the factor

loadings E(ξ2i t ). When the loadings are constant, the R2 from this regression is close

to zero, and a large R2 is evidence of variation in the factor loadings.

Under the condition that the number of variables N satisfies T /N 2 → 0, the

estimation error in the factors does not affect the limiting distribution of the test

statistics. In the analysis of the test statistic, the idiosyncratic errors are assumed

to be white noise. Serial correlation in the errors can be controlled for by basing

the test statistic on GLS estimation of the factor model suggested by Breitung and

Tenhofen (2011), and we show that the GLS version of the test has the correct size in

our simulations.

A related test is the Chow test of Breitung and Eickmeier (2011) for structural

breaks in the factor loadings. They consider a factor model in which the factor load-

ings are λ1 for t = 1, ...t∗, and λ2 for t > t∗. They find evidence of structural breaks

in a large number of variables for both US and European datasets. Eickmeier et al.

(2015) and Del Negro and Otrok (2008) suggest factor models in which the factor

2.2. TESTING FOR TIME-VARYING LOADINGS 45

loadings are time-varying.

We consider two empirical applications of our testing procedure. We use the

dataset of McCracken and Ng (2015) for the US economy, and we apply our testing

procedure for different choices for the number of factors. We find evidence of time-

varying factor loadings in over half of series, irrespective of the number of included

factors. In the second application, we consider excess returns on 100 portfolios sorted

on size and book-to-market. We find that around 80% of the portfolios are associated

with time-varying factor loadings, indicating that the portfolios have time-varying

exposures to the risk factors. Furthermore, the estimated factors are closely related to

the three risk factors of Fama and French (1993). The squared canonical correlations

between the estimated factors and the Fama French factors are all larger than 0.90.

The rest of the paper is organized as follows. In Section 2.2 we introduce the factor

model and the test statistic. Section 2.3 states the assumption for the data-generating

process and the main result on the limiting distribution of the test statistic. In Section

2.4, a Monte Carlo study shows the finite samples properties of the test statistic. In

Section 2.5 we report results for the empirical applications. Section 2.6 concludes.

2.2 Testing for time-varying loadings

Let Xi t denote the observed data at time t = 1, ...,T for observation i = 1, ..., N . We

consider a factor model with r common factors and time-varying factor loadings:


where Ft = (F1t , ...,Fr t )′ is the r -dimensional vector of common factors, and ei t is the

idiosyncratic error. We assume the factor loadings to be stationary and define the

variable:

ξi t =λi t −λi ,

where λi = E (λi t ) is the mean value of the loadings, and ξi t is a mean-zero stationary

random process. The variable ξi t separates the time-varying loadings into a constant

part and a time-varying part. The factor model can be written as:

Xi t =λ′i Ft +ξ′i t Ft +ei t . (2.1)

If the factor loadings are constant over time, the variable ξi t will be zero for all t with

zero variance, whereas in the time-varying case, ξi t will have a non-zero variance,

E(ξ2i t ) 6= 0. Under the null hypothesis we therefore assume:

H0 : E(ξ2i t ) = 0,

and we construct a test that can detect non-zero variances of ξi t , which corresponds

to time-varying factor loadings. To test the null hypothesis, we form a test statistic

using estimated factors and residuals. A regression of the squared residuals on the


squared factors gives the test statistic T R2i , where R2

i is the squared correlation co-

efficient from the regression. The test statistic has a χ2 limiting distribution with r

degrees of freedom under the null of constant loadings.

To develop some intuition for the test, consider the case of a single observed

factor, r = 1. Define the vectors F = (F1, ...,FT )′ and Xi = (Xi 1, ..., Xi T )′. As we assume

in the next section, the sample average of the squared factors converges to a positive

definite matrix, T −1 ∑t F 2

tp→ΣF > 0. Since we can observe the factors, we can consider

the OLS estimator for the loadings λi = (F ′F )−1F ′Xi . From (2.1), we have:

λi ≈λi +Σ−1F T −1

T∑t=1

F 2t ξi t +Σ−1

F T −1T∑

t=1Ft ei t .

When ξi t and ei t both have limited serial dependence, the last two terms will con-

verge to zero in probability, as both E(ξi t ) = 0 and E(ei t ) = 0. The residuals from

this regression are therefore an estimate of ξi t Ft +ei t . Define ui t := ξi t Ft +ei t , and

consider the second moment of ui t :

E(u2i t ) = E(ξi t Ft +ei t )2 = E(ξ2

i t )E(F 2t )+E(e2

i t ).

With time-varying factor loadings, the loadings variance is non-zero, E(ξ2i t ) 6= 0, and

u2i t and F 2

t will therefore be correlated. This observation shows that a regression of

u2i t on F 2

t and a constant can be used to test for time-varying loadings, because the

coefficient on F 2t will be an estimate of the variance of the loadings. Under the null,

E (ξ2i t ) = 0, and the residuals ui t will be equal to ei t . A large value of the test statistic is

therefore evidence of time-varying factor loadings.

In practice, the test statistic is constructed using estimates of the unobservable

quantities Ft , λi , and ei t . The principal components estimator gives estimates of

the factors and the constant part of the loadings (Ft , λi ), as well as estimates of the

idiosyncratic components, ei t = Xi t − λ′i Ft . The definition of the estimator is stated

in the Appendix. The test statistic is obtained as T R2i from the regression of e2

i t on the

squared principal components and a constant, and we denote this statistic as LMi .

The test statistic can be written as:

LMi = T D ′i B−1

i Di ,

with

Di = T −1T∑

t=1(e2

i t − σ2i )g (Ft F ′

t − F ′F /T ),

Bi = T −1T∑

t=1[e2

i t − σ2i ]2T −1

T∑t=1

g (Ft F ′t − F ′F /T )g (Ft F ′

t − F ′F /T )′.

where g (A) denotes the column vector of diagonal elements of a square matrix A,

and σ2i = T −1 ∑T

t=1 e2i t is the estimator for the variance of ei t . Theorem 1 in the next

2.3. ASSUMPTIONS 47

section states that the LMi statistic has a χ2 limiting distribution with r degrees of

freedom under the null hypothesis of constant factor loadings. The LMi test has

power against covariance-stationary forms of variation in the factor loadings, with

stationary autoregressions being a leading example. We assume that the number of

factor r is known. In practice, the number of factors can be estimated consistently

under the null of constant loadings, e.g. by the information criteria of Bai and Ng

(2002).

2.3 Assumptions

To establish the limiting distribution of the test statistic, we make a similar set of

assumptions as in Bai (2003). Let ‖A‖ = [tr (A′A)]1/2 denote the norm of matrix A.

The constant M ∈ (0,∞) is common to all assumptions below.

Assumption A. E‖Ft‖4 ≤ M < ∞, and T −1 ∑Tt=1 Ft F ′

tp→ ΣF for some r × r positive

definite matrix ΣF .

Assumption B. ‖λi‖ ≤ λ<∞, and ‖Λ′Λ/N −ΣΛ‖→ 0 for some positive definite ma-

trix ΣΛ.

Assumption C. There exists a positive constant M <∞ such that for all N and T :

1. E(ei t ) = 0, E |ei t |8 ≤ M .

2. E(e ′s et /N ) = E(N−1 ∑Ni=1 ei s ei t ) = γN (s, t ), |γN (s, s)| ≤ M for all s, and∑T

s=1 |γN (s, t )| ≤ M for all t .

3. E(ei t e j t ) = τi j ,t with |τi j ,t | ≤ |τi j | for some τi j and for all t . In addition∑Nj=1 |τ j i | ≤ M for all i .

4. E(ei t e j s ) = τi j ,t s , and (N T )−1 ∑Ni , j=1

∑Tt ,s=1 |τi j ,t s | ≤ M .

5. For every (s, t ), E |N−1/2 ∑Ni=1[ei s ei t −E(ei s ei t )]|4 ≤ M .

Assumption D. Ft is independent of ei s for all (i , t , s).

Assumption E. There exists a positive constant M <∞ such that for all N and T :

1. For each t ,

E‖(N T )−1/2T∑

s=1

N∑k=1

Fs [eks ekt −E(eks ekt )]‖2 ≤ M ,


2. For each t ,

E

∥∥∥∥ 1pN

N∑i=1

λi ei t

∥∥∥∥4

≤ M ,

3. For each i ,

E

∥∥∥∥ 1pT

T∑t=1

Ft ei t

∥∥∥∥2

≤ M ,

4. For each i ,

E

∥∥∥∥ 1pT

T∑t=1

ei t (Ft F ′t ⊗F ′

t )

∥∥∥∥2

≤ M .

Assumption F. The eigenvalues of the r × r matrix (ΣΛ ·ΣF ) are distinct.

Assumptions A and B imply the existence of r common factors. Assumption C al-

lows the idiosyncratic errors to exhibit limited serial correlation and cross-sectional

dependence. If ei t are i.i.d., Assumption C is satisfied. Assumption D requires the

factors and idiosyncratic errors to be independent, but dependence within groups is

allowed. In particular, Ft can be serially correlated. Assumptions A-D permit consis-

tent estimation of the factor space H ′Ft by the principal components estimator Ft ,

where H is an invertible matrix. The moment conditions in Assumption E are similar

to Assumption F in Bai (2003). Assumption F ensures that the rotation matrix H has a

unique limit. To study the limit distribution of the test statistic, we impose additional

assumptions on the idiosyncratic errors, as well as a central limit theorem.

Assumption G.

1. For all t , E(ei t ) =σ2i , E(e4

i t ) =µ4,i , and ei t and ei s are independent for t 6= s.

2. For each i , as T →∞,

B−1/2i

pT Di

d→ N (0, Ir ),

where

Di = T −1T∑

t=1(e2

i t −σ2i )g [H ′(Ft F ′

t −F ′F /T )H ],

and the asymptotic covariance matrix ofp

T Di is:

Bi = plimT→∞T −1T∑

t=1E

[(e2

i t−σ2i )2g [H ′(Ft F ′

t−F ′F /T )H ]g [H ′(Ft F ′t−F ′F /T )H ]′

]> 0.

Assumption G implies that the limiting distribution of the infeasible test statistic

T Di B−1i D ′

i is χ2 with r degrees of freedom. Since the principal components estimator

Ft is consistent for a rotation of the factors H ′Ft , the rotation matrix H appears in the

limiting distribution. Under Assumptions A-D and F, Bai (2003) show that Hp→Q−1,

where Q−1 is an invertible matrix, so H can be replaced by Q−1 in Assumption G.

2.3. ASSUMPTIONS 49

However, since the test statistic is based on the R2 from the regression of the squared

residuals on the factors, the test statistic is invariant to the scaling of the factors, and

the limiting distribution is therefore not affected by rotations of the factors.

The null distribution of the LMi test statistic is presented in the following theo-

rem.

Theorem 1. Under Assumptions A-G and if N ,T →∞ andp

T /N → 0, the statistic

LMi has a limiting χ2 distribution with r degrees of freedom for each i .

The proof is presented in the Appendix. Theorem 1 states that the LMi statistic based

on the principal components estimates Ft and ei t has the same limiting distribution

as the statistic based on the population quantities Ft and ei t . The result follows

since Di −Di =Op (δ−2N T ), where δN T = mi n

pN ,

pT . The rate condition

pT /N → 0

ensures that the estimation error in Ft does not affect the limiting distribution of

the test statistic, andp

T Di therefore has the same limiting distribution asp

T Di .

Under Assumption G.1, the asymptotic variance Bi can be consistently estimated by

Bi , and the LMi statistics therefore has the same limiting distribution as the statistic

obtained from the infeasible regression that uses Ft and ei t instead of ei t and Ft .

2.3.1 Serially correlated errors

The limiting distribution of the LMi test statistic is derived under the assumption

that ei t is i .i .d . (Assumption G.1), while Ft is allowed to exhibit serial correlation.

In practice, the idiosyncratic errors can be serially correlated if the factors do not

adequately capture the serial correlation in the data. When both the idiosyncratic

errors ei t and the factors Ft are serially correlated, the asymptotic covariance matrix

Bi is not valid. If ei t exhibits serial correlation, e2i t will also be serially correlated, and

we have:

E

[g [H ′(Ft F ′

t −F ′F /T )H ](e2i t −σ2

i )(e2i s −σ2

i )g [H ′(Fs F ′s −F ′F /T )H ]′

]6= 0 for t 6= s.

Instead the covariance matrix ofp

T Di takes the form:

B∗i = plimT→∞T −1

T∑t=1

T∑s=1

E [g [H ′(Ft F ′t−F ′F /T )H ](e2

i t−σ2i )(e2

i s−σ2i )g [H ′(Fs F ′

s−F ′F /T )H ]′].

The consequence is that the size of the test will be affected. Assumptions A-E are

sufficient to ensure that Di converges to Di , and Di converges to zero even with

serially correlated errors. The asymptotic power of the test is unaffected as serial

correlation in ei t only affects the asymptotic covariance matrix ofp

T Di .

To improve the size properties of the LMi test when Assumption G.1 is violated,

the test can be based on GLS estimation of the factor model. By fitting an auxiliary

model to the residuals, we can capture the idiosyncratic dynamics in the errors.

The estimated dynamics can then be used to perform a GLS transformation of the


factor model. The GLS residuals will resemble white noise if the auxiliary model for

the errors captures the idiosyncratic dynamics, and the LMi test based on the GLS

transformed model will therefore have better size properties.

We follow Breitung and Tenhofen (2011) and specify individual-specific AR(pi )

models for the idiosyncratic components. After obtaining the initial principal compo-

nents estimates ei t of the idiosyncratic errors, we estimate an AR(pi ) model for the

residuals by least squares:

ei t = ρi ,1ei ,t−1 + ...+ρi ,pi ei ,t−pi+ vi t .

The individual-specific lag lengths pi can be determined by information criteria.

Denote the resulting lag polynomial by ρi (L). The GLS transformed model is:

ρi (L)xi t =λ′i [ρi (L)Ft ]+e∗i t ,

where Ft is the principal components estimator of the common factors. A new esti-

mate e∗i t of the residuals is obtained from least squares regression of the GLS trans-

formed model. These residuals are serially uncorrelated if the AR(pi ) model suffi-

ciently approximates the correlation structure in ei t . To test the null hypothesis of

constant loadings, the LMi statistic is constructed from the GLS residuals e∗i t and

the GLS transformed factors ρi (L)Ft . Using the GLS transformed model to construct

the LMi statistic therefore gives a test statistic that is robust to serial correlation

in the errors. The Monte Carlo simulations in the next section show that the GLS

based statistic has an actual size very close to the nominal in the presence of serially

correlated errors.

2.4 Small sample properties

We perform a Monte Carlo study to investigate the small sample properties of the test

statistic. The simulation design is as follows:


(1−bi p L)(λi t p −λi p ) = ηi t p ,

(1−αL)ei t = vi t ,

Ft p = ρFt−1,p +ut p ,

λi ∼ i.i.d. U(0,1),

ηi t ∼ i.i.d. N(0,σ2i p (1−b2

i p )),

vt ∼ i.i.d. N(0, (1−α2)IN ),

ut p ∼ i.i.d. N(0,1−ρ2),

where i = 1, ..., N , t = 1, ...,T , and p = 1, ...,r are factor and loadings indices. We omit

the subscript p when there is no ambiguity. The processes ut p , ηi t , and vt are mu-

tually independent. We generate time-variation in the loadings by simulating them

as AR(1)’s independent over i . The constant part of the loadings is λi ∼ i.i.d. U(0,1),

and the degree of variation is determined by the variance parameter σ2i p , which is the

unconditional variance of the factor loadings, σ2i p = E(ξ2

i pt ). The parameters α and

ρ determine the degree of serial correlation in the idiosyncratic errors and factors,

2.4. SMALL SAMPLE PROPERTIES 51

respectively. In our baseline simulations we set α= 0 and ρ = 0. We also consider the

effect of having α 6= 0 and ρ 6= 0, in which case Assumption G.1 is not fulfilled and the

asymptotic covariance matrix Bi is invalid. When computing results, we discard the

first 200 observations to avoid any dependence on initial values.

Table 2.1 shows the empirical sizes of the LMi statistic. The results in (a) and (b)

are for a model with one and two factors, respectively. The data is generated with

α= 0, ρ = 0, and σ2i ,p = 0, such that the loadings are constant, and the model satisfies

Assumptions A-G. The rejection frequencies are similar for all N and T . The empirical

sizes are close to the nominal size, but slightly undersized for the majority of the

sample sizes. The number of factors does not seem to affect the empirical size of the

test. The rejection frequencies are similar in (a) and (b).

Table 2.2 shows results for a factor model with serial correlation in factors and

the idiosyncratic errors, such that Assumption G.1 is violated. We set the parameters

to α= 0.5 and ρ = 0.9. Table 2.2 (a) reports the empirical sizes of the LMi test. The

test is seen to reject the null hypothesis too frequently when the factors and errors

are serially correlated. The empirical size is larger than the nominal for all N and

T , and the size distortion generally increases with T . The results in (b) are for the

GLS transformed model. We choose the order of the autoregression for the residuals

by AIC. The rejection frequencies are now much closer to the nominal size and the

empirical sizes are similar to those in Table 2.1. The GLS transformation thus works

well for correcting for serial correlation in the idiosyncratic errors.

To study the empirical power properties of the LMi test, we simulate the model

with time-varying factor loadings and different degrees of loading variance, E(ξ2i t ) =

σ2i . The AR-parameter of the loadings is set to bi = 0.9, and the variance parameter is

σ2i = (0.1,0.5,1,1.5). The other parameters are the same as in Table 2.1. Figure 2.1 plots

the empirical power of the test for the model with one factor and time-varying factor

loadings. Table 2.3 reports the corresponding rejection frequencies. The rejection

frequencies increase monotonically with σ2i for all combinations of N and T , so the

larger the difference between the null, E(ξ2i t ) = 0, and the alternative, the higher is

the empirical power. The sample size T is also seen to increase power. The rejection

frequencies increase with T for any fixed σ2i and N . The rejection rates approach

1 as T increases for all choices of loadings variance except for σ2 = 0.1. In the case

with σ2 = 0.1, the rejection rates are around 0.35 for T = 400. This is, however, a very

limited amount of variation in the loadings, and the difference between the null

hypothesis and the alternative is small. The cross-section size has a smaller impact

on the empirical power, but does tend to increase the rejection frequencies for a given

σ2 and T .

Finally, we repeat the simulations in Figure 2.1, but with serial correlation in errors

and factors, α= 0.5 and ρ = 0.9. Figure 2.2 shows the empirical power of the LMi test,

and Figure 2.3 shows the results for the GLS transformed model. The corresponding

tables of rejection frequencies are in Tables 2.4 and 2.5, respectively. Serial correlation


generally leads to lower rejection frequencies. For T = 50, ..,200, the empirical power

is lower for the LMi test compared to the results in Figure 2.1. For the largest sample

size, serial correlation has only a minor impact. The rejection frequencies are close

to one when N and T are large. When the LMi test is based on the GLS transformed

model, we see a further reduction in the rejection frequencies. The empirical power

in Figure 2.3 is lower for all sample sizes. Otherwise, the same patterns as in Figure

2.1 are evident: Power increases with σ2i and T , whereas N has a smaller effect on the

rejection frequencies.

2.5 Empirical application

We apply our test procedure to two settings. The first is a large dataset of macroeco-

nomic variables for the US, and the second is a dataset of portfolio returns.

The macroeconomic dataset is the FRED-MD database of McCracken and Ng

(2015). The dataset contains 135 monthly variables and includes measures of real ac-

tivity, prices, money and credit aggregates, interest rates, stock prices, and exchange

rates. We perform the same pre-treatment of the data as in McCracken and Ng (2015).

Specifically, we difference the non-stationary series to stationarity, and standardize

the series to have zero mean and unit variance before extracting principal compo-

nents. The reader is referred to their paper for a closer variable description and details

of the pre-treatment of the series.

We apply our test procedure for the period 1984:2014. Breitung and Eickmeier

(2011) test for structural breaks in the factor loadings associated with the Great Mod-

eration using a similar dataset. They find evidence of structural breaks in over half

the series in 1984, and argue that it leads to an inflation in the number of factors.

Our testing procedure requires the number of factors to be constant over the sam-

ple period, and we therefore restrict the sample to 1984:2014, resulting in T = 372

observations for each variable. To determine the number of factors, we use the Bai

and Ng (2002) ICp1 criterion. The number of factors is estimated to be 10, but the

criterion is very flat for r = 7, ...,12.1 We therefore consider r = 7, ...,12 when testing

for time-varying loadings.

Table 2.6 shows the rejection rates, i.e. the share of the 135 variables for which we

reject the null hypothesis. For r = 10 factors, the rejection rate is 50%, so for half of the

variables we reject the null of constant factor loadings. If we increase or decrease the

number of factors, the rejection rates are similar. When we redo the tests based on the

GLS transformed model, the rejection rates increase slightly. For r = 10 the rejection

rate increases to 59%, with a similar increase for the other number of included factors.

In Table 2.7 we report the rejection frequencies for the individual t-statistics for the

GLS transformed model with 10 factors. The rejection rates are highest for the first

1We have also tried the method of Alessi, Barigozzi, and Capasso (2010) to determine the numberof factors. The results are very sensitive to the choice of tuning parameters and do not give any clearindication of the number of factors.

2.6. CONCLUSION 53

three factors with a rejection rate around 40%. For the remaining factors, the test also

rejects for a non-trivial share of the variables. If we include more or fewer factors,

the rejection rates for the individual t-statistics are similar. There is thus substantial

evidence of time-varying factor loadings in the macroeconomic series.

For the second application we consider returns on portfolios. The dataset is from

Kenneth French’s website and consists of excess returns on 100 portfolios sorted on

size and book-to-market. Data descriptions and details on the sorting of portfolios

can be found in Fama and French (1993). The data includes T = 636 observations

and covers 1963:1 to 2015:12. The ICp1 criterion results in 6 factors. We also consider

r = 1, ...,5 as these are common choices in the asset pricing literature.2

The results of the tests are shown in Table 2.8. The rejection rates using the LMi

test are larger than 0.80 for r = 2, ...,6 factors, and 0.46 when only a single factor is

included. The results based on GLS estimation are similar. The rejection rates are

slightly lower, but we still reject for the majority of the variables. The results for the

individual t-statistics in Table 2.9 also show high rejection rates. We get similar results

for the t-statistics with fewer factors included. We thus identify time-varying factor

loadings for a large share of the asset portfolios.

This implies that some portfolios have time-varying exposure to the underlying

risk factors. The estimated factors bear a strong resemblance to the three risk factors

of Fama and French (1993): the market excess return, small minus big factor (SMB),

and high minus low factor (HML). The squared canonical correlations between the

three Fama French factors and the 6 factors in our analysis are 0.993, 0.951, and 0.917,

respectively. Bai and Ng (2006) also find that the Fama French factors are strong

proxies for systematic risk. They find canonical correlations of 0.992, 0.917, 0.832 for

the period 1960-1996.

2.6 Conclusion

In this paper we propose a simple procedure to test for stationary variations in factor

loadings. The test is based on principal components estimation of the factors and is

constructed as T R2 from a regression of the squared residuals on the squared factors.

We show that under the assumption of an approximate factor model, the limiting

distribution of the test statistic is unaffected by the estimation error of the common

factors. The test statistic converges to a χ2 random variable with degrees of freedom

equal to the number of factors. The limiting distribution is therefore the same as if

the factors could be observed. Furthermore, the test can be based on GLS estimation

of the factor model such that serial correlation in the idiosyncratic errors is removed.

When testing for time-varying factor loadings in a large macroeconomic dataset,

we find evidence for time-varying factor loadings in around half of the series. When

2The method of Alessi et al. (2010) tends to pick 2-4 factors, depending on the choice of tuningparameters.


applying the test to returns on portfolios, we find that the factor loadings are time-

varying for 80% of the portfolios. These portfolios therefore have a time-varying

exposure to the risk embedded in the underlying factors. Furthermore, the factors

have a strong relation with the three Fama French factors with squared canonical

correlations all larger than 0.90.

Acknowledgements

The author acknowledges support from The Danish Council for Independent Re-

search (DFF 4003-00022) and CREATES - Center for Research in Econometric Analysis

of Time Series (DNRF78), funded by the Danish National Research Foundation.

2.6. CONCLUSION 55

Figure 2.1: Empirical power (average rejection frequencies).

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(a) N = 20

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(b) N = 50

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(c) N = 100

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(d) N = 150

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(e) N = 200

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(f ) N = 400

NOTE: Figures (a)-(f) plot the average rejections frequencies of rejection of the LMi test for themodel with factor loadings variances σ2 = (0.1,0.5,1,1.5). Actual observations are marked with an"x". The lines are piecewise linear interpolations.


Figure 2.2: Empirical power (average rejection frequencies).Serially correlated errors – LMi .

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(a) N = 20

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(b) N = 50

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(c) N = 100

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(d) N = 150

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(e) N = 200

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(f ) N = 400

NOTE: Figures (a)-(f) plot the average frequencies of rejection of the LMi test for the model withfactor loadings variances σ2 = (0.1,0.5,1,1.5). The errors and factors are serially correlated withAR-parameters, α = 0.5 and ρ = 0.9, respectively. Actual observations are marked with an "x".The lines are piecewise linear interpolations.

2.6. CONCLUSION 57

Figure 2.3: Empirical power (average rejection frequencies).Serially correlated errors – LMi –GLS.

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(a) N = 20

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(b) N = 50

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(c) N = 100

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(d) N = 150

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(e) N = 200

50 100 150 200 250 300 350 400

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

σ2=0.1

σ2=0.5

σ2=1

σ2=1.5

(f ) N = 400

NOTE: Figures (a)-(f) plot the average frequencies of rejection of the LMi test based on GLSestimation for the model with factor loadings variancesσ2 = (0.1,0.5,1,1.5). The errors and factorsare serially correlated with AR-parameters, α= 0.5 and ρ = 0.9, respectively. Actual observationsare marked with an "x". The lines are piecewise linear interpolations.


Table 2.1: Empirical sizes (average rejection frequencies).

(a) 1 factor

N T = 50 T = 100 T = 150 T = 200 T = 400

20 0.037 0.040 0.041 0.057 0.04350 0.039 0.037 0.048 0.041 0.050

100 0.036 0.044 0.047 0.044 0.039150 0.035 0.043 0.042 0.043 0.044200 0.042 0.047 0.039 0.036 0.043400 0.035 0.038 0.045 0.040 0.055

(b) 2 factors

N T = 50 T = 100 T = 150 T = 200 T = 400

20 0.049 0.044 0.051 0.048 0.05850 0.043 0.048 0.045 0.039 0.051

100 0.044 0.039 0.049 0.044 0.053150 0.035 0.047 0.044 0.056 0.050200 0.048 0.036 0.047 0.044 0.050400 0.039 0.040 0.051 0.057 0.055

NOTE: The table reports the average rejection frequencies ofthe LMi test for the factor model with 1 factor (a) and 2 factors(b), and constant factor loadings, E(ξ2

i t ) = 0. The nominal sizeis 0.05 and the results are based on 2000 replications.

2.6. CONCLUSION 59

Table 2.2: Empirical sizes (average rejection frequencies).Serially correlated errors.

(a) LMi

N T = 50 T = 100 T = 150 T = 200 T = 400

20 0.051 0.073 0.076 0.076 0.09550 0.065 0.076 0.077 0.092 0.098

100 0.068 0.072 0.079 0.076 0.094150 0.067 0.075 0.086 0.076 0.092200 0.065 0.070 0.100 0.091 0.089400 0.070 0.077 0.079 0.082 0.105

(b) LMi – GLS

N T = 50 T = 100 T = 150 T = 200 T = 400

20 0.042 0.043 0.033 0.049 0.04450 0.035 0.037 0.045 0.045 0.049

100 0.035 0.040 0.043 0.040 0.052150 0.032 0.052 0.042 0.050 0.049200 0.033 0.046 0.042 0.046 0.045400 0.038 0.036 0.037 0.048 0.050

NOTE: The table reports the average rejection frequencies ofthe LMi test for the factor model with 1 factor and constantfactor loadings, E(ξ2

i t ) = 0. The errors and factors are seriallycorrelated with AR-parameterα= 0.5 and ρ = 0.9, respectively.Results in (a) for the LMi test, and results in (b) are for the GLStransformed model. The nominal size is 0.05 and the resultsare based on 2000 replications.


Table 2.3: Empirical power (average rejection frequencies).

(a) N = 20

T 50 100 150 200 400

σ2

0.1 0.071 0.117 0.141 0.155 0.3010.5 0.183 0.358 0.508 0.628 0.8831.0 0.252 0.455 0.637 0.731 0.9511.5 0.282 0.491 0.635 0.754 0.955

(b) N = 50

T 50 100 150 200 400

σ2

0.1 0.080 0.130 0.163 0.201 0.3490.5 0.227 0.443 0.607 0.728 0.9551.0 0.312 0.594 0.761 0.877 0.9951.5 0.368 0.614 0.777 0.878 0.994

(c) N = 100

T 50 100 150 200 400

σ2

0.1 0.089 0.130 0.180 0.215 0.3720.5 0.246 0.471 0.656 0.790 0.9741.0 0.335 0.648 0.833 0.907 0.9981.5 0.384 0.693 0.862 0.948 1.000

(d) N = 150

T 50 100 150 200 400

σ2

0.1 0.070 0.123 0.174 0.200 0.3670.5 0.239 0.501 0.668 0.787 0.9761.0 0.345 0.660 0.842 0.937 0.9991.5 0.420 0.715 0.877 0.964 1.000

(e) N = 200

T 50 100 150 200 400

σ2

0.1 0.068 0.135 0.170 0.195 0.3590.5 0.247 0.498 0.681 0.812 0.9761.0 0.358 0.688 0.873 0.940 0.9991.5 0.423 0.757 0.902 0.975 1.000

(f ) N = 400

T 50 100 150 200 400

σ2

0.1 0.067 0.120 0.171 0.210 0.3580.5 0.246 0.497 0.689 0.817 0.9781.0 0.391 0.701 0.881 0.949 1.0001.5 0.417 0.761 0.921 0.973 1.000

NOTE: The table reports the average frequencies of rejection of the LMi test for the model withfactor loadings variance E(ξ2

i t ) =σ2. The results are based on 2000 replications.

2.6. CONCLUSION 61

Table 2.4: Empirical power (average rejection frequencies).Serially correlated errors – LMi .

(a) N = 20

T 50 100 150 200 400

σ2

0.1 0.067 0.092 0.129 0.142 0.2520.5 0.109 0.202 0.326 0.406 0.7041.0 0.124 0.273 0.378 0.524 0.7841.5 0.174 0.309 0.426 0.528 0.809

(b) N = 50

T 50 100 150 200 400

σ2

0.1 0.073 0.110 0.149 0.169 0.2720.5 0.125 0.237 0.366 0.460 0.7971.0 0.171 0.313 0.469 0.615 0.9021.5 0.171 0.359 0.506 0.625 0.905

(c) N = 100

T 50 100 150 200 400

σ2

0.1 0.076 0.108 0.144 0.168 0.2770.5 0.128 0.307 0.415 0.531 0.8361.0 0.180 0.372 0.537 0.689 0.9291.5 0.208 0.401 0.579 0.691 0.949

(d) N = 150

T 50 100 150 200 400

σ2

0.1 0.080 0.113 0.149 0.180 0.3160.5 0.127 0.261 0.410 0.554 0.8431.0 0.186 0.352 0.540 0.686 0.9401.5 0.211 0.377 0.578 0.696 0.952

(e) N = 200

T 50 100 150 200 400

σ2

0.1 0.092 0.126 0.146 0.167 0.3180.5 0.125 0.265 0.412 0.513 0.8391.0 0.175 0.364 0.540 0.668 0.9541.5 0.207 0.396 0.577 0.713 0.953

(f ) N = 400

T 50 100 150 200 400

σ2

0.1 0.079 0.117 0.151 0.168 0.2810.5 0.132 0.270 0.410 0.533 0.8311.0 0.176 0.356 0.553 0.692 0.9461.5 0.204 0.405 0.570 0.697 0.961

NOTE: The table reports the average frequencies of rejection of the LMi test for the model with factorloadings variance E(ξ2

i t ) =σ2. The errors and factors are serially correlated with AR-parameters,α= 0.5 and ρ = 0.9, respectively. The results are based on 2000 replications.


Table 2.5: Empirical power (average rejection frequencies).Serially correlated errors – LMi –GLS.

(a) N = 20

T 50 100 150 200 400

σ2

0.1 0.032 0.059 0.053 0.074 0.1020.5 0.057 0.118 0.172 0.221 0.4081.0 0.088 0.181 0.262 0.344 0.5741.5 0.100 0.223 0.321 0.389 0.652

(b) N = 50

T 50 100 150 200 400

σ2

0.1 0.048 0.062 0.079 0.075 0.1230.5 0.074 0.139 0.211 0.285 0.5451.0 0.110 0.236 0.358 0.453 0.7591.5 0.124 0.278 0.405 0.507 0.815

(c) N = 100

T 50 100 150 200 400

σ2

0.1 0.056 0.059 0.075 0.090 0.1420.5 0.083 0.178 0.258 0.346 0.6341.0 0.117 0.268 0.399 0.528 0.8521.5 0.151 0.308 0.488 0.590 0.902

(d) N = 150

T 50 100 150 200 400

σ2

0.1 0.041 0.065 0.073 0.097 0.1440.5 0.084 0.162 0.251 0.343 0.6341.0 0.121 0.269 0.414 0.543 0.8481.5 0.151 0.321 0.479 0.621 0.914

(e) N = 200

T 50 100 150 200 400

σ2

0.1 0.050 0.061 0.072 0.090 0.1400.5 0.079 0.171 0.264 0.347 0.6411.0 0.120 0.277 0.425 0.553 0.8871.5 0.147 0.318 0.501 0.641 0.923

(f ) N = 400

T 50 100 150 200 400

σ2

0.1 0.045 0.063 0.071 0.090 0.1510.5 0.086 0.199 0.273 0.361 0.6781.0 0.105 0.281 0.436 0.588 0.8921.5 0.155 0.334 0.510 0.638 0.938

NOTE: The table reports the average frequencies of rejection of the LMi test based on GLS estimationfor the model with factor loadings variance E (ξ2

i t ) =σ2. The errors and factors are serially correlatedwith AR-parameters, α= 0.5 and ρ = 0.9, respectively. The results are based on 2000 replications.

2.6. CONCLUSION 63

Table 2.6: Rejection rates for LM-statistics – macro data.

Number of factors: r = 7 r = 8 r = 9 r = 10 r = 11 r = 12

rej % LM 0.50 0.50 0.49 0.50 0.58 0.54rej % LM–GLS 0.55 0.54 0.56 0.59 0.64 0.64

NOTE: ’rej % LM ’ is the rejection rate of the N individual LM statistics, and ’rej %LM–GLS’ is the rejection rates for the GLS transformed model. The significancelevel is 5%.

Table 2.7: Rejection rates for t-statistics – macro data.

F1 F2 F3 F4 F5

rej % t–GLS 0.43 0.39 0.38 0.13 0.25

F6 F7 F8 F9 F10

rej % t–GLS 0.28 0.12 0.33 0.26 0.24

NOTE: ’rej % t–GLS’ is the rejection rate of the N indi-vidual t– statistics on the 10 factors for the GLS trans-formed model. The significance level is 5%.

Table 2.8: Rejection rates for LM-statistics – portfolio data.

Number of factors: r = 1 r = 2 r = 3 r = 4 r = 5 r = 6

rej % LM 0.46 0.82 0.86 0.86 0.83 0.87rej % LM–GLS 0.46 0.81 0.81 0.80 0.79 0.78

NOTE: ’rej % LM ’ is the rejection rate of the N individual LM statistics, and’rej % LM–GLS’ is the rejection rates for the GLS transformed model. Thesignificance level is 5%.

Table 2.9: Rejection rates for t-statistics – portfolio data.

F1 F2 F3 F4 F5 F6

rej % t–GLS 0.42 0.56 0.40 0.19 0.32 0.29

NOTE: ’rej % t–GLS’ is the rejection rate of the N individual t-statistics on the 6 factors for the GLS transformed model. Thesignificance level is 5%.


2.7 References

Alessi, L., Barigozzi, M., Capasso, M., 2010. Improved penalization for determining

the number of factors in approximate factor models. Statistics & Probability Letters

80 (23), 1806–1813.

Bai, J., 2003. Inferential theory for factor models of large dimensions. Econometrica

71, 135–171.



Bai, J., Ng, S., 2006. Evaluating latent and observed factors in macroeconomics and

finance. Journal of Econometrics 131 (1), 507–537.

Bernanke, B., Boivin, J., Eliasz, P., 2005. Factor augmented vector autoregressions

(FVARs) and the analysis of monetary policy. Quarterly Journal of Economics

120 (1), 387–422.

Breitung, J., Eickmeier, S., 2011. Testing for structural breaks in dynamic factor models.

Journal of Econometrics 163 (1), 71–84.

Breitung, J., Tenhofen, J., 2011. GLS estimation of dynamic factor models. Journal of

the American Statistical Association 106 (495), 1150–1166.

Del Negro, M., Otrok, C., 2008. Dynamic factor models with time-varying parame-

ters: measuring changes in international business cycles. Staff report 326, Federal

Reserve Bank of New York, New York.

Eickmeier, S., Lemke, W., Marcellino, M., 2015. Classical time varying factor-

augmented vector auto-regressive models – estimation, forecasting and structural

analysis. Journal of the Royal Statistical Society: Series A (Statistics in Society)

178 (3), 493–533.

Fama, E. F., French, K. R., 1993. Common risk factors in the returns on stocks and

bonds. Journal of Financial Economics 33 (1), 3–56.

Giannone, D., Reichlin, L., Sala, L., 2006. VARs, common factors and the empirical

validation of equilibrium business cycle models. Journal of Econometrics 132 (1),

257–279.

McCracken, M., Ng, S., 2015. FRED-MD: A monthly database for macroeconomic

research. Forthcoming, Journal of Business and Economic Statistics.

Stock, J., Watson, M., 2002. Macroeconomic forecasting using diffusion indexes. Jour-

nal of Business and Economic Statistics 20, 147–162.

A.1. APPENDIX 65

A.1 Appendix

Let X = (X1, ..., XT )′ be the T × N matrix of observations, and let VN T be the r ×r diagonal matrix of the r largest eigenvalues of (N T )−1X X ′ in decreasing order.

The principal components estimator F is obtained asp

T times the eigenvectors

corresponding to the largest r eigenvalues of the matrix X X ′. By the definition of

eigenvalues and eigenvectors, we have (N T )−1X X ′F = FVN T or (N T )−1X X ′FV −1N T =

F , where F ′F /T = Ir , and H = (Λ0′Λ0/N )(F ′F /T )V −1N T is the r ×r rotation matrix. The

following results are based on the identity (see Bai (2003)):

Ft −H ′Ft =V −1N T

(T −1

∑s

FsγN (s, t )+T −1∑

sFsζst +T −1

∑s

Fsηst +T −1∑

sFsξst

),

(A.1)

where

- ζst = e ′s etN −γN (s, t ),

- ηst = F 0′s Λ

0′et /N ,

- ξst = F 0′t Λ

0′es /N .

Lemma A.3 in Bai (2003) implies that ‖V −1N T ‖ = Op (1) and ‖H‖ = Op (1). We also

have T −1 ∑t ‖Ft − H ′Ft‖2 = Op (δ−2

N T ) from Lemma A.1 in Bai (2003) where δN T =mi n

pN ,

pT . As stated in Bai and Ng (2002), p 198, ‖Ft − H ′Ft‖2 = Op (δ−2

N T ) if∑s γN (s, t)2 ≤ M for all t , which we show in Lemma A.1 below. These results will

be used extensively in the following. It should be noted that our Assumptions D and E

differ slightly from the corresponding assumtions in Bai (2003), but are still sufficient

to arrive at the same results. The proof of Theorem 1 requires the following lemmas.

Lemma A.1. Under Assumption C, we have for all t and some M ≤∞:∑sγN (s, t )2 ≤ M .

Proof: As in Bai and Ng (2002), letρ(s, t ) = γN (s, t )/[γN (s, s)γN (t , t )]1/2. Then |ρ(s, t )| ≤1. We can write:∑

sγN (s, t )2 =∑

sρ(s, t )2γN (s, s)γN (t , t )

≤∑s|ρ(s, t )||γN (s, s)γN (t , t )|1/2|γN (s, s)|1/2|γN (t , t )|1/2

≤ M∑

s|ρ(s, t )||γN (s, s)γN (t , t )|1/2 = M

∑s|γN (s, t )| ≤ M 2,

for all t by Assumption C.2.


Lemma A.2. Under Assumptions A-D, the r × r matrix satisfies

E

∥∥∥∥ 1pN T

T∑t=1

N∑k=1

Ftλ′k ekt

∥∥∥∥2

≤ M .

Proof: We can write:

E

∥∥∥∥ 1pN T

T∑t=1

N∑k=1

Ftλ′k ekt

∥∥∥∥2

= (N T )−1∑t ,s

∑k,l

tr E(el sλl F ′s Ftλ

′k ekt )

≤ (N T )−1∑t ,s

∑k,l

E(el s ekt )E(F ′s Ft )λ′

kλl

≤ λ2(N T )−1∑t ,s

∑k,l

|E(el s ekt )|(E‖Fs‖2)1/2(‖Ft‖2)1/2

≤ λ2M(N T )−1∑t ,s

∑k,l

|τkl ,t s | ≤ M ,

which follows from Assumptions C.4 and D.

Lemma A.3. Under Assumptions A-E, we have:

T −1∑

t‖(Fs −H ′Ft )ei t‖2 =Op (δ−2

N T ).

Proof: From the identity (A.1) we have:

(Ft −H ′Ft )ei t =V −1N T

(T −1

∑s

FsγN (s, t )ei t +T −1∑

sFsζst ei t

+T −1∑

sFsηst ei t +T −1

∑s

Fsξst ei t

).

Using Loève’s inequality gives:

T −1∑

t‖(Ft −H ′Ft )ei t‖2 ≤ 4‖V −1

N T ‖2(T −1

∑t‖T −1

∑s

FsγN (s, t )ei t‖2

+T −1∑

t‖T −1

∑s

Fsζst ei t‖2 +T −1∑

t‖T −1

∑s

Fsηst ei t‖2

+T −1∑

t‖T −1

∑s

Fsξst ei t‖2)= 4‖V −1

N T ‖2(I + I I + I I I + IV ).

Consider I :

I = T −1∑

t‖T −1

∑s

FsγN (s, t )ei t‖2 ≤ T −3∑

t

(∑s‖Fs‖|γN (s, t )||ei t |

)2

≤ T −2∑

te2

i t

(T −1

∑s‖Fs‖2

)(∑sγN (s, t )2

)= r T −2

∑t

e2i t

(∑sγN (s, t )2

)≤ r MT −2

∑t

e2i t =Op (T −1),

A.1. APPENDIX 67

by Lemma A.1, and the fact that T −1 ∑s ‖Fs‖2 = r .

For I I we have:

I I = T −1∑

t‖T −1

∑s

Fsζst ei t‖2 ≤ T −3∑

t

(∑s‖Fs‖|ζst ||ei t |

)2

≤ T −2∑

t

(T −1

∑s‖Fs‖2

)(∑sζ2

s,t e2i t

)= r T −2

∑t

∑sζ2

s,t e2i t

= r N−1T −2∑

t

∑s|N−1/2

∑i

[ei t ei s −E(ei s ei t )]|2e2i t .

The last term can be bounded in expectation:

E

(|N−1/2

∑i

[ei t ei s −E(ei s ei t )]|2e2i t

)≤ E |N−1/2

∑i

[ei t ei s −E(ei s ei t )]|4E(e4i t ) ≤ M 2,

for all s, t by Assumptions C.1 and C.5. Thus I I =Op (N−1).

For I I I we have:

I I I = T −1∑

t‖T −1

∑s

Fsηst ei t‖2 = N−1T −1∑

t

∥∥∥∥T −1∑

sFs F ′

s

(Λ′etp

N

)ei t

∥∥∥∥2

≤ N−1T −1∑

t

∥∥∥∥Λ′etpN

ei t

∥∥∥∥2

‖T −1∑

sFs F ′

s‖2

≤ N−1T −1∑

t


ei t

∥∥∥∥2(T −1

∑s‖Fs‖2

)(T −1

∑s‖Fs‖2

)≤ r Op (1)N−1

(T −1

∑t


∥∥∥∥4)1/2(T −1

∑t

e4i t

)1/2

=Op (N−1),

by Assumption E.2.

For IV we have:

IV = T −1∑

t‖T −1

∑s

Fsξst ei t‖2 = N−1T −1∑

t

∥∥∥∥T −1∑

sFs F ′

t

(Λ′esp

N

)ei t

∥∥∥∥2

≤ N−1T −1∑

t

(T −1

∑s‖Fs‖‖Ft‖

∥∥∥∥Λ′espN

∥∥∥∥|ei t |)2

= N−1T −1∑

t‖Ft‖2e2

i t

(T −1

∑s‖Fs‖


∥∥∥∥)2

N−1(T −1

∑t‖Ft‖2e2

i t

)(T −1

∑s‖Fs‖2

)(T −1

∑s


∥∥∥∥2)= r N−1Op (1) =Op (N−1),

as T −1 ∑t ‖Ft‖2e2

i t is bounded in expectation. Thus I + I I + I I I + IV = Op (T −1)+Op (N−1) =Op (δ−2

N T ).


T −1∑

t‖(Fs −H ′Ft )F ′

t‖2 =Op (δ−2N T ).



(Ft −H ′Ft )F ′t =V −1

N T

(T −1

∑s

FsγN (s, t )F ′t +T −1

∑s

Fsζst F ′t

+T −1∑

sFsηst F ′

t +T −1∑

sFsξst F ′

t

).


T −1∑

t‖(Ft −H ′Ft )F ′

t‖2 ≤ 4‖V −1N T ‖2

(T −1

∑t‖T −1

∑s

FsγN (s, t )F ′t‖2

+T −1∑

t‖T −1

∑s

Fsζst F ′t‖2 +T −1

∑t‖T −1

∑s

Fsηst F ′t‖2

+T −1∑

t‖T −1

∑s

Fsξst F ′t‖2

)= 4‖V −1

N T ‖2(I + I I + I I I + IV ).

Consider I :

I = T −1∑

t‖T −1

∑s

FsγN (s, t )F ′t‖2 ≤ T −3

∑t

(∑s‖Fs‖|γN (s, t )|‖Ft‖

)2

≤ T −2∑

t‖Ft‖2

(T −1

∑s‖Fs‖2

)(∑sγN (s, t )2

)≤ r MT −2

∑t‖Ft‖2 =Op (T −1),

by Lemma A.1 and Assumption A.

For I I we have:

I I = T −1∑

t‖T −1

∑s

Fsζst F ′t‖2 ≤ T −3

∑t

(∑s‖Fs‖|ζst |‖Ft‖

)2

≤ T −1∑

t‖Ft‖2

(T −1

∑s‖Fs‖2

)(T −1

∑sζ2

s,t

)= r N−1T −1

∑t‖Ft‖2

(T −1

∑s|N−1/2

∑i

[ei t ei s −E(ei t ei s )|2)=Op (N−1),

by Assumptions A and C.5.

For I I I we have:

I I I = T −1∑

t‖T −1

∑s

Fsηst F ′t‖2 = N−1T −1

∑t

∥∥∥∥T −1∑

sFs F ′

s

(N−1/2Λ′et F ′

t

)∥∥∥∥2

≤ N−1T −1∑

t

∥∥∥∥N−1/2Λ′et F ′t

∥∥∥∥2

‖T −1∑

sFs F ′

s‖2

≤ N−1T −1∑

t

∥∥∥∥N−1/2Λ′et F ′t

∥∥∥∥2(T −1

∑s‖Fs‖2

)(T −1

∑s‖Fs‖2

)= r Op (1)N−1T −1

∑t

∥∥∥∥N−1/2Λ′et F ′t

∥∥∥∥2

.

A.1. APPENDIX 69

The last term can be bounded in expectation:

T −1∑

tE

∥∥∥∥N−1/2Λ′et F ′t

∥∥∥∥2

= T −1N−1∑

t

∑i

∑k

E(ei t ekt )λ′iλk E(F ′

t Ft )

≤ λ2T −1∑

tE‖Ft‖2N−1

∑i

∑k|E(ei t ekt )| ≤ λ2T −1

∑t

E‖Ft‖2N−1∑

i

∑k|τi k |

≤ λ2MT −1∑

tE‖Ft‖2 =Op (1),

by Assumptions A, B, and C.3. Thus, I I I =Op (N−1).

For IV we have:

IV = T −1∑

t‖T −1

∑s

Fsξst F ′t‖2 = N−1T −1

∑t

∥∥∥∥T −1∑

sFs F ′

t

(Λ′esp

N

)F ′

t

∥∥∥∥2

≤ N−1T −1∑

t

(T −1

∑s‖Fs‖‖Ft‖


∥∥∥∥‖Ft‖)2

= N−1T −1∑

t‖Ft‖4

(T −1

∑s‖Fs‖


∥∥∥∥)2

≤ N−1(T −1

∑t‖Ft‖4

)(T −1

∑s‖Fs‖2

)(T −1

∑s


∥∥∥∥2)= r N−1Op (1) =Op (N−1),

by Assumptions A and E.2. Thus I + I I + I I I + IV =Op (T −1)+Op (N−1) =Op (δ−2N T ).


T −1∑

t‖(Fs −H ′Ft )F ′

t‖2 =Op (δ−2N T ).


(Ft −H ′Ft )F ′t =V −1

N T

(T −1

∑s

FsγN (s, t )F ′t +T −1

∑s

Fsζst F ′t

+T −1∑

sFsηst F ′

t +T −1∑

sFsξst F ′

t

).


T −1∑


t‖2 ≤ 4‖V −1N T ‖2

(T −1

∑t‖T −1

∑s

FsγN (s, t )F ′t‖2

+T −1∑

t‖T −1

∑s

Fsζst F ′t‖2 +T −1

∑t‖T −1

∑s

Fsηst F ′t‖2

+T −1∑

t‖T −1

∑s

Fsξst F ′t‖2

)= 4‖V −1

N T ‖2(I + I I + I I I + IV ).

Consider I :

I = T −1∑

t‖T −1

∑s

FsγN (s, t )F ′t‖2 ≤ T −3

∑t

(∑s‖Fs‖|γN (s, t )|‖Ft‖

)2

≤ T −2∑

t‖Ft‖2

(T −1

∑s‖Fs‖2

)(∑sγN (s, t )2

)≤ r 2MT −1 =Op (T −1),


by Lemma A.1.

For I I we have:

I I = T −1∑

t‖T −1

∑s

Fsζst F ′t‖2 ≤ T −3

∑t

(∑s‖Fs‖|ζst |‖Ft‖

)2

≤ T −1∑

t‖Ft‖2

(T −1

∑s‖Fs‖2

)(T −1

∑sζ2

s,t

)= r 2N−1

(T −1

∑s|N−1/2

∑i

[ei t ei s −E(ei t ei s )|2)=Op (N−1),

by Assumption C.5.

For I I I we have:

I I I = T −1∑

t‖T −1

∑s

Fsηst F ′t‖2 ≤ T −1

∑t‖Ft‖2

(T −1

∑s‖Fs‖|ηst |

)2

≤(T −1

∑t‖Ft‖2

)(T −1

∑s‖Fs‖2

)(T −1

∑sη2

st

)= r 2T −1

∑sη2

st .

The last term is bounded in expectation:

T −1∑

sE(N−1F ′

sΛ′et )2 ≤ N−1E‖N−1/2Λ′et‖2T −1

∑s

E‖Fs‖2 = N−1Op (1),

by Assumption E.2, so I I I is Op (N−1).

For IV we have:

IV = T −1∑

t‖T −1

∑s

Fsξst F ′t‖2 = N−1T −1

∑t

∥∥∥∥T −1∑

sFs

(N−1/2F ′

tΛ′es

)F ′

t

∥∥∥∥2

= N−1T −1∑

t

∥∥∥∥T −1∑

sFs

(N−1/2e ′sΛFt

)F ′

t

∥∥∥∥2

≤ N−1(T −1

∑t‖Ft Ft‖2

)∥∥∥∥T −1∑

sFs

(N−1/2e ′sΛ

)∥∥∥∥2

≤ N−1(T −1

∑t‖Ft F ′

t‖2)(

T −1∑

s‖Fs‖2

)(T −1

∑s‖N−1/2e ′sΛ‖2

)≤ 2r Op (1)N−1

(T −1

∑t‖Ft (Ft −H ′Ft )′‖2

)+2r Op (1)N−1

(T −1

∑t‖Ft F ′

t H‖2)

=Op (N−1δ−2N T )+Op (N−1) =Op (N−1),

from Lemma A.4 above. Thus I + I I + I I I + IV =Op (T −1)+Op (N−1) =Op (δ−2N T ).

A.1. APPENDIX 71

Lemma A.6. Under Assumptions A-E we have:

T −1∑

t(Ft −H ′Ft )ei t Ft F ′

t =Op (δ−2N T ).


T −1∑


t =V −1N T

(T −2

∑t

∑s

FsγN (s, t )ei t Ft F ′t +T −2

∑t

∑s

Fsζst ei t Ft F ′t

+T −2∑

t

∑s

Fsηst ei t Ft F ′t +T −2

∑t

∑s

Fsξst ei t Ft F ′t

)=V −1

N T (I + I I + I I I + IV ).

For I we can write:

I = T −2∑

t

∑s

(Fs −H ′Fs )γN (s, t )ei t Ft F ′t +T −2

∑t

∑s

H ′FsγN (s, t )ei t Ft F ′t .


T −2∑

s‖Fs −H ′Fs‖

(∑tγN (s, t )2

)1/2(∑t

e2i t‖Ft‖4

)1/2

≤ T −1/2(T −1

∑s‖Fs −H ′Fs‖2

)1/2(T −1

∑s

∑tγN (s, t )2T −1

∑t

e2i t‖Ft‖4

)1/2

= T −1/2Op (δ−1N T )Op (1),

where Op (1) follows as E(e2i t )E‖Ft‖4 = Op (1) by Assumptions A and C.1, and from∑

s γN (s, t )2 ≤ M for all t by Lemma A.1.

The second term can be bounded in expectation (ignore H):

T −2∑

t

∑s|γN (s, t )|E |ei t |(E‖Fs‖2)1/2(E‖Ft‖4)1/2 ≤ MT −2

∑t

∑s|γN (s, t )| =Op (T −1),

from Assumption C.2. Thus I is Op (T −1/2δ−1N T ).

For I I we write:

T −2∑

t

∑s

(Fs −H ′Fs )ζst ei t Ft F ′t +T −2

∑t

∑s

H ′Fsζst ei t Ft F ′t .


T −2∑

s‖Fs −H ′Fs‖‖

∑tζst ei t Ft F ′

t‖ ≤(T −1

∑s‖Fs −H ′Ft‖2

)1/2(T −3

∑s‖∑

tζst ei t Ft F ′

t‖2)1/2

≤ N−1/2Op (δ−1N T )

(T −2

∑s

∑t|N−1/2

∑k

[eks ekt −E(eks ekt )]|2T −1∑

te2

i t‖F ′t‖4

)1/2

= N−1/2Op (δ−1N T )Op (1),


by Assumption C.5.

For the second term we have:

1pN T

T −1∑

t

(1pN T

∑s

∑k

Fs [eks ekt −E(eks ekt )]

)ei t Ft F ′

t

≤ 1pN T

(T −1

∑t

∥∥∥∥ 1pN T

∑s

∑k


∥∥∥∥2)1/2(T −1

∑t

e2i t‖Ft‖4

)1/2

=Op

(1pN T

)Op (1),

from Assumption E.1. Thus I I is Op (N−1/2δ−1N T ).

We rewrite I I I as:

T −2∑

t

∑s

(Fs −H ′Fs )ηst ei t Ft F ′t +H ′T −2

∑t

∑s

Fsηst ei t Ft F ′t .

For the first term, we write:

N−1/2T −2∑

s(Fs −H ′Fs )F ′

s

∑t

(Λ′etp

N

)ei t Ft F ′

t

≤ N−1/2(T −1

∑s‖(Fs −H ′Fs )F ′

s‖2)1/2(

T −3∑

s‖∑

t

(Λ′etp

N

)ei t Ft F ′

t‖2)1/2

.

The first parenthesis is Op (δ−1N T ) by Lemma A.4. The term inside the second parenthe-

sis is bounded by:

T −1∑

s

(T −1

∑t


∥∥∥∥2)(T −1

∑t

e2i t‖Ft‖4

)=Op (1),

by Assumption E.2. The first term is thus N−1/2Op (δ−1N T ).

The second term can be written as:(T −1

∑s

Fs F ′s

)(N−1T −1

∑t

∑kλk ekt ei t Ft F ′

t

).

The first parenthesis is Op (1), and the second is bounded in expectation by:

N−1T −1∑

t

∑kλk E(ekt ei t )E‖Ft‖2 ≤ λN−1T −1

∑t

∑k|τi k | =Op (N−1),

by Assumption C.3, and I I I is thus Op (N−1/2δ−1N T ).

We rewrite IV as:

T −2∑

t

∑s

Fsξst ei t Ft F ′t = T −2

∑t

∑s

Fs (F ′tΛ

′es /N )ei t Ft F ′t = T −2

∑t

∑s

Fs (e ′sΛ/N )Ft ei t Ft F ′t

= T −2∑

t

∑s

(Fs −H ′Fs )(e ′sΛ/N )Ft ei t Ft F ′t +H ′T −2

∑t

∑s

Fs (e ′sΛ/N )Ft ei t Ft F ′t .

A.1. APPENDIX 73


N−1/2T −2∑


∥∥∥∥N−1/2e ′sΛ∥∥∥∥∑

t|ei t |‖Ft‖3

≤ N−1/2(T −1


)1/2(T −1

∑s

∥∥∥∥N−1/2e ′sΛ∥∥∥∥2)1/2

T −1∑

t|ei t |‖Ft‖3,

which is N−1/2Op (δ−1N T ).

For the second term we write:

1pN T

(1pN T

∑s

∑k

Fs eksλ′k

)(T −1

∑t

Ft ei t Ft F ′t

)=Op

(1pN T

),

by Lemma A.2. Thus IV is Op (N−1/2δ−1N T ). We therefore have that I + I I + I I I + IV =

Op (N−1/2δ−1N T )+Op (T −1/2δ−1

N T ) =Op (δ−2N T ).

Lemma A.7. Under Assumptions A-E we have:

T −1∑

t(Ft −H ′Ft )e2

i t F ′t =Op (δ−2

N T ).


T −1∑

t(Ft −H ′Ft )e2

i t F ′t =V −1

N T

(T −2

∑t

∑s

FsγN (s, t )e2i t F ′

t +T −2∑

t

∑s

Fsζst e2i t F ′

t

+T −2∑

t

∑s

Fsηst e2i t F ′

t +T −2∑

t

∑s

Fsξst e2i t F ′

t

)=V −1

N T (I + I I + I I I + IV ).

We rewrite I as:

T −2∑

t

∑s


t = T −2∑

t

∑s

(Fs −H ′Fs )γN (s, t )e2i t F ′

t +H ′T −2∑

t

∑s


t .


T −2∑


∑t|γN (s, t )|e2

i t‖Ft‖

≤ T −2∑


(∑t|γN (s, t )|2

)1/2(∑t

e4i t‖Ft‖2

)1/2

≤ T −1/2(T −1


)1/2(T −1

∑s

∑t|γN (s, t )|2T −1

∑t

e4i t‖Ft‖2

)1/2

= T −1/2Op (δ−1N T ),

by Assumptions A and C.1 and Lemma A.1.


The second term is bounded in expectation by:

T −2∑

t

∑s|γN (s, t )|E(e2

i t )

(E‖Ft‖2

)1/2(E‖Fs‖2

)1/2

≤ MT −1(T −1

∑t

∑s|γN (s, t )|

)=Op (T −1),

by Assumption C.2. Thus I is T −1/2Op (δ−1N T ).

For I I we have:

T −2∑

t

∑s

Fsζst e2i t F ′

t = T −2∑

t

∑s

(Fs −H ′Fs )ζst e2i t F ′

t +H ′T −2∑

t

∑s

Fsζst e2i t F ′

t .


T −2∑

s‖Fs −H ′Fs‖‖

∑tζst e2

i t F ′t‖ ≤

(T −1

∑s‖Fs −H ′Ft‖2

)1/2(T −3

∑s‖∑

tζst e2

i t F ′t‖2

)1/2

≤ N−1/2Op (δ−1N T )

(T −2

∑s

∑t|N−1/2

∑k

[eks ekt −E(eks ekt )]|2T −1∑

te4

i t‖F ′t‖2

)1/2

= N−1/2Op (δ−1N T ),

by Assumption C.5.

The second term is:

1pN T

T −1∑

t

(1pN T

∑s

∑k


)e2

i t F ′t

≤ 1pN T

(T −1

∑t

∥∥∥∥ 1pN T

∑s

∑k


∥∥∥∥2)1/2(T −1

∑t

e4i t‖Ft‖2

)1/2

=Op

(1pN T

)Op (1),

by Assumption E.1, so I I is Op (N−1/2δ−1N T ).

For I I I we have:

T −2∑

t

∑s

Fsηst e2i t F ′

t = T −2∑

t

∑s

(Fs −H ′Fs )ηst e2i t F ′

t +H ′T −2∑

t

∑s

Fsηst e2i t F ′

t .

We write the first term as:

N−1/2(T −1

∑s

(Fs −H ′Fs )F ′s

)(T −1

∑t

(Λ′etp

N

)e2

i t F ′t

).

By Lemma B.2 in Bai (2003), the first parenthesis is Op (δ−2N T ). The second parenthesis

is bounded by: (T −1

∑t


∥∥∥∥2)1/2(T −1

∑t

e4i t‖Ft‖2

)1/2

=Op (1),

A.1. APPENDIX 75

by Assumption E.2. The first term of I I I is thus N−1/2Op (δ−2N T ).

The second term can be written as:(T −1

∑s

Fs F ′s

)(T −1N−1

∑t

∑kλk ekt F ′

t e2i t

).

The first parenthesis is Op (1). We can bound the second parenthesis in expectation:

1pN T

E

(1pN T

∑t

∑kλk ekt F ′

t

)e2

i t

≤ 1pN T

(E

∥∥∥∥ 1pN T

∑t

∑kλk ekt F ′

t

∥∥∥∥2

E(e4i t )

)1/2

=Op

(1pN T

),

by Lemma A.2. Thus I I I is Op (N−1/2δ−1N T ).

For the IV we can write:

T −2∑

t

∑s

Fsξst e2i t F ′

t = T −2∑

t

∑s

Fs (F ′tΛ

′es /N )e2i t F ′

t = T −2∑

t

∑s

Fs (e ′sΛ/N )Ft e2i t F ′

t

= T −2∑

t

∑s

(Fs −H ′Fs )(e ′sΛ/N )Ft e2i t F ′

t +H ′T −2∑

t

∑s

Fs (e ′sΛ/N )Ft e2i t F ′

t .


N−1/2T −2∑


∥∥∥∥N−1/2e ′sΛ∥∥∥∥∑

te2

i t‖Ft‖2

≤ N−1/2(T −1


)1/2(T −1

∑s

∥∥∥∥N−1/2e ′sΛ∥∥∥∥2)1/2

T −1∑

te2

i t‖Ft‖2,

which is N−1/2Op (δ−1N T ).


1pN T

(1pN T

∑s

∑k

Fs eksλ′k

)(T −1

∑t

Ft e2i t F ′

t

)=Op

(1pN T

),

by Lemma A.2, and IV is therefore Op (N−1/2δ−1N T ). Collecting results thus gives that

I + I I + I I I + IV =Op (N−1/2δ−1N T )+Op (T −1/2δ−1

N T ) =Op (δ−2N T ).

Lemma A.8. Under Assumptions A-D, we have:

a. ‖T −1(F −F H)′ei‖2 =Op (δ−2N T ),

b. ‖T −1(F −F H)′F‖2 =Op (δ−2N T ),

c. ‖T −1(F −F H)′F‖2 =Op (δ−2N T ).


Proof: For a we have:

‖T −1(F −F H)′ei‖2 = ‖T −1∑

t(Ft −H ′Ft )ei t‖2

≤(T −1

∑t‖Ft −H ′Ft‖2

)(T −1

∑t

e2i t

)=Op (δ−2

N T )Op (1),

as T −1 ∑t e2

i t = Op (1) by Assumption C.1. The proof of b and c follows in the same

way by using T −1 ∑t ‖Ft‖2 =Op (1) and T −1 ∑

t ‖Ft‖2 = r .


‖λi −H−1λi‖2 =Op (δ−2N T ).

Proof: Following in Bai (2003), p 165, we can write λi −H−1λi as:

λi −H−1λi = T −1F ′(F H − F )H−1λi +T −1(F −F H)′ei +T −1H ′F ′ei .

The norm is thus bounded by:

‖λi −H−1λi‖2 ≤ 3‖T −1F ′(F H − F )H−1λi‖2 +3‖T −1(F −F H)′ei‖2 +3‖T −1H ′F ′ei‖2

≤ 3‖T −1F ′(F H − F )‖2‖H−1‖2‖λi‖2 +3‖T −1(F −F H)′ei‖2

+3‖H‖2‖T −1F ′ei‖2 =Op (δ−2N T ),

which follows from Lemmas A.8.a and A.8.c, and since ‖T −1F ′ei‖2 = T −1‖T −1/2F ′ei‖2 =T −1Op (1) by Assumption E.3.

Lemma A.10. Under Assumptions A-E we have for all t and i :

a. ei t = ei t +Op (δ−1N T ),

b. Ci t =Ci t +Op (δ−1N T ).

Proof: We start with a. We can write the principal components residual as:

ei t = xi t − (λi −H−1λi )′Ft −λ′i (H−1)′Ft

= xi t − (λi −H−1λi )′(Ft −H ′Ft )− (λi −H−1λi )′H ′Ft −λ′i (H−1)′(Ft −H ′Ft )−λ′

i (H−1)′H ′Ft

= xi t −λ′i Ft − (λi −H−1λi )′(Ft −H ′Ft )− (λi −H−1λi )′H ′Ft −λ′

i (H−1)′(Ft −H ′Ft )

= ei t − (λi −H−1λi )′(Ft −H ′Ft )− (λi −H−1λi )′H ′Ft −λ′i (H−1)′(Ft −H ′Ft ).

(A.2)

For the second term we have:

|(λi −H−1λi )′(Ft −H ′Ft )| ≤ ‖λi −H−1λi‖‖Ft −H ′Ft‖ =Op (δ−1N T )Op (δ−1

N T ),

A.1. APPENDIX 77

by Lemma A.9. The third term is bounded by:

|(λi −H−1λi )′H ′Ft | ≤ ‖λi −H−1λi‖‖H‖‖Ft‖ =Op (δ−1N T )Op (1)Op (1),

by Lemma A.9 and Assumption A. The last term is bounded by:

|λ′i (H−1)′(Ft −H ′Ft )| ≤ ‖λi‖‖H−1‖‖Ft −H ′Ft‖ =Op (1)Op (1)Op (δ−1

N T ),

by Assumption B. We thus have that:

ei t = ei t +Op (δ−1N T ).

The proof of b follows immediately by noting:

Ci t −Ci t = λ′i Ft −λ′

i Ft = (xi t −λ′i Ft )− (xi t − λ′

i Ft ) = ei t − ei t =Op (δ−1N T ).

Lemma A.11. Under Assumtions A-E we have

T −1∑

te2

i t = T −1∑

te2

i t +Op (δ−2N T ).

Proof: We can write T −1 ∑t e2

i t as:

T −1∑

te2

i t = T −1∑

t(xi t − Ci t )2 = T −1

∑t

(xi t −Ci t +Ci t − Ci t )2

= T −1∑

t(xi t −Ci t )2 +T −1

∑t

(Ci t − Ci t )2 +2T −1∑

tei t (Ci t − Ci t )2

= T −1∑

te2

i t +T −1∑

t(Ci t − Ci t )2 +2T −1

∑t

(Ci t − Ci t )ei t .

We start with T −1 ∑t (Ci t − Ci t )ei t . Using (A.2) we can write:

T −1∑

t(Ci t − Ci t )ei t =−T −1

∑t

(λi +H−1λi )′(Ft −H ′Ft )ei t −T −1∑

t(λi −H−1λi )′H ′Ft ei t

−T −1∑

tλ′

i (H−1)′(Ft −H ′Ft )ei t .

For the first term we write:

−T −1∑

t(λi +H−1λi )′(Ft −H ′Ft )ei t =−(λi +H−1λi )′T −1

∑t

(Ft −H ′Ft )ei t .

Bai (2003), p 165, shows that (λi +H−1λi ) =Op (T −1/2), and Lemma B.1, also in Bai

(2003), states that T −1 ∑t (Ft−H ′Ft )ei t =Op (δ−2

N T ). The first term is thus Op (T −1/2)Op (δ−2N T ).


The second term is

−T −1∑

t(λi −H−1λi )′H ′Ft ei t =−(λi −H−1λi )′H ′T −1

∑t

Ft ei t =Op (T −1/2)Op (T −1/2),

again from Bai (2003), p 165 and Assumption E.3.

The last term is:

−T −1∑

tλ′

i (H−1)′(Ft −H ′Ft )ei t =−λ′i (H−1)′T −1

∑t

(Ft −H ′Ft )ei t =Op (δ−2N T ),

by Assumption B and Lemma B.1 in Bai (2003). Thus T −1 ∑t (Ci t−Ci t )ei t =Op (T −1/2)Op (δ−2

N T )+Op (T −1/2)Op (T −1/2)+Op (δ−2

N T ) =Op (δ−2N T ).

For T −1 ∑t (Ci t − Ci t )2 we again use (A.2). By Loève’s inequality we get:

T −1∑

t(Ci t − Ci t )2 = T −1

∑t

((λi −H−1λi )′(Ft −H ′Ft )

+ (λi −H−1λi )′H ′Ft +λ′i (H−1)′(Ft −H ′Ft ))2

≤ 3T −1∑

t‖(λi −H−1λi )′(Ft −H ′Ft )‖2 +3T −1

∑t‖(λi −H−1λi )′H ′Ft‖2

+3T −1∑

t‖λ′

i (H−1)′(Ft −H ′Ft )‖2.

The first term is:

T −1∑

t‖(λi−H−1λi )′(Ft−H ′Ft )‖2 ≤ ‖λi−H−1λi‖2T −1

∑t‖Ft−H ′Ft‖2 =Op (δ−2

N T )Op (δ−2N T ),

by Lemma A.9.

The second term is:

T −1∑

t‖(λi −H−1λi )′H ′Ft‖2 ≤ ‖λi −H−1λi‖2‖H‖2T −1

∑t‖Ft‖2 =Op (δ−2

N T ),


The last term is:

T −1∑

t‖λ′

i (H−1)′(Ft −H ′Ft )‖2 ≤ ‖λi‖2‖H−1‖2T −1∑

t‖Ft −H ′Ft‖2 =Op (δ−2

N T ),

by Assumption B. We thus have T −1 ∑t (Ci t − Ci t )2 =Op (δ−2

N T ).

Lemma A.12. Under Assumtions A-E we have:

T −1∑

te4

i t = T −1∑

te4

i t +Op (T /N 2)+Op (δ−1N T ).

A.1. APPENDIX 79

Proof: We can write T −1 ∑t e4

i t as:

T −1∑

te4

i t = T −1∑

t(xi t − Ci t )4 = T −1

∑t

(xi t −Ci t +Ci t − Ci t )4

= T −1∑

t(ei t +Ci t − Ci t )4 = T −1

∑t

e4i t +T −1

∑t

(Ci t − Ci t )4

+6T −1∑

te2

i t (Ci t − Ci t )2 +4T −1∑

te3

i t (Ci t − Ci t )+4T −1∑

tei t (Ci t − Ci t )3

= T −1∑

te4

i t + I + I I + I I I + IV.

Using Loève’s inequality, I can be written as:

T −1∑

t(Ci t − Ci t )4 = T −1

∑t

[(λi −H−1λi )′(Ft −H ′Ft )

+ (λi −H−1λi )′H ′Ft +λ′i (H−1)′(Ft −H ′Ft )

]4

≤ 27T −1∑

t[(λi −H−1λi )′(Ft −H ′Ft )]4 +27T −1

∑t

[(λi −H−1λi )′H ′Ft ]4

+27T −1∑

t[λ′

i (H−1)′(Ft −H ′Ft )]4.


T −1∑

t[(λi −H−1λi )′(Ft −H ′Ft )]4 ≤ ‖λi −H−1λi‖4max

t‖Ft −H ′Ft‖2T −1

∑t‖Ft −H ′Ft‖2

=Op (δ−4N T )Op (δ−2

N T )maxt

‖Ft −H ′Ft‖2.

By Proposition 2 in Bai (2003), maxt

‖Ft −H ′Ft‖2 =Op (T −1)+Op (T /N ), so the term is

Op (δ−6N T )(Op (T −1)+Op (T /N )).

The second term is:

T −1∑

t[(λi −H−1λi )′H ′Ft ]4 ≤ ‖λi −H−1λi‖4‖H‖4T −1

∑t‖Ft‖4 =Op (δ−4

N T ),


The third term is:

T −1∑

t[λ′

i (H−1)′(Ft −H ′Ft )]4 ≤ ‖λi‖4‖H−1‖4T −1∑

t‖Ft −H ′Ft‖4

≤Op (1)maxt

‖Ft −H ′Ft‖2T −1∑

t‖Ft −H ′Ft‖2

= [Op (T −1)+Op (T /N )]Op (δ−2N T ).

Now [Op (T −1)+Op (T /N )]Op (δ−2N T ) =Op (T −1)+Op (T /N 2), so I =Op (T −1)+Op (T /N 2).


For I I we have:

T −1∑

te2

i t (Ci t − Ci t )2 = T −1∑

te2

i t

[(λi −H−1λi )′(Ft −H ′Ft )+ (λi −H−1λi )′H ′Ft

+λ′i (H−1)′(Ft −H ′Ft )

]2

= T −1∑

t

[(λi −H−1λi )′(Ft −H ′Ft )ei t

+ (λi −H−1λi )′H ′Ft ei t +λ′i (H−1)′(Ft −H ′Ft )ei t

]2

≤ 3T −1∑

t‖(λi −H−1λi )′(Ft −H ′Ft )ei t‖2

+3T −1∑

t‖(λi −H−1λi )′H ′Ft ei t‖2 +3T −1

∑t‖λ′

i (H−1)′(Ft −H ′Ft )ei t‖2.


T −1∑

t‖(λi −H−1λi )′(Ft −H ′Ft )ei t‖2 ≤ ‖λi −H−1λi‖2T −1

∑t‖(Ft −H ′Ft )ei t‖2


N T ),

by Lemmas A.3 and A.9.

The second term is:

T −1∑

t‖(λi −H−1λi )′H ′Ft ei t‖2 ≤ ‖λi −H−1λi‖2‖H‖2T −1

∑t‖Ft ei t‖2 =Op (δ−2

N T )Op (1),

as the last term is bounded in expectation T −1 ∑t E‖Ft ei t‖2 ≤ T −1 ∑

t E‖Ft‖2E(e2i t )

≤ T −1 ∑t M =Op (1).

The third term is:

T −1∑

t‖λ′

i (H−1)′(Ft −H ′Ft )ei t‖2 ≤ ‖λi‖2‖H−1‖2T −1∑

t‖(Ft −H ′Ft )ei t‖2 =Op (1)Op (δ−2

N T ),

by Assumption B and Lemma A.3. So I I =Op (δ−2N T ).

The term I I I can be written as:

T −1∑

te3

i t (Ci t − Ci t ) ≤(T −1

∑t

e6i t

)1/2(T −1

∑t

(Ci t − Ci t )2)1/2

=Op (δ−1N T ),

since E (e8i t ) ≤ M , and the second term was show to be Op (δ−2

N T ) in the proof of Lemma

A.11.

For IV we have:

T −1∑

tei t (Ci t − Ci t )3 ≤

(T −1

∑t

e2i t (Ci t − Ci t )2

)1/2(T −1

∑t

(Ci t − Ci t )4)1/2

=(Op (δ−2

N T )

)1/2(Op (T −1)+Op (T /N 2)

)1/2

=Op (δ−2N T ),

A.1. APPENDIX 81

which follows from I and I I . We now have that I+I I+I I I+IV =Op (T −1)+Op (T /N 2)+Op (δ−2

N T )+Op (δ−1N T )+Op (δ−2

N T ) =Op (T /N 2)+Op (δ−1N T ).

Lemma A.13. Under Assumptions A-E:

a. T −1 ∑t Ft F ′

t e2i t −T −1 ∑

t Ft F ′t e2

i t =Op (δ−2N T ),

b. T −1 ∑t Ft F ′

t e2i t −T −1 ∑

t H ′Ft F ′t He2

i t =Op (δ−2N T ).

Proof: As in the proof of A.11, we use that e2i t = e2

i t + (Ci t − Ci t )2 +2(Ci t − Ci t )ei t . We

can therefore write a as:

T −1∑

tFt (e2

i t −e2i t )F ′

t = T −1∑

tFt (Ci t − Ci t )2F ′

t +2T −1∑

tFt (Ci t − Ci t )ei t F ′

t .

For the first term we have:

T −1∑

tFt (Ci t − Ci t )2F ′

t ≤ T −1∑

t‖(Ci t − Ci t )F ′

t‖2 =

T −1∑

t‖(λi −H−1λi )′(Ft −H ′Ft )F ′

t + (λi −H−1λi )′H ′Ft F ′t +λ′

i (H−1)′(Ft −H ′Ft )F ′t‖2

≤ 3T −1∑

t‖(λi −H−1λi )′(Ft −H ′Ft )F ′

t‖2 +3T −1∑

t‖(λi −H−1λi )′H ′Ft F ′

t‖2

+3T −1∑

t‖λ′

i (H−1)′(Ft −H ′Ft )F ′t‖2 = I + I I + I I I .

We can bound I by:

‖λi −H−1λ′i‖2T −1

∑t‖(Ft −H ′Ft )F ′

t‖2 =Op (δ−2N T )Op (δ−2

N T ),

by Lemmas A.5 and A.9 .

For I I we can write:

T −1∑

t‖(λi −H−1λi )′H ′Ft F ′

t‖2 ≤ ‖λi −H−1λi‖2‖H‖2T −1∑

t‖Ft F ′

t‖2

≤ ‖λi −H−1λi‖2‖H‖2T −1∑

t‖Ft (Ft −H ′Ft )′+Ft F ′

t H‖2

≤ 2‖λi −H−1λi‖2‖H‖2T −1∑

t‖Ft (Ft −H ′Ft )′‖2

+2‖λi −H−1λi‖2‖H‖4T −1∑

t‖Ft‖4 =Op (δ−2

N T )Op (δ−2N T )+Op (δ−2

N T )Op (1) =Op (δ−2N T ),

from Lemmas A.4 and A.9, and Assumption A.

For I I I we have:

T −1∑

t‖λ′

i (H−1)′(Ft −H ′Ft )F ′t‖2 ≤ ‖λi‖2‖H−1‖2T −1

∑t‖(Ft −H ′Ft )F ′

t‖2 =Op (δ−2N T ),


by Lemma A.5, so we have T −1 ∑t Ft (Ci t − Ci t )2F ′

t =Op (δ−4N T )+Op (δ−2

N T ) =Op (δ−2N T ).


T −1∑


t = T −1∑

t(Ft −H ′Ft )(Ci t − Ci t )ei t F ′

t

+T −1∑

tH ′Ft (Ci t − Ci t )ei t (Ft −H ′Ft )′+T −1

∑t

H ′Ft (Ci t − Ci t )ei t F ′t H .

We can bound T −1 ∑t (Ft −H ′Ft )(Ci t − Ci t )ei t F ′

t by:(T −1

∑t

(Ci t − Ci t )2e2i t

)1/2(T −1

∑t‖(Ft −H ′Ft )F ′

t‖2)1/2


N T ),

which follows as the term in the first parenthesis was shown to be Op (δ−2N T ) in the

proof of Lemma A.12, and the term in the second parenthesis is also Op (δ−2N T ) by

Lemma A.5. By similar calculations we also get T −1 ∑t H ′Ft (Ci t −Ci t )ei t (Ft −H ′Ft )′ =

Op (δ−2N T ).

For T −1 ∑t H ′Ft (Ci t − Ci t )ei t F ′

t H we can write (ignore H as it is Op (1)):

T −1∑


t = T −1∑

tFt

[(λi +H−1λi )′(Ft −H ′Ft )+ (λi −H−1λi )′H ′Ft

+λ′i (H−1)′(Ft −H ′Ft )

]ei t F ′

t = T −1∑

tFt (λi +H−1λi )′(Ft −H ′Ft )ei t F ′

t

+T −1∑

tFt (λi −H−1λi )′H ′Ft ei t F ′

t +T −1∑

tFtλ

′i (H−1)′(Ft −H ′Ft )ei t F ′

t .


‖λi +H−1λi‖(T −1

∑t‖(Ft −H ′Ft )ei t‖2

)1/2(T −1

∑t‖Ft‖4

)1/2

=Op (T −1/2)(δ−1N T )Op (1) =Op (δ−2

N T ),


For the second term we apply the vec operator:

T −1∑

t

(vec[Ft (λi −H−1λi )′H ′Ft ei t F ′

t ]

)′= T −1

∑t

(ei t (Ft F ′

t ⊗Ft )vec[(λi −H−1λi )′H ′])′

= T −1∑

t(λi −H−1λi )′H ′(Ft F ′

t ⊗F ′t )ei t = T −1/2(λi −H−1λi )′H ′

(T −1/2

∑t

(Ft F ′t ⊗F ′

t )ei t

)=Op (T −1/2)Op (T −1/2) =Op (T −1),

by Assumption E.4.

A.1. APPENDIX 83

For the third term we can write:

T −1∑

tλ′

i (H−1)′(Ft −H ′Ft )ei t Ft F ′t =λ′

i (H−1)′T −1∑


t =Op (1)Op (δ−2N T ),

by Assumption B and Lemma A.6. We therefore have that a is Op (δ−4N T )+Op (T −1)+

Op (δ−2N T ) =Op (δ−2

N T ).

For b we can write:

T −1∑

tFt F ′

t e2i t −T −1

∑t

H ′Ft F ′t He2

i t = T −1∑

t(Ft F ′

t −H ′Ft F ′t H)e2

i t

= T −1∑

t

((Ft −H ′Ft )F ′

t +H ′Ft (Ft −H ′Ft )

)e2

i t

= T −1∑

t(Ft −H ′Ft )F ′

t e2i t +T −1

∑t

H ′Ft (Ft −H ′Ft )e2i t

= T −1∑

t(Ft −H ′Ft )(Ft −H ′Ft )′e2

i t +T −1∑

t(Ft −H ′Ft )F ′

t He2i t +T −1

∑t

H ′Ft (Ft −F ′t H)′e2

i t .

From Lemma A.7, the last two terms are Op (δ−2N T ). The first term is bounded by:

‖T −1∑

t(Ft −H ′Ft )ei t ei t (Ft −H ′Ft )‖ ≤ T −1

∑t‖(Ft −H ′Ft )ei t‖2 =Op (δ−2

N T ),

from Lemma A.3. Thus b is Op (δ−2N T ).

Proof of Theorem 1: The proof consists of two steps. First we show that Di = Di +Op (δ−2

N T ). This implies thatp

T Di has the same limiting distribution asp

T Di ifpT /N → 0. The second step shows that Bi is a consistent estimator for Bi , and this

implies that T Di B−1i D ′

i has a χ2 distribution with r degrees of freedom from Assump-

tion G.

First consider Di = T −1 ∑Tt=1(e2

i t−σ2i )g (Ft F ′

t−F ′F /T ). We can ignore σ2i as T −1 ∑T

t=1 g (Ft F ′t−

F ′F /T ) = 0. We will show that the r × r matrix T −1 ∑Tt=1 e2

i t (Ft F ′t − F ′F /T ) matrix con-

verges to T −1 ∑Tt=1 e2

i t (H ′Ft F ′t H −H ′F ′F H/T ), and this will imply that Di converges

to Di as the g (Ft F ′t − F ′F /T ) is the vector of diagonal elements of Ft F ′

t − F ′F /T . We

can write:

T −1T∑

t=1e2

i t (Ft F ′t − F ′F /T ) = T −1

T∑t=1

e2i t Ft F ′

t −T −1T∑

t=1e2

i t F ′F /T.

From Lemma A.13 we have:

T −1T∑

t=1e2

i t Ft F ′t = T −1

T∑t=1

e2i t H ′Ft F ′

t H +Op (δ−2N T ).

From Lemma A.11 we have that T −1 ∑Tt=1 e2

i t = T −1 ∑Tt=1 e2

i t +Op (δ−2N T ), and for F ′F /T

we have:

F ′F /T = F ′(F −F H)/T + (F −F H)′F H/T +H ′F ′F H/T

= H ′F ′F H/T +Op (δ−2N T ),


from Lemmas B.2 and B.3 in Bai (2003), so T −1 ∑Tt=1 e2

i t F ′F /T = T −1 ∑Tt=1 e2

i t H ′F ′F H/T+Op (δ−2

N T ), and we therefore have that Di = Di +Op (δ−2N T ).

Next we need to show that Bi is a consistent estimator for Bi . Under Assumption G.1

and as Ft and ei t are assumed to be independent, we have that:

Bi = plimT→∞T −1T∑

t=1E

[(e2

i t −σ2i )2g [H ′(Ft F ′

t −F ′F /T )H ]g [H ′(Ft F ′t −F ′F /T )H ]′

]

= plimT→∞T −1T∑

t=1E(e2

i t −σ2i )2E

[g [H ′(Ft F ′


]

= (µ4,i −σ4i )plimT→∞T −1

T∑t=1

E

[g [H ′(Ft F ′


].

From Lemma A.11 we have that σ2i = T −1 ∑T

t=1 e2i t = T −1 ∑T

t=1 e2i t +Op (δ−2

N T ), so σ4i

can be consistently estimated by σ4i . Lemma A.12 states that T −1 ∑

t e4i t = T −1 ∑

t e4i t +

Op (T /N 2)+Op (δ−1N T ), so T −1 ∑

t e4i t is consistent for µ4,i . As the final step we show

that

T −1∑

tg (Ft F ′

t − F ′F /T )g (Ft F ′t − F ′F /T )′

= T −1T∑

t=1g [H ′(Ft F ′

t −F ′F /T )H ]g [H ′(Ft F ′t −F ′F /T )H ]′+Op (δ−1

N T )

which will imply that Bi is consistent for Bi .

We can write g (Ft F ′t − F ′F /T ) as:

g (Ft F ′t − F ′F /T ) =g [(Ft −H ′Ft )F ′

t +H ′Ft (Ft −H ′Ft )′+H ′Ft F ′t H − F ′F /T ]

=g [(Ft −H ′Ft )F ′t +H ′Ft (Ft −H ′Ft )′]+ g [H ′Ft F ′

t H − F ′F /T ]

=g [(Ft −H ′Ft )F ′t +H ′Ft (Ft −H ′Ft )′]

+g [H ′Ft F ′t H −H ′F ′F H/T ]

+g [H ′F ′F H/T − F ′F /T ].

A.1. APPENDIX 85

We therefore get:

T −1∑

tg (Ft F ′

t − F ′F /T )g (Ft F ′t − F ′F /T )′

= T −1∑

tg [H ′Ft F ′

t H −H ′F ′F H/T ]g [H ′Ft F ′t H −H ′F ′F H/T ]′

+T −1∑

tg [(Ft −H ′Ft )F ′

t +H ′Ft (Ft −H ′Ft )′]g [(Ft −H ′Ft )F ′t +H ′Ft (Ft −H ′Ft )′]′

+T −1∑

tg [H ′F ′F H/T − F ′F /T ]g [H ′F ′F H/T − F ′F /T ]′

+2T −1∑


t +H ′Ft (Ft −H ′Ft )′]g [H ′Ft F ′t H −H ′F ′F H/T ]′

+2T −1∑


t +H ′Ft (Ft −H ′Ft )′]g [H ′F ′F H/T − F ′F /T ]

+2T −1∑

tg [H ′Ft F ′

t H −H ′F ′F H/T ]g [H ′F ′F H/T − F ′F /T ]

= T −1∑

tg [H ′Ft F ′

t H −H ′F ′F H/T ]g [H ′Ft F ′t H −H ′F ′F H/T ]′+ I + I I + I I I + IV + IV.

We start with I . Recalling that g (A) is the vector of diagonal elements of A, we have

for any square matrix A:

‖g (A)‖2 = g (A)′g (A) ≤ vec(A)′vec(A) = tr (A′A) = ‖A‖2.

We can therefore write:

T −1∑


t +H ′Ft (Ft −H ′Ft )′]g [(Ft −H ′Ft )F ′t +H ′Ft (Ft −H ′Ft )′]′

≤ T −1∑

t

∥∥∥∥g [(Ft −H ′Ft )F ′t +H ′Ft (Ft −H ′Ft )′]

∥∥∥∥2

≤ 2T −1∑


t‖2 +2T −1∑

t‖H ′Ft (Ft −H ′Ft )′‖2 =Op (δ−2

N T ),


For I I we have:

T −1∑

tg [H ′F ′F H/T − F ′F /T ]g [H ′F ′F H − F ′F /T ]′

≤ ‖H ′F ′F H/T − F ′F /T ‖2 = ‖T −1F ′(F −F H)+T −1(F −F H)′F H‖2 =Op (δ−2N T ),

by Lemmas A.8.b and A.8.c.

For I I I we have:

T −1∑


t +H ′Ft (Ft −H ′Ft )′]g [H ′Ft F ′t H −H ′F ′F H/T ]′

≤(T −1

∑t‖(Ft −H ′Ft )F ′

t +H ′Ft (Ft −H ′Ft )′‖2)1/2(

T −1∑

t‖H ′Ft F ′

t H −H ′F ′F H/T ‖2)1/2

≤Op (δ−1N T )

(2‖H ′F ′F H/T ‖2T −1

∑t‖H ′Ft F ′

t H‖2)1/2

=Op (δ−1N T )Op (1),


from I above and Assumption A.

For IV we have:

T −1∑


t +H ′Ft (Ft −H ′Ft )′]g [H ′F ′F H/T − F ′F /T ]

= g [H ′F ′F H/T − F ′F /T ]T −1∑


t +H ′Ft (Ft −H ′Ft )′].

We can write H ′F ′F H/T −F ′F /T = T −1F ′(F −F H )+T −1(F −F H )′F H . From Lemmas

B.2 and B.3 in Bai (2003), these terms are Op (δ−2N T ), so IV =Op (δ−2

N T )Op (δ−2N T ).

Finally, for V we have:

T −1∑

tg [H ′Ft F ′

t H −H ′F ′F H/T ]g [H ′F ′F H/T − F ′F /T ]

= g [H ′F ′F H/T − F ′F /T ]T −1∑

tg [H ′Ft F ′

t H −H ′F ′F H/T ] =Op (δ−2N T ),

again from Lemmas B.2 and B.3 in Bai (2003). Thus I + I I + I I I + IV +V =Op (δ−2N T )+

Op (δ−4N T )+Op (δ−1

N T ) =Op (δ−1N T ), and the Theorem 1 follows.

C H A P T E R 3FOREIGN EXCHANGE RATES AND

MACROECONOMIC FACTORS: EVIDENCE FROM

TIME-VARYING LOADINGS

Eric HillebrandAarhus University and CREATES


Giovanni UrgaCass Business School

Abstract

We examine the relationship between variations in exchange rates and macroeco-

nomic fundamentals through a model that allows for time-varying effects. The funda-

mentals are extracted as factors from a large macroeconomic dataset, and the relation

between exchange rates and fundamentals is estimated as time-varying factor load-

ings. Our results show that the factor loadings on the macroeconomic factors vary

considerably over time with frequent sign changes. The factor loadings on a US real

activity indicator exhibit large increases during the financial crisis, and the loadings

on inflation and housing indicators are also found to be highly unstable. The model

explains a large share of the variations and directional changes in exchange rates.

87

88

CHAPTER 3. FOREIGN EXCHANGE RATES AND MACROECONOMIC FACTORS: EVIDENCE FROM


3.1 Introduction

In this paper, we study the unstable relationship between exchange rates and macroe-

conomic fundamentals. Survey evidence shows that the weights financial market

participants place on macroeconomic fundamentals as drivers of exchange rate fluc-

tuations vary considerably over time. Cheung and Chinn (2001) and Cheung, Chinn,

and Marsh (2004) document that the relative importance that US foreign exchange

traders attach to fundamentals changes over time. Using a more recent survey in

which financial market participants are asked to rate the current importance of fun-

damentals, Fratzscher, Rime, Sarno, and Zinna (2015) show that the survey weights

add explanatory power to exchange rate models. Sarno and Valente (2009) find that

the predictive information contained in fundamentals changes frequently over time.

There is thus ample evidence that the relationship between changes in exchange

rates and macroeconomic fundamentals is highly unstable. A possible explanation of

the weak explanatory power of fundamentals is given by Meese and Rogoff (1983a,b,

1988). They suggest that the presence of parameter instability could be the cause.

Rossi (2006) finds widespread parameter instability in exchange rate models confirm-

ing the unstable relationship between exchange rates and fundamentals.

We specify a model for the unstable relationship between exchange rates and fun-

damentals using the framework of Engel and West (2005). The model is derived from

an interest parity condition and an equation that relates the interest differential to

observed and unobserved macroeconomic fundamentals. The two equations can be

reduced to a single stochastic difference equation, where the exchange rate depends

on the present value of expected future fundamentals. The effect on the exchange rate

takes the form of a linear combination of the observed fundamentals. This linear rela-

tion between the exchange rate and the observed fundamentals is further affected by

unobservable fundamentals. These unobservable fundamentals are transitory shocks

that change the weights in the linear combination of observable fundamentals. The

presence of unobservable fundamentals can be given a number of interpretations.

They can be viewed as transitory shocks that affect the equilibrium relation between

the observed fundamentals and the exchange rate. Another interpretation is that

market participants change the weights they attach to observable fundamentals as

indicated by the survey evidence. The model gives a derivative of the exchange rate

with respect to the observable fundamentals that is time-varying. Through the unob-

servable shocks, the weights on the observable fundamentals change over time, and

the relationship between fundamentals and the exchange rate is therefore unstable.

A related paper is the scapegoat theory of Bacchetta and Van Wincoop (2013).

They propose a model where the exchange rate is driven by a linear combination

of macroeconomic fundamentals, some of which are unobservable. They show that

when there is uncertainty about the structural parameters of the model, the changes

in the exchange rate are driven by expectations of the structural parameters, and not

the structural parameters themselves. These expectations can be highly unstable,


and this results in the derivative of the exchange rate with respect to the observable

fundamentals being time-varying. Their model thus gives predictions about the

behaviour of exchange rates similar to ours.

In our empirical exercise, we consider the US exchange rate against 14 differ-

ent currencies for the period 1995-2014, and we estimate the time-varying relation

between the exchange rate and the observable fundamentals. We extract a set of

macroeconomic factors from a large macroeconomic dataset to represent the in-

formation available in observable fundamentals. The factors can be interpreted as

indicators of real activity, inflation, and variations in the housing market, respectively.

The factors act as the observable fundamentals in the model. The unstable weights

on the fundamentals are estimated by specifying time-varying factor loadings that

follow stationary autoregressions. The estimated paths of the loadings are thus an

estimate of the weights that agents attach to the macroeconomic indicators over

time.

The empirical results show that the weights on the factors are highly unstable

with large variations and frequent sign changes. The factor loadings on the US real

activity indicator exhibit large increases during the financial crises for all currencies.

The inflation and housing indicators also have highly variable weights. The relative

importance of the factors in determining the exchange rate thus varies significantly.

The model performs very well in explaining the variations in the exchange rates.

For the GBP and EUR exchange rates, the squared correlation with the model fit

is 0.66 and 0.77, respectively, and the proportion of correct sign matches between

the realized change in the exchange rates and the model fit is 0.83 and 0.89. In

comparison, when the factor loadings are treated as constants, the model explains

only a minor part of the exchange rate variations, and the coefficients are insignificant.

The results therefore show clear evidence that the relationship between changes in

exchanges rates and fundamentals is unstable.

The rest of the paper is organised as follows. In Section 3.2, we present the model

and show that it leads to an unstable relationship between exchange rate fluctuations

and variations in observed fundamentals. We then show that the model can be written

in the form of a factor model with time-varying loadings, where the time-varying

loadings represent the effect of variations in fundamentals on the exchange rate.

Section 3.3 describes the data and the macroeconomic indicators. In Section 3.4, we

discuss the empirical results. Section 3.5 concludes.

3.2 Model and Estimation

We consider the class of models of Engel and West (2005) where the exchange rate can

be described by a single stochastic difference equation. The value of the exchange

rate depends on the present value of expected future macroeconomic fundamentals.

90



Specifically, the relation is:

∆st =∞∑

j=0

(1

µ

) j

E [∆(Ft+ j +bt+ j )|It ]− 1

µ

∞∑j=0

(1

µ

) j

E [φt+ j |It ], (3.1)

where st is the log of the exchange rate measured as the domestic price per unit of

foreign currency, E(·|It ) is the expectation of the representative agent conditional

on It , the information set available at time t , and µ> 1. The value of the exchange

rate is determined by the expected future values of the observed macroeconomic

fundamentals, Ft , and by unobserved fundamentals, bt . Finally,φt can be interpreted

as the risk premium.

Eq. (3.1) results from two equations that describe how the exchange rate and the

fundamentals are related. The first equation is an uncovered interest parity condition

(UIP):

E(∆st+1|It ) = it − i∗t +φt , (3.2)

where it and i∗t are the domestic and foreign nominal one-period interest rates. The

expected change in the exchange rate is thus equal to the interest rate differential

between the domestic and foreign country in addition to possible risk premia.

The other equation relates the interest rate differential and the exchange rate with

the observed and unobserved macroeconomic fundamentals. Following Bacchetta

and Van Wincoop (2013) we write it in the form1

it − i∗t =µ∆st −µ∆(Ft +bt ). (3.3)

Eq. (3.3) is considered to be a reduced form macroeconomic model for exchange

rates. Engel and West (2005) discuss several examples that lead to an equation of this

form. Combining Eq. (3.2) and (3.3) leads to:

E(∆st+1|It ) =µ∆st −µ∆(Ft +bt )+φt

µ∆st = E(∆st+1|It )+µ∆(Ft +bt )−φt

∆st =∆(Ft +bt )− 1

µφt + 1

µE(∆st+1|It ).

(3.4)

By recursive substitution of ∆st and assuming no bubbles, Eq. (3.2) and (3.3) lead to

Eq. (3.1).

We specify the effect of the observed and unobserved fundamentals on the ex-

change rate to be of the form:

∆(Ft +bt ) = f ′t (β+κt ). (3.5)

The vector ft = ( f1t , ..., fr t )′ contains stationary observed macroeconomic funda-

mentals that determine the exchange rate, and the vector κt = (κ1t , ...,κr t )′ contains

1Note that Bacchetta and Van Wincoop (2013) specify their model for the levels of exchange rates. Inan earlier version of that paper (Bacchetta and Van Wincoop (2009)) they show that it also holds for firstdifferences.


mean-zero stationary unobservable fundamentals. The unobservable fundamentals

κt can be interpreted as transitory shocks that affect the equilibrium relation between

the macroeconomic fundamentals and the exchange rate, f ′tβ, or as shifts in expecta-

tions of market participants. In the long run, the relationship between fundamentals

and the exchange rate is reflected through the parameter vector β= (β1, ...,βr ). In the

short run, the transitory shocks κt can shift the relative importance of the fundamen-

tals in determining the exchange rate. For example, if at some future time t + s there

is a shock to the first element of κt+s such that κ1,t+s > 0, the first fundamental f1,t+s

will receive a larger weight β1 +κt+s > β1, and the exchange rate will be relatively

more affected by variations in the first fundamental f1,t+1 than in period with no

shocks.

To derive the effect of changes in the observed fundamentals on the exchange rate,

consider the scalar case with a single observed and unobserved fundamental driving

the exchange rate. The following result generalizes readily to multiple fundamentals.

We assume that both ft and κt follow AR(1) processes:

ft = ρ f ft−1 + vt ,

κt = ρκκt−1 +ut ,

where |ρ f |, |ρκ| < 1, and the innovations vt ,ut have mean zero and variances σ2f and

σ2κ, respectively. We have that E ( ft+ j |It ) = ρ j

f ft and E (κt+ j |It ) = ρ jκκt . Assuming that

ft and κt are uncorrelated and ignoring φt , Eq. (3.1) becomes:

∆st =∞∑

j=0

(1

µ

) j

ρjf ftβ+

∞∑j=0

(1

µ

) j

ρjκ ftκt = ft

(µ

µ−ρ fβ+ µ

µ−ρκκt

). (3.6)

The derivative of the exchange rate with respect to the observed fundamentals is:

∂∆st

∂ ft= µ

µ−ρ fβ+ µ

µ−ρκκt . (3.7)

The effect of variations in the macroeconomic fundamentals on the exchange rate

is equal to the constant part µµ−ρ f

β and the time-varying part µµ−ρκκt . The presence

of unobserved fundamentals therefore leads to an unstable relationship between

fundamentals and the exchange rate as reflected in the transitory component in the

derivative.

The unstable relationship between exchange rates and fundamentals has also

been considered by Bacchetta and Van Wincoop (2013) using a similar model as

our Eq. (3.1). They consider a case where the agents do not know the values of the

structural parameters β in the linear combination f ′tβ that determines the effect

of fundamentals on the exchange rate and show that this results in a time-varying

derivative of the exchange rate with respect to fundamentals as in Eq. (3.7). The

agents gradually learn about the parameters over time by forming expectations of the

structural parameters, and these parameters can vary significantly over time. They

92



derive the following expression for the effect of a change in the p th fundamental on

the exchange rate:

∂st

∂ fpt= θβp + (1−θ)E(βp |It )+ (1−θ) f ′

t∂E(β|It )

∂ fpt,

where we have changed their notation to match ours. The two first terms are a

weighted average of the true structural parameters and the agents’ expectation of

them. The last term is time-varying and reflects the gradual learning about the pa-

rameter β. The agents observe the signal f ′tβ+bt , where bt contains unobserved

fundamentals. As they do not know the value of β, they cannot determine whether

a large value of the signal f ′tβ+bt is due to changes in fundamentals ft combined

with a larger value of β than expected, or changes in unobservables bt . It therefore

becomes rational for the agents to attribute at least some weight to a larger β, which

raises their expectations of the structural parameter β, and this results in the time-

varying relationship between fundamentals and the exchange rate even though the

structural parameters are constant. The Bacchetta and Van Wincoop (2013) model

thus leads to a similar relationship between the fundamentals and the exchange rate

as our model.

Combining Eq. (3.6) with the autoregressive process for the unobservable shocks

κt gives the system:

κt = ρκκt−1 +ut ,

∆st = ft

(µ

µ−ρ fβ+ µ

µ−ρκκt

).

(3.8)

We aim to estimate of the relation between exchange rates and fundamentals, µµ−ρ f

β+µ

µ−ρκκt . We therefore write the system (3.8) in the following state space representa-

tion:

λt − λ= b(λt−1 − λ)+ηt ,

∆st = f ′tλt +εt ,

ηt ∼ i .i .d .(0,σ2η),

εt ∼ i .i .d .(0,σ2ε).

(3.9)

The measurement error εt can be interpreted as an estimate of the risk premium

which was omitted from Eq.(3.6) for simplicity.2 The state vector λt is estimating the

relation between the macroeconomic fundamentals ft and the exchange rate st , and

by comparing (3.8) and (3.9) it is clear that λt = µµ−ρ f

β+ µµ−ρκκt . We can therefore

map the parameters of the state space representation (3.9) to the parameters of the

2If we include the risk premium in Eq. (3.6) and assume that it follows the AR processφt = ρφφt−1+εt

we get ∆st = ft

(µ

µ−ρ fβ+ µ

µ−ρκ κt

)− 1µ−ρφ φt . The measurement error in Eq. (3.9) is then εt =− 1

µ−ρφ φt .


system (3.8) as:

λ= E(λt ) = E

(µ

µ−ρ fβ+ µ

µ−ρκκt

)= µ

µ−ρ fβ,

σ2η

1−b2 = var (λt ) = var (µ

µ−ρ fβ+ µ

µ−ρκκt ) =ω2var (κt ) =ω2 σ2

u

1−ρ2κ

,

where ω= µµ−ρκ . The autocorrelation parameter ρκ of the process κt is the same as

the autocorrelation parameter b of λt . Estimation of the state space system (3.9)

will therefore give estimates of the parameter vector µµ−ρ f

β and estimates of the

unobserved shock process scaled by ω.

The state space representation (3.9) generalizes readily to the multivariate case

with r observed fundamentals ft = ( f1t , ..., fr t )′ and state vector λt = (λ1t , ...,λr t )′ as

follows:

B(L)(λt − λ) = ηt ,

∆st = f ′tλt +εt ,

ηt ∼ i .i .d .(0,Q),

εt ∼ i .i .d .(0,σ2ε),

(3.10)

where λ = E(λt ) is the unconditional mean of the state vector λt , and B(L) = I −B 0

i ,1L− ...−B 0i ,q Lq is a q th-order lag polynomial with roots outside the unit circle. The

state innovations ηt has covariance matrix E(ηtη′t ) =Q.

In our empirical exercise, we use a set of macroeconomic indicators to represent

the information available in the observable fundamentals. The indicators are ex-

tracted as the first r principal components from a large panel of macroeconomic data

series X t = (X1t , ..., XN t )′, t = 1, ...,T . That is, we assume that Xi t has an approximate

factor structure of the form:

Xi t =α′i t ft +εi t ,

where ft is an r ×1 vector of common factors, αi t is the corresponding time-varying

factor loadings, and εi t are idiosyncratic errors. The approximate factor structure

allows the idiosyncratic errors to have limited cross-sectional correlation. The num-

ber of factors, r , is significantly smaller than the number of series, N , such that the

information in the large number of macroeconomic series is summarized by the r

dimensional factors, ft . We use the principal components estimator to estimate the

factors. The principal components estimator treats the loadings as being constant

over time, αi t ≡αi , and solves the minimization problem:

( f , αi ) = minf ,αi

(N T )−1N∑

i=1

T∑t=1

(Xi t −α′i ft )2,

where f is the T ×r matrix of common factors, and α is r ×1 vector of factor loadings.

By imposing the restriction f ′ f /T = Ir and concentrating out αi , the minimization

problem is equivalent to maximizing tr ( f ′(X X ′) f ), where X = (X1, ..., XT )′ is the

94



T ×N matrix of observations. The resulting estimator f is given byp

T times the

eigenvectors corresponding to the r largest eigenvalues of the T ×T matrix X X ′.The solution is not unique: any orthogonal rotation of f is also a solution. Bates

et al. (2013) show that the principal components estimator gives consistent factor

estimates when there is instability in the factor loadings.

Having obtained the principal components estimates ft , we estimate the parame-

ters of the state space model (3.10) by forming the likelihood function:

LT (∆s| f ;θ) =−1

2log(2π)− 1

2Tlog|Σ|− 1

2T(∆s −E(∆s))′Σ−1(∆s −E(∆s)),

where ∆s = (∆s1, ...,∆sT )′ with mean E(∆s) = f λ and variance matrix V ar (∆s) = Σ.

The parameter vector is θ = (B(L), λ,Q,σ2ε) and is estimated as:

θ = argmaxθ

LT (∆s| f ;θ).

The likelihood can be calculated with the Kalman filter as (3.10) is a linear state space

system. Mikkelsen, Hillebrand, and Urga (2015) show that the maximum likelihood

estimator θ is consistent for the parameters of the time-varying factor loadings λt .

The estimates of the factor loadings λt for t = 1, ...,T are computed with the state

smoother. The estimation procedure therefore gives estimates of the time-varying

relationship between the factors and the exchange rate in Eq. (3.7).

To access the relative contribution of the time-varying loadings in explaining

variations in the exchange rate, we also estimate a version of the model where the

factor loadings are constant. If the unobservable shocks κt are absent in Eq. (3.5),

the model in (3.8) reduces to ∆st = ft

(µ

µ−ρ f

), consistent with a present value model

for exchange rates with constant parameters. The relation between changes in the

exchange rate and fundamentals in this model is

(µ

µ−ρ f

), which can be estimated by

a regression of ∆st on the factors ft . We denote the estimate of

(µ

µ−ρ f

)by λOLS . The

least squares model thus serves as a benchmark to access to relative contribution

of the time-varying factor loadings for the models’ performance. By comparing the

model fit from the time-varying model λ′t ft with model fit from the constant loadings

model λ′OLS ft , we can measure to what extent the time-varying factor loadings add

to the explanatory power of the factors.

3.3 Data

We use monthly averages of the US dollar exchange rate against a set of 14 currencies

spanning the period January 1995 - December 2014. The exchange rates are: the

Australian dollar (AUD), the Brazilian real (BRL), the Canadian dollar (CAD), the

Danish krone (DKK), the Indian rupee (INR), the Mexican peso (MXN), the New

3.3. DATA 95

Zealand dollar (NZD), the Norwegian krone (NOK), the South African rand (ZAR),

the Swedish krona (SEK), the Swiss franc (CHF), the British pound (GBP), and the

Euro (EUR).3 The data is taken from WRDS, and consists of exchange rates from the

Federal Reserve Bank.

Table 3.1 reports summary statistics for the first difference of the 14 log exchange

rates. The mean percentage changes are close to zero for all the exchange rates with

standard deviations varying from 1.7% to 4.1%. The largest variations occur for the

Brazilian real where the largest monthly changes are a 24.2% depreciation and 11.3%

appreciation against the US dollar. The autocorrelations are all positive at the one

month horizon, and the Box-Pierce tests show that the first 3 autocorrelations are

significant at the 5% level for all exchange rates except the Mexican peso.

For the macroeconomic indicators, we use the FRED-MD database of McCracken

and Ng (2015)4 which consists of 125 monthly series sorted into 8 broad categories: 1:

output & income, 2: labour market, 3: housing, 4: orders & inventories, 5: money &

credit, 6: interest rates, 7: prices, and 8: stock market. The data series are transformed

to stationarity before extracting principal components. We use the same transforma-

tions as McCracken and Ng (2015) and refer to their paper for transformation codes

and a closer variable description.

For the estimation of the model, we use one and three factors, respectively.5

Results using 5 factors can be found in the Appendix. In Figure 3.1, we plot squared

correlations of the principal components with the macro series sorted into variable

groups. The first factor correlates strongly with measures of output and employment,

and also exhibits large correlations with variables associated with manufacturing

orders and capacity utilization. The correlations with price variables and money and

credit aggregates are small. We can therefore interpret the first factor as an indicator

of real activity. The second and third factors display similar correlation patterns. Both

factors load heavily on series for housing starts and new housing permits, as well

as price variables and credit aggregates. The correlations with the remaining series

are small. These factors are thus associated with variations in the housing markets

and inflation. A large share of the variations in the macroeconomic series is naturally

associated with the financial crises, and this is reflected in the second and third factor.

The time series of macroeconomic indicators are shown in Figure 3.2. The effect

3Prior to 1999, the exchange rate for the ECU is used in place of the Euro, i.e. an weighted average ofthe Austrian Schilling, Belgian and Luxembourg Francs, Finnish Markka, French Franc, German Mark,Irish Pound, Italian Lira, Netherlands Guilder, Portuguese Escudo, and Spanish Peseta.

4The data was downloaded on 04/01/2016 from the FRED-MD website:https://research.stlouisfed.org/econ/mccracken/fred-databases/ and consists of the most recentvintage available at that date. As in McCracken and Ng (2015) we remove 5 series to get a balanced panel.In addition, we remove 5 exchange rate series from the dataset to avoid including variations from exchangerates in fundamentals.

5We have estimated the number of factors in the macroeconomic dataset using various methods. TheBai and Ng (2002) ICp1 criterion picks 9 factors. The criterion of Kapetanios (2004) results in 6 factors.Using the method of Alessi et al. (2010) on the two criteria results in 1 and 3 factors, respectively. Theresults are, however, sensitive to the choice of tuning parameters.

96



of the subprime crisis on the US economy is clearly evident in the factors. The real

activity indicator in (a) exhibits a large decline that starts during 2007 and reaches its

trough in December 2008. The early 2000s recession is also reflected in the smaller

drop in the real factor in this period. The second and third factors show a great

amount of variability. Both factors exhibit large variations from mid-2008 to mid-

2009 after which they vary around a lower level. This reflects the decrease in new

housing starts and housing permits as well as the lower inflation rates in the US from

2009 to the end of the sample period.

3.4 Empirical Results

For each exchange rate, we estimate the model in (3.10) and report results for the

estimated factor loadings λt as well as the estimated model fit, λ′t ft . The state equa-

tions for the loadings are univariate autoregressions of order one in all results. We

also report the fitted values from the least squares regression of the exchange rate

changes on the factors, λ′OLS ft . To compare the models performance, we report the

squared correlations (R2) between the changes in the exchange rate and the model

fit. We also report the hit rate (HR), which is the proportion of times the fitted value

has the same sign as the realized change in the exchange rates. For the discussion of

the empirical results, we focus on the exchange rates for GBP and EUR.

Figure 3.3 (a) plots the exchange rate changes for GBP together with the model fit

from the time-varying and constant loadings models, respectively. The time-varying

loadings model tracks the large depreciation and appreciation of the GBP during the

financial crises remarkably well, using only the US real activity indicator. The model

is able to explain 45.6% of the variation in GBP (see Table 3.2) and correctly signs the

directional changes for 78.66% of the observations (see Table 3.3). In contrast, the

R2 from the least squares regression in only 1.6%, showing almost no explanatory

power when the factor loadings are taken to be constant. The hit rate is 54.81%, which

is only marginally better than a naive model that determines the sign change at

random with probability 0.5. The time-varying factor loadings thus add substantial

explanatory power compared to the constant loadings model. Figure 3.3 (b) shows

the estimated factor loadings on the real activity indicator. The time-varying weights

on the indicator are highly variable with frequent sign changes. In particular, the

loadings exhibit a large increase followed by a rapid decrease during the financial

crisis, which leads to the strong correlation between the exchange rate and the factor

during this period. In comparison, the least squares factor loading is close to zero

and insignificant, showing almost no relationship with the real activity factor. The

time-varying factor loadings are far outside the confidence bands for the constant

loading during the financial crisis period.

When we add the housing and inflation factors to the model, we see an improve-

ment in the performance of the model. Figure 3.4 (a) shows the fit of the model

3.4. EMPIRICAL RESULTS 97

using the three factors. We now track more of the variation outside the financial

crisis period through the inclusion of the inflation and housing indicators. The R2

for the time-varying loadings model increases to 66%, and the hit rate increases over

4 percentage points to 82.85%. With constant factor loadings, we see an increase

in the hit rate of a similar magnitude, but the share of correctly signed directional

changes is still considerable lower than for the time-varying loadings model. The R2

is also markedly lower at 8.8%. The factor loadings on the first factor hardly change

when adding the second and third factor. The estimated loadings are very similar

in Figure 3.3 (b) and 3.4 (b), both for the time-varying and constant factor loadings.

The time-varying weights on the two housing and inflation factors are both highly

unstable, but exhibit quite different behaviour. The factor loadings on the second

factor are anti-persistent with a negative AR-parameter, while the loadings on the

third factor has a positive AR-parameter. The loadings on the third factor exhibit a

large decline at the end of the nineties followed by a steady increase in the early 2000s.

The least squares loadings on the second and third factors are close to zero, with

only the second loading being marginally significant. The model with time-varying

weights on the macroeconomic indicators thus substantially outperforms the model

with constant weights, both in terms of the R2 and directional accuracy.

We now move to the results for the EUR exchange rate. Figure 3.5 displays the

estimated factor loadings and model fit for the models with the US real activity

factor. The time-varying loadings model is again seen to capture the variations in

the exchange rate accurately during the financial crisis. The R2 is 43.7%, and the

rate of correctly signed directional changes is 72.38%. In comparison, the constant

loadings model has an R2 that is zero to three decimals, and the fit of the model in

Figure 3.6 is almost a flat line at zero. The hit rate with constant factor loadings is

54.39%, again not much better than a naive model. The time-varying loadings model

gives a large weight to the real activity indicator during 2008 (Figure 3.5 (b)). The

estimated factor loadings exhibit a large spike during 2008, and also exhibit frequent

variations and sign changes outside this period. The constant factor loading is not

significantly different from zero, and the time-varying loadings are outside the least

squares confidence bounds for a large part of the sample period.

The model fit for the time-varying loadings model increases markedly with the

inclusion of the housing and inflation indicators. The R2 increases to 77.1%, and

the hit rate is 89.12%, so the model signs the realized change in the exchange rate

correctly for almost 90% of the observations. The improvement in the model fit is

clearly evident in Figure 3.6 (a). The model tracks the changes in the EUR exchange

rate closely, albeit exhibiting less variability than the actual changes. In particular,

the variations during the sovereign debt crisis are captured very accurately by the

inclusion of the housing and inflation indicators. The model with constant factor

loadings does not perform better when the housing and inflation indicators are

added. The model fit in Figure 3.6 (a) shows only minor variations around zero. The

98



R2 is 0.4%, and the hit rate even decreases compared to the model with only one

factor. The estimated factor loadings are plotted in Figure 3.6 (b)-(d). The constant

factor loadings are all insignificant as expected from the poor fit for this model. The

time-varying factor loadings on the first factor are similar to the results in Figure 3.5

(b). The weights on the housing and inflation factors in Figure 3.6 (c) and (d) are both

highly unstable. The loadings frequently changes sign and often vary outside the

confidence bounds for the constant factor loadings. The model with time-varying

loadings outperforms the constant loadings model by an even larger margin than

for the GDP exchange rate, and the results show clear evidence in favour of unstable

weights on the macroeconomic indicators.

We briefly discuss the estimation results for the remaining exchange rates. Figures

3.7-3.10 show the results for the models with a single factor. The exchange rate

changes and model fit are plotted in the left column, and the estimated factor loadings

are plotted in the right column. For all currencies, there is a large increase in the

weight on the real activity indicator during the financial crisis period. The models

with time-varying factor loadings track the changes in the exchange rate accurately

in this period for all currencies except for the CHF rate, where the model fit only

exhibits minor variations. The constant factor loadings are all insignificant, and

these models do not explain a significant part of the variations in exchange rates.

Comparing the models’ performance on the basis of the R2 in Table 3.2, the models

with time-varying factor loadings all outperform the models with constant loadings.

The R2s with constant factor loadings are all less than 1%. The R2s for the models with

time-varying loadings show R2s varying from 15.4% for CHF to 55.7% for AUD. The

hit rates are reported in Table 3.3. The shares of correctly signed directional changes

are all the largest for the models with time-varying factor loadings. We even see some

hit rates being lower than 50% for the models with constant factor loadings.

Figures 3.11-3.14 show the results for the models with all three factors included.

The exchange rate changes and model fit are again shown in the left column. The

right column plots the time-varying factor loadings. We omit the constant loadings

to avoid cluttering of the graphs. The estimated factor loadings are quite unstable

for most currencies. MXN stands out with factor loadings on the second and third

factor not exhibiting any variation. Furthermore, BRL, JPY, and ZAR each have a

single loading that is estimated as being constant, but the general conclusion is that

the factor loadings are highly unstable. The models with time-varying loadings all

outperform the constant loadings models on the basis of R2 and hit rates. The R2

are substantially larger, and the hit rates also clearly dominate for the models with

time-varying loadings.

If we compare our results with the scapegoat theory of Bacchetta and Van Win-

coop (2013), our findings are broadly consistent with the scapegoat explanation.

Their model predicts that the weights on macroeconomic variables increase (become

scapegoats) when, at the same time, there is a large movement in the exchange rate,

3.5. CONCLUSIONS 99

and the macroeconomic variables deviate from their long run equilibrium values.

In particular, the large drop in the real activity indicator around 2009 in Figure 3.2

coincides with the large variations in exchange rates during the same period. This is

exactly when the estimated factor loadings all exhibit large increases, showing that

the weights on the real activity indicator increase at this time as predicted by the

scapegoat theory.

In summary, the results clearly show that the relationship between exchange rates

and macroeconomic factors is highly unstable. The estimated factor loadings change

frequently over time, and the relative importance of each factor in explaining the

variations in exchange rates can shift dramatically. In contrast, when the weights on

the factors are treated as constants, the models show only minor explanatory power.

The model with time-varying factor loadings outperform the constant loadings model

substantially on the basis of R2 and hit rate.

3.5 Conclusions

In this paper, we study the relationship between exchange rates and macroeconomic

fundamentals. Using the framework of Engel and West (2005), we specify a present

value model for the exchange rate that leads to an unstable relationship between

changes in the exchange rate and macroeconomic fundamentals. The weights on the

macroeconomic fundamentals change through the presence of unobservable shocks

to these weights. The model is applied to the US exchange rate versus 14 different

currencies. We extract a set of macroeconomic factors from a large macroeconomic

dataset to represent the information available in the fundamentals. The macroeco-

nomic factors have the interpretation of a US real activity indicator and housing and

inflation indicators. The unstable weights on the factors are estimated by specifying

time-varying factor loadings. The results show that the estimated factor loadings are

highly unstable. The loadings on the factors exhibit large variations and frequent

sign changes. In particular, the weights on the US real activity indicator show large

increases during the financial crisis. We benchmark our results against a model with

constant loadings on the factors. The model with time-varying factor loadings is

far superior in explaining both the variations in exchange rates and the directional

changes. We thus provide clear evidence of unstable weights on macroeconomic

fundamentals.

Acknowledgements

Jakob Guldbæk Mikkelsen and Eric Hillebrand acknowledge support from The Danish

Council for Independent Research (DFF 4003-00022) and CREATES - Center for

Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish

National Research Foundation.

100



Table 3.1: Summary statistics for exchange rates.

Currency Mean Std.dev. Min. Max. ρ(1) ρ(2) ρ(3) QBP

AUD 0.000 0.029 -0.173 0.071 0.360 0.019 0.048 0.000BRL -0.005 0.041 -0.242 0.113 0.422 0.029 -0.019 0.000CAD 0.001 0.017 -0.113 0.060 0.276 0.079 0.050 0.000DKK -0.000 0.024 -0.078 0.062 0.304 -0.036 0.001 0.000INR -0.003 0.017 -0.066 0.059 0.321 -0.004 0.010 0.000JPY -0.001 0.027 -0.081 0.105 0.304 0.125 0.033 0.000MXN -0.004 0.028 -0.177 0.089 0.148 -0.086 0.021 0.068NZD 0.001 0.028 -0.103 0.074 0.330 0.092 0.142 0.000NOK -0.000 0.025 -0.132 0.057 0.367 0.046 0.001 0.000ZAR -0.005 0.036 -0.192 0.105 0.320 -0.005 0.018 0.000SEK -0.000 0.026 -0.108 0.071 0.349 0.006 0.036 0.000CHF 0.001 0.026 -0.117 0.082 0.194 -0.013 -0.012 0.028GDP -0.000 0.020 -0.095 0.060 0.271 0.052 0.082 0.000EUR -0.000 0.023 -0.078 0.062 0.297 -0.035 -0.003 0.000

NOTE: The table reports summary statistics for the first difference of the log US exchange rates for the period Jan 1995-Sep 2015. For each exchange rate we report the mean, standard deviation (Std.dev.), minimum (Min.), maximum(Max.), autocorrelations at 1 month ρ(1), 2 months ρ(2), and 3 months ρ(3), and the p-value of the Box-Pierce QBPtest.

Table 3.2: Model performance – R2.

1 Factor 3 Factors

Time-varying Constant Time-varying ConstantCurrency factor loadings factor loadings factor loadings factor loadings

AUD 0.557 0.002 0.794 0.007BRL 0.308 0.001 0.677 0.014CAD 0.482 0.004 0.846 0.009DKK 0.425 0.000 0.708 0.004INR 0.341 0.004 0.753 0.004JPY 0.207 0.008 0.278 0.011MXN 0.322 0.005 0.318 0.009NZD 0.359 0.000 0.691 0.013NOK 0.474 0.000 0.665 0.016ZAR 0.430 0.001 0.475 0.024SEK 0.435 0.004 0.799 0.012CHF 0.154 0.002 0.472 0.010GBP 0.456 0.016 0.664 0.088EUR 0.437 0.000 0.771 0.004

NOTE: The table reports the squared correlations between changes in the exchange rate and the model fit with time-varying and constant loadings, respectively, for the model with one and three factors.

3.5. CONCLUSIONS 101

Table 3.3: Model performance – HR(%).

1 Factor 3 Factors



NOTE: The table reports the hit rate in percent, i.e. the proportion of times the sign of the fitted values is the same asthe sign of the actual changes for the models with time-varying and constant factor loadings.

102



Figure 3.1: Marginal R2 between factors and macro series.

(a) 1st factor.

1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) 2nd factor.

1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c) 3rd factor.

1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

NOTE: Figures (a) - (c) show the R2 from a regression of the series on the the first, second, andthird factor, respectively. The series are grouped into 1: output & income, 2: labour market,3: housing, 4: orders & inventories, 5: money & credit, 6: interest rates, 7: prices, and 8: stockmarket.


Figure 3.2: Factors.

(a) 1st factor.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-5

-4

-3

-2

-1

0

1

(b) 2nd factor.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-3

-2

-1

0

1

2

(c) 3rd factor.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-3

-2

-1

0

1

2

3

NOTE: Figures (a) - (c) plot the time series of principal components.

104



Figure 3.3: Results for GBP with 1 factor.

(a) Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Data

time-varying

constant

(b) 1st factor loading.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025 Time-varying

constant

conf. band

NOTE: Figure (a) plots the changes in the exchange rate (black line), the model fit with time-varying loadings (blue line), and the model fit with constant loadings (red line). Figure (b)plots the estimated factor loadings: time-varying factor loadings (black line), constant factorloading (red line), and 95% confidence intervals for the constant factor loadings based onrobust standard errors (HAC).


Figure 3.4: Results for GBP with 3 factors.


1997 2000 2002 2005 2007 2010 2012

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time-varying

constant

conf. band

(c) 2nd factor loading.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.002

0

0.002

0.004

0.006

0.008

0.01

Time-varying

constant

conf. band

(d) 3rd factor loading.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.007

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002Time-varying

constant

conf. band

NOTE: Figure (a) plots the changes in the exchange rate (black line), the model fit with time-varying loadings (blue line), and the model fit with constant loadings (red line). Figures (b)-(d)plot the estimated factor loadings: time-varying factor loadings (black line), constant factorloading (red line), and 95% confidence intervals for the constant factor loadings based onrobust standard errors (HAC).

106



Figure 3.5: Results for EUR with 1 factor.


1997 2000 2002 2005 2007 2010 2012

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015 -0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time-varying

constant

conf. band

NOTE: Figure (a) plots the changes in the exchange rate (black line), the model fit with time-varying loadings (blue line), and the model fit with constant loadings (red line). Figure (b)plots the estimated factor loadings: time-varying factor loadings (black line), constant factorloading (red line), and 95% confidence intervals for the constant factor loadings based onrobust standard errors (HAC).


Figure 3.6: Results for EUR with 3 factors.


1997 2000 2002 2005 2007 2010 2012

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time-varying

constant

conf. band

(c) 2nd factor loading.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Time-varying

constant

conf. band

(d) 3rd factor loading.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.01

-0.005

0

0.005

0.01

Time-varying

constant

conf. band

NOTE: Figure (a) plots the changes in the exchange rate (black line), the model fit with time-varying loadings (blue line), and the model fit with constant loadings (red line). Figures (b)-(d)plot the estimated factor loadings: time-varying factor loadings (black line), constant factorloading (red line), and 95% confidence intervals for the constant factor loadings based onrobust standard errors (HAC).

108



Figure 3.7: Results with 1 factor.

(a) AUD: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.15

-0.1

-0.05

0

0.05

Data

time-varying

constant

(b) AUD: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.01

0

0.01

0.02

0.03

0.04Time-varying

constant

conf. band

(c) BRL: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1 Data

time-varying

constant

(d) BRL: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Time-varying

constant

conf. band

(e) CAD: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant

(f ) CAD: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025 Time-varying

constant

conf. band

NOTE: The figures in the left column plot the changes in the exchange rate (black line), the modelfit with time-varying loadings (blue line), and the model fit with constant loadings (red line).The figures in the right column plot the estimated factor loadings: time-varying factor loadings(black line), constant factor loading (red line), and 95% confidence intervals for the constant factorloadings based on robust standard errors (HAC).



(a) DKK: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant

(b) DKK: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time-varying

constant

conf. band

(c) INR: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.06

-0.04

-0.02

0

0.02

0.04

Data

time-varying

constant

(d) INR: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.01

-0.005

0

0.005

0.01

0.015Time-varying

constant

conf. band

(e) JPY: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1 Data

time-varying

constant

(f ) JPY: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006 Time-varying

constant

conf. band


110




(a) MXN: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.15

-0.1

-0.05

0

0.05

Data

time-varying

constant

(b) MXN: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035Time-varying

constant

conf. band

(c) NZD: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant

(d) NZD: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025Time-varying

constant

conf. band

(e) NOK: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Data

time-varying

constant

(f ) NOK: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.03

-0.02

-0.01

0

0.01

0.02

0.03Time-varying

constant

conf. band




(a) ZAR: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.15

-0.1

-0.05

0

0.05

0.1Data

time-varying

constant

(b) ZAR: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05 Time-varying

constant

conf. band

(c) SEK: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant

(d) SEK: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025Time-varying

constant

conf. band

(e) CHF: Exchange rate and model fit.

1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08Data

time-varying

constant

(f ) CHF: factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

Time-varying

constant

conf. band


112



Figure 3.11: Results with 3 factors.


1997 2000 2002 2005 2007 2010 2012

-0.15

-0.1

-0.05

0

0.05

Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.01

0

0.01

0.02

0.03

0.04

λt,1

λt,2

λt,3


1997 2000 2002 2005 2007 2010 2012

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1 Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04λ

t,1

λt,2

λt,3


1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

λt,1

λt,2

λt,3

NOTE: The figures in the left column plot the changes in the exchange rate (black line), the modelfit with time-varying loadings (blue line), and the model fit with constant loadings (red line). Thefigures in the right column plot the time-varying factor loadings.




1997 2000 2002 2005 2007 2010 2012

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

λt,1

λt,2

λt,3


1997 2000 2002 2005 2007 2010 2012

-0.06

-0.04

-0.02

0

0.02

0.04

Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02λ

t,1

λt,2

λt,3


1997 2000 2002 2005 2007 2010 2012-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1 Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008λ

t,1

λt,2

λt,3


114





1997 2000 2002 2005 2007 2010 2012

-0.15

-0.1

-0.05

0

0.05

Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

λt,1

λt,2

λt,3


1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

λt,1

λt,2

λt,3


1997 2000 2002 2005 2007 2010 2012

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.01

0

0.01

0.02

0.03λ

t,1

λt,2

λt,3





1997 2000 2002 2005 2007 2010 2012

-0.15

-0.1

-0.05

0

0.05

0.1Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05λ

t,1

λt,2

λt,3


1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025λ

t,1

λt,2

λt,3


1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

λt,1

λt,2

λt,3


116



3.6 References

Alessi, L., Barigozzi, M., Capasso, M., 2010. Improved penalization for determining

the number of factors in approximate factor models. Statistics & Probability Letters

80 (23), 1806–1813.

Bacchetta, P., Van Wincoop, E., 2009. On the unstable relationship between exchange

rates and macroeconomic fundamentals. NBER Working paper No 15008.

Bacchetta, P., Van Wincoop, E., 2013. On the unstable relationship between exchange

rates and macroeconomic fundamentals. Journal of International Economics 91 (1),

18–26.





177, 289–304.

Cheung, Y.-W., Chinn, M. D., 2001. Currency traders and exchange rate dynamics:

a survey of the US market. Journal of International Money and Finance 20 (4),

439–471.

Cheung, Y.-W., Chinn, M. D., Marsh, I. W., 2004. How do UK-based foreign exchange

dealers think their market operates? International Journal of Finance & Economics

9 (4), 289–306.


Economy 113 (3).

Fratzscher, M., Rime, D., Sarno, L., Zinna, G., 2015. The scapegoat theory of exchange

rates: the first tests. Journal of Monetary Economics 70, 1–21.

Kapetanios, G., 2004. A new method for determining the number of factors in factor

models with large datasets. Working Paper, Department of Economics, Queen Mary,

University of London.

McCracken, M., Ng, S., 2015. FRED-MD: A monthly database for macroeconomic

research. Forthcoming, Journal of Business and Economic Statistics.

Meese, R., Rogoff, K., 1983a. Empirical exchange rate models of the seventies: Do they

fit out of sample? Journal of International Economics 14 (1), 3–24.

Meese, R., Rogoff, K., 1983b. The out-of-sample failure of empirical exchange rate

models: sampling error or misspecification? In: Exchange Rates and International

Macroeconomics. University of Chicago Press, 67–112.

3.6. REFERENCES 117

Meese, R., Rogoff, K., 1988. Was it real? the exchange rate-interest differential relation

over the modern floating-rate period. Journal of Finance 43 (4), 933–948.

Mikkelsen, J., Hillebrand, E., Urga, G., 2015. Maximum likelihood estimation of time-

varying loadings in high-dimensional factor models. CREATES research paper

2015-61.

Rossi, B., 2006. Are exchange rates really random walks? some evidence robust to

parameter instability. Macroeconomic Dynamics 10 (01), 20–38.

Sarno, L., Valente, G., 2009. Exchange rates and fundamentals: Footloose or evolving

relationship? Journal of the European Economic Association 7 (4), 786–830.

118



A.1 Appendix: Results with 5 factors

Table A.1: Model performance – R2 and HR(%) for the model with 5 factors.

R2 HR(%)



NOTE: The table reports the model fit statistics for the models with 5 factors. The columns R2 report the squaredcorrelations, and the columns HR(%) report the hit rates.

A.1. APPENDIX: RESULTS WITH 5 FACTORS 119

Figure A.1: Results for GBP with 5 factors.


1997 2000 2002 2005 2007 2010 2012

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Data

time-varying

constant

(b) Factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.01

-0.005

0

0.005

0.01

λt,1

λt,2

λt,3

λt,4

λt,5

NOTE: Figure (a) plots the changes in the exchange rate (black line), the model fit with time-varying loadings (blue line), and the model fit with constant loadings (red line). Figure (b) plotsthe time-varying factor loadings.

120



Figure A.2: Results for EUR with 5 factors.


1997 2000 2002 2005 2007 2010 2012

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant

(b) Factor loadings.

1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015λ

t,1

λt,2

λt,3

λt,4

λt,5

NOTE: Figure (a) plots the changes in the exchange rate (black line), the model fit with time-varying loadings (blue line), and the model fit with constant loadings (red line). Figure (b) plotsthe time-varying factor loadings.


Figure A.3: Results with 5 factors.


1997 2000 2002 2005 2007 2010 2012

-0.15

-0.1

-0.05

0

0.05

Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.01

0

0.01

0.02

0.03

λt,1

λt,2

λt,3

λt,4

λt,5


1997 2000 2002 2005 2007 2010 2012

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1 Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

λt,1

λt,2

λt,3

λt,4

λt,5


1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.01

-0.005

0

0.005

0.01

0.015

0.02 λt,1

λt,2

λt,3

λt,4

λt,5


122





1997 2000 2002 2005 2007 2010 2012

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015λ

t,1

λt,2

λt,3

λt,4

λt,5


1997 2000 2002 2005 2007 2010 2012

-0.06

-0.04

-0.02

0

0.02

0.04

Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

λt,1

λt,2

λt,3

λt,4

λt,5


1997 2000 2002 2005 2007 2010 2012-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1 Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

λt,1

λt,2

λt,3

λt,4

λt,5





1997 2000 2002 2005 2007 2010 2012

-0.15

-0.1

-0.05

0

0.05

Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015λ

t,1

λt,2

λt,3

λt,4

λt,5


1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02λ

t,1

λt,2

λt,3

λt,4

λt,5


1997 2000 2002 2005 2007 2010 2012

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02 λt,1

λt,2

λt,3

λt,4

λt,5


124





1997 2000 2002 2005 2007 2010 2012

-0.15

-0.1

-0.05

0

0.05

0.1Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05 λt,1

λt,2

λt,3

λt,4

λt,5


1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025λ

t,1

λt,2

λt,3

λt,4

λt,5


1997 2000 2002 2005 2007 2010 2012

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08Data

time-varying

constant


1995 1997 2000 2002 2005 2007 2010 2012 2015

-0.03

-0.02

-0.01

0

0.01

0.02λ

t,1

λt,2

λt,3

λt,4

λt,5


DEPARTMENT OF ECONOMICS AND BUSINESS ECONOMICS AARHUS UNIVERSITY

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