Dynamic Conditional Scores Models: a review...May 17, 2018 · AFN/USD exchange rate _. We refer to the related work of Fry (1974), who analyses this pattern and attributes its causes

Dynamic Conditional Scores Models: a reviewSZABOLCS BLAZSEK (FRANCISCO MARROQUIN UNIVERSITY)

ADRIAN LICHT (FRANCISCO MARROQUIN UNIVERSITY)

GESG SEMINAR, UNIVERSIDAD FRANCISCO MARROQUIN, 17 MAY 2018 1

GESG seminar, Universidad Francisco Marroquín, 17 May 2018

StructureStructureStructureStructure

1. Summary

2. Differences between classical and Dynamic Conditional Score (DCS) time series models

3. Applications

4. General conclusions


1. Summary1. Summary1. Summary1. Summary


SummarySummarySummarySummaryWe review the recent class of DCS time series models that are introduced by Creal, Koopman and Lucas (2013) and Harvey (2013).

We show how DCS models are related to the classical observation-driven and parameter-driven models.

We also show how DCS models differ from classical time series models.

We compare the statistical performances of DCS models and classical models in the followings applications:


SummarySummarySummarySummary

1) We compare the statistical performances of the QAR (quasi-autoregressive) (Harvey 2013) plus Beta-t-EGARCH (exponential generalized autoregressive conditional heteroscedasticity) (Harvey and Chakravarty 2008; Harvey 2013) model with the classical AR (Box and Jenkins 1976) plus t-GARCH (Engle 1982; Bollerslev 1986, 1987; Taylor 1986) model.

We use data on daily returns on the DAX equity index.


SummarySummarySummarySummary

2) We compare the score-driven local level, score-driven seasonality plus Beta-t-EGARCH model with the classical local level, classical dynamic seasonality plus GARCH model, applied to the daily AFN/USD (Afghani/United States Dollar) currency exchange data.

3) We compare the multivariate dynamic model of location named as QVAR (quasi-vector autoregressive) model (Harvey 2013; Blazsek, Escribano and Licht 2017) with the classical VAR model (Sims 1980, 1986; Sims, Goldfeld and Sachs 1982; Bernanke 1986; Lütkepohl2005), applied to the monthly United States (US) inflation rate and US unemployment rate data.


2. Differences between 2. Differences between 2. Differences between 2. Differences between classical and DCS time classical and DCS time classical and DCS time classical and DCS time series modelsseries modelsseries modelsseries models


Classical Gaussian signal plus noise model Classical Gaussian signal plus noise model Classical Gaussian signal plus noise model Classical Gaussian signal plus noise model (Harvey 1989)(Harvey 1989)(Harvey 1989)(Harvey 1989)

This time series model decomposes the dependent variable into the local level c + �� and irregular (noise) components and irregular (noise) components and irregular (noise) components and irregular (noise) components ��:

�� = � + �� + �� (1)

�� = �� + !" " = 1, … , & (2)

where ��~()*(0, ,-.) and !�~()*(0, ,/

.) are i.i.d. error terms, 0 is the constant parameter and 1 is the first-order dynamic parameter.

Substituting Equation (2) into (1) gives the reduced-form ARMA(1,1) model:

�� = � 1 − � + �� − �� + (�� + !�) (3)


Classical Gaussian signal plus noise model Classical Gaussian signal plus noise model Classical Gaussian signal plus noise model Classical Gaussian signal plus noise model (Harvey 1989)(Harvey 1989)(Harvey 1989)(Harvey 1989)

This model is estimated by using the maximum likelihood (ML) method (Davidson and MacKinnon 2003), for which the likelihood function is computed with the Kalman filter technique (Kalman 1960; Harvey 1989).

It is noteworthy that when the data generating process involves

a heavy-tailed distribution for �� , then the Gaussian signal plus noise model will include extreme observations into the local level component 0 + �� (instead of the irregular component ��)....


Dynamic Student’s Dynamic Student’s Dynamic Student’s Dynamic Student’s tttt location model location model location model location model (Harvey 2013)(Harvey 2013)(Harvey 2013)(Harvey 2013)The dynamic Student’s t location model (also named as QAR model) is similar to the Gaussian signal plus noise model, since it decomposes the dependent variable into local level and irregular components. QAR(1) is updated by the score function with respect to location:

�� = � + �� + �� = �� + ,4� (4)

�� = �� + 56�� (5)

6� =789:

7;9:< (6)

where 4�~" = is an i.i.d. error term with the Student’s t distribution.


Dynamic Student’s Dynamic Student’s Dynamic Student’s Dynamic Student’s tttt location model location model location model location model (Harvey 2013)(Harvey 2013)(Harvey 2013)(Harvey 2013)The conditional distribution of �� is the non-standardized Student’s tdistribution ��~"[� + �� , ,, =], where � + �� is the dynamic location parameter, , is the scale parameter and = is the degrees of freedom parameter. The updating variable 6� is proportional to the conditional score with respect to ��, as follows:

@ABC(D:|DF,…,D:GF)

@H:= 6� ×

7;

78< (7)

� is the constant parameter and � is the first-order dynamic parameter.

The QAR(1) model is estimated by using the ML method.


3. Applications3. Applications3. Applications3. Applications


First application: Dynamic models of First application: Dynamic models of First application: Dynamic models of First application: Dynamic models of expected return and volatilityexpected return and volatilityexpected return and volatilityexpected return and volatility

Motivation:

(i) When a financial or economic crisis impacts the markets, the volatility of equity indexes increases significantly. For those periods, the appropriate measurement and forecasting of volatility is important for financial investors and analysts.

(ii) The value of financial derivatives on an equity index is influenced by the volatility of the underlying index. Examples of those financial derivatives are futures and options on indices, and exchange traded funds (ETFs) related to those indices.


First application: Dynamic models of First application: Dynamic models of First application: Dynamic models of First application: Dynamic models of expected return and volatility expected return and volatility expected return and volatility expected return and volatility

In this application, we compare the statistical performances of the AR(1) plus t-GARCH(1,1) and the QAR(1) plus Beta-t-EGARCH(1,1) models applied to the DAX equity index.

We control for possible serial correlation in the mean, by using dynamic specifications for the expected return. We also control for possible serial correlation in the variance, by using dynamic models of volatility.



The classical AR(1) plus t-GARCH(1,1) model is:

�� = �� + �� = �� + J�4� (8)

for " = 1, … , &, where the error term is K� ~�(L), �� is the time-varying conditional location parameter (i.e. conditional

expected return), �� denotes the unexpected return, and M� is

the time-varying conditional scale parameter that drives conditional volatility. The conditional volatility of N� is

,� = J� ×7

7�.(9)



For AR(1):

�� = � + �� (10)

We use t-GARCH(1,1) with leverage effects (Glosten, Jagannathan and Runkle 1993):

J� = O + PJ�� + Q�� . + Q∗��

. )(�� < 0) (11)

where T∗ is a measure of the leverage effects, and )(·) is the indicator function that takes the value one if the argument is true and zero otherwise.



The QAR plus Beta-t-EGARCH model is:

�� = �� + �� = �� + exp(J�)4� (12)

where 4�~" = denotes the i.i.d. error term. The interpretation of �� and �� is the same as for the AR plus GARCH model. The conditional volatility of N� is

,� = exp J�7

7�.(13)



For QAR(1):

�� = � + �� + 5WH,�� (14)

where the X�,� is the conditional score with respect to ��

that is given by

WH,� =7YZ[(\:)9:

7;9:< (15)



We use Beta-t-EGARCH(1,1) with leverage effects:

J� = O + PJ�� + QW\,�� + Q∗sgn(�� − �� )(W\,�� + 1) (16)

where sgn(·) is the signum function, and XM,� is the conditional score with respect to M� that is given by:

W\,� =(7; )9:

<

7;9:< − 1 (17)

An advantage of the use of both QAR(1) and Beta-t-EGARCH(1,1) models is that the updating terms X�,� and XM,� discount the impact of the new information K� on location and scale, respectively.


First application: Dynamic models of First application: Dynamic models of First application: Dynamic models of First application: Dynamic models of expected return and volatility expected return and volatility expected return and volatility expected return and volatility DATA:

We use data from the DAX equity index for period 5th January 1988 to 29th December 2017.

The DAX index is a market capitalization weighted average of the prices of 30 large German companies, traded on the Frankfurt Stock Exchange (source: Yahoo Finance).

We use the log-return time series N� = ]^(_�/_��a) with " =1, … , &, where b denotes the number of observations, and _�denotes the adjusted closing value of the DAX index for day �.





RESULTS:

For the AR(1) model, the new information is transformed according to a linear function. Hence, the new information is not discounted by the AR model. On the other hand, for the QAR(1) model, the new information is discounted according to the non-linear score function.

For the t-GARCH(1,1) model, the new information is transformed according to a quadratic function. Hence, the new information is not discounted by the t-GARCH(1,1) model (in fact, the new information is accentuated by the GARCH model). On the other hand, for the Beta-t-EGARCH(1,1) model, the new information is discounted according to the non-linear score function.




We find that, for both models, all conditions of the asymptotic properties of the ML estimator are supported.

All likelihood-based metrics suggest that QAR-Beta-t-EGARCH is superior to AR-t-GARCH. In addition, we also estimate the cd of the Mincer-Zarnowitz (1969) (MZ) regression model; this MZ regression model measures the in-sample volatility forecast performance for each model.

According to the cd of the MZ regression model, the in-sample volatility forecast performance of the QAR plus Beta-t-EGARCH model is superior to that of the AR plus t-GARCH model.



Second application:Second application:Second application:Second application: Dynamic local level, dynamic Dynamic local level, dynamic Dynamic local level, dynamic Dynamic local level, dynamic seasonality and dynamic volatility modelsseasonality and dynamic volatility modelsseasonality and dynamic volatility modelsseasonality and dynamic volatility modelsMotivation:

At least partly, this application is motivated by the fact that there is a very limited literature on modelling the seasonality of the AFN/USD exchange rate _�.

We refer to the related work of Fry (1974), who analyses this pattern and attributes its causes mainly to the seasonality of agricultural exports from Afghanistan (with maximum approximately from September to November) and to the related subsequent USD inflows to Afghanistan approximately in December and January. From Fry (1974, p. 251) we refer:



September to

December

Second application:Second application:Second application:Second application: Dynamic local level, dynamic Dynamic local level, dynamic Dynamic local level, dynamic Dynamic local level, dynamic seasonality and dynamic volatility modelsseasonality and dynamic volatility modelsseasonality and dynamic volatility modelsseasonality and dynamic volatility models

In this application, we compare the classical local level, classical dynamic seasonality plus normal-GARCH model and the score-driven local level, score-driven seasonality plus Beta-t-EGARCH model. We use data for the AFN/USD exchange rate.



Models:

Firstly, for the classical local level, classical dynamic seasonality plus normal-GARCH(1,1) model, the daily value of the AFN/USD exchange rate is formulated as

e� = �� + f� + �� = �� + f� + J�4� (18)

where 4� ~((0,1) i.i.d. noise term component of the exchange rate.



The dynamic components are specified as follows:

The local level component �� is

�� = �� + 5�� (19)

which is updated by the first lag of the irregular component �� .

The seasonality component g� is

f� = *�hi� (20)

where *� = *lmB,� , … , *nYo,�his a vector of dummy variables



and the dynamic parameters p� are

i� = i�� + q�� (21)

where q� is a 12 × 1 vector of dynamic parameters, for which each element sof q� is parameterized as

qt� = qt if *t� = 1 and qt� = −qt/(12 − 1) if *t� = 0. (22)

Thus, in this formulation, qt� with s = Jan, … , Dec are parameters to be

estimated.

The scale component M� is according to the normal-GARCH(1,1) model:

J� = O + PJ�� + Q�� . (23)



Secondly, for the score-driven local level, score-driven seasonality plus Beta-t-EGARCH model, the daily value of the AFN/USD exchange rate is formulated as

e� = �� + f� + �� = �� + f� + exp(J�)4� (24)

where 4�~"[exp = + 2] is the i.i.d. error term.

The dynamic components are specified as follows:

The local level component �� is

�� = �� + 5WH,�� (25)

where the updating term is the score function with respect to ��



is given by

WH,� =YZ[(wx)yx

yx<;YZ[(z);.

(26)

The seasonality component g� is

f� = *�hi� (27)

where *� = *{|},� , … , *~��,�his a vector of dummy variables and the

dynamic parameters p� are

i� = i�� + q�WH,�� (28)

where q� is a 12 × 1 vector of dynamic parameters,



for which each element s of q� is parameterized as

qt� = qt if *t� = 1 and qt� = −qt/(12 − 1) if *t� = 0. (29)

Thus, in this formulation, qt� with s = Jan, … , Dec are parameters to be

estimated.

Scale component M� is specified according to Beta-t-EGARCH model:

J� = O + PJ�� + QW\,�� (30)

where the score function is given by:

W\,� =[YZ[ z ;�]yx

<

YZ[(z);.;yx< − 1 (31)



DATA:

We use the information of the AFN/USD exchange rate for period 1st March 2007 to 7th July 2017 (source: Bloomberg).

The models are estimated for the daily closing exchange rate e� for days " = 1, … , &, where & denotes the number of observations.





RESULTS:

We analyse the local level components of the classical and the DCS model.

We also analyse the seasonality components of the classical and the DCS models.






We analyse the discounting of extreme observations in the noise (i.e. new information arriving to the market). For the classical model, the new information is transformed according to a linear function. Hence, new information is not discounted by the classical model. On the other hand, for the DCS model, the new information is discounted according to the non-linear score function.

For the normal-GARCH(1,1) model, the new information is transformed according to a quadratic function. Hence, new information is not discounted by the normal-GARCH(1,1) model. On the other hand, for the Beta-t-EGARCH(1,1) model, the new information is discounted according to the non-linear score function.




We find that, for both models, all conditions of the asymptotic properties of the ML estimator are supported.

The statistical performance of both models is evaluated by using the following likelihood-based performance criteria: LL, AIC, BIC and HQC.


Third application:Third application:Third application:Third application: Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of locationlocationlocationlocation

Motivation:

Nordhaus (1975) proposes a macroeconomic model of inflation and unemployment rates about government-policies, for which the objective of the government is to win the next elections.

That paper presents that the objective of winning the next elections may create political cycles of economic recessions and economic expansions:


Third application:Third application:Third application:Third application: Multivariate dynamic Multivariate dynamic Multivariate dynamic Multivariate dynamic models of locationmodels of locationmodels of locationmodels of locationMotivation:

After winning the elections, the government tends to establish policies that keep the inflation rate at a low level, at the expense of a high unemployment rate.

Subsequently, government policies are modified during the period preceding the next elections, in order to increase the level of inflation and decrease the unemployment rate. The governments pursue these economic policies because, on the date of elections, the voters discount the costs of past economic outcomes (e.g. high unemployment rate) (see also Findley 2015).



In this application, we present an application of the QVAR(1) multivariate DCS model (Harvey 2013; Blazsek, Escribano and Licht 2017).

We use QVAR(1) in order to study the dynamic interaction effects between the US inflation and unemployment rates.

We also compare the statistical performances of QVAR(1) and VAR(1) model.


Third application:Third application:Third application:Third application: Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of locationlocationlocationlocationModels:

Firstly, the VAR(a) model for the vector of dependent variables � �, … , �� ′is given by:

�� = �� + �� = �� + �� 4� (32)

�� is the conditional mean of N�|(Na, … , N��a) that is specified as

�� = � + �� (33)

where 0 is a vector of constant parameters and � is a matrix of dynamic parameters, �� is a vector of contemporaneously correlated reduced-form error terms, K� is a vector of contemporaneously uncorrelated structural-form error terms, and ��a is a lower triangular scaling matrix.



The structural-form linear infinite vector moving average (VMA) representation of N� is

�� = ∑ �t��t�� + ∑ �t��

t�� 4��t (34)

The impulse response function )��t =@D:��

@9:for s = 0, 1, … , ∞ is

given by:

)��t = �t�� (35)



Secondly, the QVAR(a) model for the vector of dependent variables � �, … , �� ′ is given by:

�� = � + �� + �� = � + �� + �� 4� (36)

where ��~��(�, �, L) i.i.d. with the multivariate Student’s tdistribution is the reduced-form error term.

Σ is positive definite and = > 2. The interpretation of all parameters and variables of QVAR(1) is the same as that of VAR(1). �� is specified as

�� = �� + �W�� (37)

where X� is the vector of score functions with respect to ��.


Third application:Third application:Third application:Third application: Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of locationlocationlocationlocationThe log conditional density of �� is

ln� ��|� , … , �� =

lnΓ7;�

.− lnΓ

7

.−

�

.−

.ln Σ −

7;�

.ln 1 +

-:��GF-:

7(38)

The partial derivative of the log of the conditional density with respect to �� is

@�}C D:|DF,…,D:GF

@H:=

7;�

7�� × 1 +

-:��GF-:

7�� =

7;�

7�� × W� (39)

The second equality of the previous equation defines the score function X� that updates the conditional mean of N�.


Third application:Third application:Third application:Third application: Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of locationlocationlocationlocationThe structural-form nonlinear VMA representation of �� is

�� = � + ∑ �t� = − 2 = /.�� 9:GFG�

7�.;9:GFG�� 9:GFG�

�t�� +

7

7�.

/.�� 4� (40)

The impulse response function )��t =@D:��

@9:is given by:

)��t� =7

7�.

/.�� for s = 0 and

)��t� = �t� = − 2 = /.�� *�� t for s > 1

*� = @

�:

�G<��:��:

@9:(41)



Data:

We use monthly data on the US inflation rate Na� and the US unemployment rate Nd� for period 1st January 1948 to 1st December 2017 (source: Federal Reserve Bank of St. Louis).

The models are estimated for the time series �� = (� �, �.�)′for " = 1, … , &.




Third application:Third application:Third application:Third application: Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of Multivariate dynamic models of locationlocationlocationlocationResults:

For VAR(1) and QVAR(1), we analysed the IRFs. For VAR(1), the IRF shows non-significant dynamic effects of unemployment shocks on inflation and negative significant dynamic effects of inflation shocks on unemployment. For QVAR(1), the IRF shows significant negative dynamic effects of unemployment shocks on inflation and also negative dynamic effects of inflation shocks on unemployment.

Thus, the IRF estimates of QVAR(1) support the theory of Nordhaus, with respect to the negative interaction effects between the US inflation and unemployment rates. On the other hand, the IRF estimated of VAR(1) do not support completely the theory presented in Nordhaus.





Results:

We find that, for both models, all conditions of covariance stationarity are supported. It is noteworthy that for QVAR(1) all conditions of consistency and asymptotic normality of the ML estimator are satisfied.

For both VAR(1) and QVAR(1), we use Escanciano-Lobato (2009) martingale difference sequence (MDS) test with optimal lag order for the estimates of 4� (the use of this test is motivated by Harvey 2013). We find that the MDS null hypothesis is never rejected.



4. General conclusions4. General conclusions4. General conclusions4. General conclusions


General conclusionsGeneral conclusionsGeneral conclusionsGeneral conclusions(i) We have explained the main differences between the classical and DCS time series models, by comparing the Gaussian signal plus noise model and the dynamic Student’s t location model, respectively, to highlight the main differences between the classical and DCS models.

(ii) We have provided practical applications of three DCS models for financial and economic data.

(iii) We have presented the estimation results and model diagnostics for all classical and DCS models in our applications. Those results suggest that the statistical performance of each DCS model is superior to that of the corresponding classical time series model.


Thank you for yourattention!

[email protected]@[email protected]@UFM.EDU

[email protected]@[email protected]@UFM.EDU


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Dynamic Conditional Scores Models: a review...May 17, 2018 · AFN/USD exchange rate _. We refer to the related work of Fry (1974), who analyses this pattern and attributes its causes