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Time-varying correlation and common structures in volatility

Liu, Yang

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Citation for published version (APA):Liu, Y. (2016). Time-varying correlation and common structures in volatility.

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Yang Liu

Universiteit van Amsterdam

Time-V

arying Correlation and Comm

on Structures in Volatility Yang Liu

671

Time-Varying Correlation and Common Structures in Volatility

This thesis studies time series properties of the covariance structure of multivariate asset returns. First, the time-varying feature of correlation is investigated at the intraday level with a new correlation model incorporating the intraday correlation dynamics. Second, the thesis develops a multivariate factor model where the common factors are imposed directly on volatility. Third, the pricing implications of the volatility factors are shown by applications on option returns. Yang Liu holds a Bachelor degree in Economics from the Central University of Finance and Economics in Beijing and an M.Phil. in Finance from the Tinbergen Institute. In September 2013, he joined Amsterdam School of Economics at the University of Amsterdam as a Ph.D. student. His research interests are in the area of multivariate volatility models with applications in risk management, empirical asset pricing and portfolio design.

Time-Varying Correlation and Common

Structures in Volatility

ISBN 978 90 5170 960 5

Cover design: Crasborn Graphic Designers bno, Valkenburg a.d. Geul

This book is no. 671 of the Tinbergen Institute Research Series, established through

cooperation between Rozenberg Publishers and the Tinbergen Institute. A list of

books which already appeared in the series can be found in the back.

Time-Varying Correlation and Common

Structures in Volatility

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Universiteit van Amsterdam

op gezag van de Rector Magnificus

prof. dr. ir. K.I.J. Maex

ten overstaan van een door het College voor Promoties ingestelde commissie,

in het openbaar te verdedigen in de Agnietenkapel

op donderdag 17 november 2016, te 14.00 uur

door

Yang Liu

geboren te Liaoning, China

Promotiecommissie :

Promotor : Prof. dr. H. P. Boswijk Universiteit van Amsterdam

Overige leden : Prof. dr. D. J. C. van Dijk Erasmus Universiteit Rotterdam

Prof. dr. S. J. Koopman Vrije Universiteit Amsterdam

Prof. dr. F. R. Kleibergen Universiteit van Amsterdam

Prof. dr. R. J. A. Laeven Universiteit van Amsterdam

dr. S. A. Broda Universiteit van Amsterdam

Faculteit: Economie en Bedrijfskunde

Acknowledgments

It has been almost five years since I first landed at Schiphol Airport, where I was

warmly greeted by a heavy coffee scent. I remember that I stood right next to a

pickup service point having no clue about properly communicating in English nor

about why I suddenly became the dwarf among the giants. Now that I think about

it, I should have been more excited then, because what was lying ahead turned out

to be the best time of my life so far, for example, I got a 9 for the Derivatives course,

I visited Italy, Chelsea won the Champions League, and Game of Thrones still keeps

some characters alive. Now I am writing the thesis, those small excitements have

already faded away after a beer or two. What is left on my mind are the things that

I am so lucky to encounter.

I am indebted to my supervisor Prof. dr. Peter Boswijk, without whom I would

still be the slapdash MPhil graduate that usually writes tons of typos in a report.

I feel obliged to acknowledge his confidence in my capability, his insightful com-

ments, and most importantly, his encouragements throughout my studies and ca-

reer. My gratefulness also goes to other committee members, Prof. dr. Dick van

Dijk, Prof. dr. Siem Jan Koopman, Prof. dr. Frank Kleibergen, Prof. dr. Roger

Laeven, and dr. Simon Broda for their time and advice on reviewing this disserta-

tion. Furthermore, I would like to thank the Tinbergen Institute for offering me the

ideal first step to pursue my career in quantitative finance.

My Ph.D. life, as exciting as it sounds and believe it or not, would have been

very boring without all the friendships bonding from Amsterdam, Beijing, New

York, Seoul, Shanghai etc. I sincerely thank: Zhaokun Zhang, Oliver Liu, Cash

Li, Xuehan Zhang and Hao Fang for the happiness and laughters we shared and

bragged about; Arturas Juodis, Merrick Li, Simin He, Jindi Zheng, Zhiling Wang,

Swapnil Singh, Ilke Aydogan, Alexandra Rusu, Tristan Linke, Rutger Poldermans,

Yueshen Zhou, Xiye Yang, among others, for being the best colleagues in either TI

or UvA; Zhi Li, Chao Zhao, Dongxu Song, Junbang Niu, Shiyong Zhang, Chenlu

Zhang, Jinghan Zhou, Ming Li, and Ye Ji for our good old times at the CUFE;

Han Gao for all the nicknames we invented and nearly merchandised; Xin Xia

for just being there as my brother. I would also like to thank Thomas Hufener,

Robin van Boxsel, Matthew Newbon, Martijn van Tongeren, Jerry Tworek, Michal

Garmulewicz, Mahdi Jaghoori, and Erkki Silde for providing the excellent working

environment in our IV office, where magic is happening every hour and every day.

Last but not least, I would like to thank Andrei Lalu for being my best friend and

for making our office a place where I would rather sit and do nothing than staying

at home playing video games. I can only hope that his English is not dragged down

to my level after our daily conversations. I will not be surprised by however high

his achievement can be in the future, because I know how ridiculously brilliant he

is. May my best wishes extend to Adelina.

My special thanks go to a special lady, Xiao Xiao, who, using her magic, has

made Rotterdam a home for me. Meeting and starting a new life with her in

the Netherlands has cleared any doubts I once held about coming to this country,

because it is, and will always be the best decision I ever made.

Finally, I am privileged to have a loving family supporting me from the small

and quiet town of Chaoyang. I thank my parents for being my life advisers, who

are sometimes, or maybe most of the time, more involved than needed. To my

sister, though I am not a fan of being woken up at 3 a.m., I enjoyed every minute

we spent on and off the phone. The biggest thank you goes to my grandparents

for everything they have done for me, and especially for keeping my sister busy

so that she cannot call me at 3 a.m. too frequently. This thesis is dedicated to them.

谨向我的母亲孟晓阳和我的父亲刘志平表达我最深切的感恩和爱戴。

Yang Liu

Amsterdam

April 2016

Contents

1 Introduction 1

2 Correlation Aggregation in Intraday Financial Data 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The Gaussian Copula GAS Model . . . . . . . . . . . . . . . . . . . . 8

2.3 Correlation Aggregation under the GAS Model . . . . . . . . . . . . 10

2.3.1 Relation between daily and intraday correlation . . . . . . . . 11

2.3.2 Dynamics of intraday conditional correlation . . . . . . . . . 14

2.3.3 Correlation aggregation . . . . . . . . . . . . . . . . . . . . . . 15

2.3.4 HFGAS vs. Intraday GAS . . . . . . . . . . . . . . . . . . . . . 16

2.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Bivariate estimation . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 Multivariate generalization . . . . . . . . . . . . . . . . . . . . 18

2.5 Monte Carlo Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.1 Data simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.2 The validity of the linearized Fisher transformation . . . . . . 21

2.5.3 Parameter robustness and model fit check (correctly specified) 23

2.5.4 Model fit comparison: misspecified . . . . . . . . . . . . . . . 26

2.6 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6.1 Data description . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6.2 In-sample performance . . . . . . . . . . . . . . . . . . . . . . 32

2.6.3 Out-of-sample performance . . . . . . . . . . . . . . . . . . . . 36

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

i

ii Contents

3 Score-Driven Variance-Factor Models 43

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 Variance-factor GAS models . . . . . . . . . . . . . . . . . . . 45

3.2.2 Conditions for parameter identification . . . . . . . . . . . . . 48

3.2.3 Comparison with the DCC model . . . . . . . . . . . . . . . . 51

3.2.4 Comparison with factor GARCH models . . . . . . . . . . . . 52

3.2.4.1 Case 1: Λt = diag(D2t ) . . . . . . . . . . . . . . . . . 53

3.2.4.2 Case 2: Λt = log(diag(D2t )) . . . . . . . . . . . . . . 54

3.2.5 Estimation and diagnostic tests . . . . . . . . . . . . . . . . . . 55

3.3 Monte Carlo Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Empirical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.1 Model fit and VaR coverage . . . . . . . . . . . . . . . . . . . . 60

3.4.1.1 In-sample fit comparison . . . . . . . . . . . . . . . . 61

3.4.1.2 Out-of-sample performance: simulations and VaR

coverage . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.2 Fitting the CAPM with time-varying beta . . . . . . . . . . . . 72

3.4.2.1 Model setting and Fama-Macbeth regression results 72

3.4.2.2 Portfolio sorting . . . . . . . . . . . . . . . . . . . . . 76

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Factor Premia in Variance Risk 79

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Variance-Factor Model and Variance Risk Premium . . . . . . . . . . 85

4.2.1 The variance-factor model . . . . . . . . . . . . . . . . . . . . . 85

4.2.2 The two-factor case . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2.3 Reasons for two-factor model . . . . . . . . . . . . . . . . . . . 88

4.2.3.1 Reason 1: Idiosyncratic volatility comovement . . . 88

4.2.3.2 Reason 2: Variance risk premia of individual stocks 92

4.2.4 The second factor: the VR factor . . . . . . . . . . . . . . . . . 94

Contents iii

4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3.1 Filtering under the physical measure . . . . . . . . . . . . . . 96

4.3.2 Filtering under the risk-neutral measure . . . . . . . . . . . . 101

4.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4.1 Data description . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.4.2 Empirical results under the physical measure . . . . . . . . . 106

4.4.3 Empirical results under the risk-neutral measure . . . . . . . 109

4.4.4 Implications on the term structure of variance . . . . . . . . . 114

4.5 Option Portfolio Design . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.5.1 Straddle returns for different maturities . . . . . . . . . . . . . 116

4.5.2 VRP sorting strategy . . . . . . . . . . . . . . . . . . . . . . . . 118

4.5.3 Empirical performance . . . . . . . . . . . . . . . . . . . . . . 119

4.5.4 Robustness checks . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.5.5 Implications on dispersion trade . . . . . . . . . . . . . . . . . 124

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5 Summary 129

6 Samenvatting 131

Bibliography 133

1 | Introduction

One of the cornerstones of financial econometrics lies in the estimation of return

covariance structures. Empirical evidence on the temporal or contemporaneous

dependencies for a multivariate asset class has been a sustained force pushing the

development of multivariate volatility and correlation models. My dissertation

focuses on two prominent features of the return covariance matrix, namely, the

time-varying correlation and the volatility comovement.

11/2007 05/2008 11/2008 05/2009 11/2009

0.4

0.5

0.6

0.7

0.8

0.9

S&P500 Price S&P500 Implied Correlation

Figure 1.1: CBOE S&P 500 Implied Correlation.

This figure shows the CBOE S&P 500 Implied Correlation Index (ICJ January 2010). The S&P

500 Index level is depicted in the grey area (divided by 1800 to fit in the graph). Sample period:

November 2007 – November 2009.

The recent decade saw the development of dynamic correlation models in both

academia and industry, where scholars and practitioners have come to the consen-

sus that the correlation between two asset returns exhibits significant time varia-

tion. Figure 1.1 shows the CBOE S&P 500 Implied Correlation Index measuring the

average option-implied correlation between the S&P 500 Index and its 50 largest

1

2 Chapter 1. Introduction

components. The time variation can easily be detected, especially during the 2008

crisis when the correlation peaks suddenly and remains at a relatively high level for

months before showing signs of mean-reversion. The second chapter of this disser-

tation, based on Boswijk and Liu (2014), investigates further into the time-varying

feature of correlation by developing a new class of correlation models where the

dynamics of conditional correlation exist at the intraday level. This chapter also de-

rives the link between the correlations at different frequencies followed by a tempo-

ral aggregation procedure that successfully accommodates intraday dynamics into

a daily recurrence equation. The validity of this aggregation process and the su-

periority of the resulting HFGAS model are supported by Monte Carlo simulation

evidence. An empirical application on the intraday currency returns of GBP/USD

and EUR/USD shows good in-sample performance of the HFGAS model. The

advantage of using the HFGAS model is further illustrated by an out-of-sample

analysis of portfolio optimization.

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.81%-20% & 80%-100%20%-40% & 60%-80%40%-60%

Figure 1.2: Quintile plot of the annualized volatilities of the Dow Jones components.

This figure shows the quintile plot of the annualized volatilities of all the components in the Dow

Jones Industrial Average Index. Sample period: January 2001 – December 2010.

Unlike correlation, the time variation in volatility is deeply rooted in the early

theoretical and empirical finance literature. Bearing systematic risk, the volatilities

of individual stocks follow a strong factor structure. Figure 1.2 shows the quintile

plot of the conditional volatilities of all the components of the Dow Jones Industrial

Average Index, where one can see a strong level of comovement. The third chapter,

3

adapted from Boswijk and Liu (2015), explores this factor structure in stock re-

turn volatilities by developing a class of multivariate models with common factors

in the conditional variance series based on the Generalized Autoregressive Score

model of Creal et al. (2011). The model distinguishes itself from other multivariate

GARCH models by having the advantage that it separately estimates the condi-

tional variances and correlations and at the same time, preserves a strong factor

structure in the conditional variances. The chapter provides detailed model speci-

fications, conditions for parameter identification, and factor-updating mechanisms.

Empirical applications are designed to predict the Value-at-Risk for different equity

portfolios and to estimate the time-varying market betas for individual stocks.

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.1

0.2

0.3

0.4

0.5

0.6Variance Risk Premium (S&P500)

Figure 1.3: Variance risk premium of the S&P 500 Index options.

This figure shows the annualized variance risk premium time series plot for the S&P 500 Index

options. The measure is calculated by taking the difference between the model-free implied variance

and the realized variance. Sample period: January 2001 – December 2010.

The common dynamic structure in variances is the driver of the recent popular

risk measure called variance risk premium, which reflects the premium investors

demand for bearing the risk of the uncertainty in future variances. Figure 1.3 shows

the time series plot of the variance risk premium for the S&P 500 Index options,

namely, the difference between its implied variance and the realized variance. The

figure clearly indicates a non-negative variance risk premium as the implied vari-

ance is most of the time higher than the realized variance. In the fourth chapter

which is based on Liu and Xiao (2016), the variance-factor model developed in

4 Chapter 1. Introduction

Chapter 3 is parsimoniously tailored to a two-factor case to study the dynamics

of the variance risk premia embedded in both the index and individual stock op-

tions. The two factors can be identified independently as a market variance (MV)

factor and a variance residual (VR) factor, respectively governing the short-term

and long-term dynamics of individual stock variances. The chapter shows that the

MV factor resembles the shape of the index variance and carries a positive pre-

mium, which has an excess spillover effect on individual stock variances. On the

contrary, the VR factor serves to compensate the excess spillover by carrying a neg-

ative premium. The differences in the factor premia and factor memories suggest

that an option portfolio with long positions on long-term individual stock straddles

and short positions on short-term index straddles generates significant positive re-

turns. The strategy can be further enhanced by choosing straddles according to the

model-predicted variance risk premia. The advantage of this model-based strategy

is demonstrated in both in- and out-of-sample analyses.

2 | Correlation Aggregation in Intra-

day Financial Data

2.1 Introduction

We introduce a new class of daily correlation models assuming that the correlation

parameter is varying at the intraday level. The intraday dynamics are modeled by

a GAS-type (Generalized Autoregressive Score model of Creal et al. (2013)) recur-

rence equation, which is then temporally aggregated to obtain a daily formulation.

Incorporating the daily recurrence equation in a Gaussian copula structure renders

a class of High-Frequency copula GAS models, which we label HFGAS models.

Modeling the correlation structure of a group of assets has been a major in-

terest in academia and the financial industry. Empirical evidence such as the

time-varying historical correlation patterns and the correlation risk premium de-

scribed in Driessen et al. (2009) indicate that the allowing for time-varying corre-

lations explains financial market comovement better than the assumption of con-

stant correlations. To study the time-varying feature of correlation has been the

focus of a voluminous body of literature, such as Engle (2002), Tse and Tsui (2002)

and Christodoulakis and Satchell (2002), in which the DCC (dynamic conditional

correlation) models are introduced and developed. Another class of models first

introduced by Patton (2006) considers using copula functions to capture the assets’

dependence structures. Lee and Long (2009) extend the subject by introducing

the copula-GARCH model which integrates the generalized autoregressive condi-

5

6 Chapter 2. Correlation Aggregation in Intraday Financial Data

tional heteroskedasticity (GARCH) model of Bollerslev (1986) with various copula

functions.

In both classes of models, the daily conditional correlation is updated using a

rolling-window sample correlation calculated by daily returns as the driving force.

With the increasing availability of intraday data, it is tempting to study whether

intraday information, usually in the form of realized measures, improves model

performance. A recent example is the multivariate high-frequency-based volatility

(HEAVY) model of Noureldin et al. (2012), in which the inclusion of the realized

covariance matrix in a multivariate GARCH model improves the robustness of the

estimation and achieves significant forecast gains, especially for short horizons. In

a more direct approach, Salvatierra and Patton (2014) augment a wide range of

dynamic copula models under the GAS model framework with several intraday

realized measures. This new class of models is labeled GRAS models. Empirical

studies show that the GRAS models significantly outperform daily copula-GAS

models, both in- and out-of-sample.

The common feature of the augmented realized measures in the HEAVY mo-

dels and the GRAS models is that they are constructed daily and that they weigh

intraday observations equally regardless of their recording time. We differentiate

our work from the previous literature on exploiting intraday information in corre-

lation modeling by assuming that the dynamics of the correlation parameter in

the HFGAS model exist at the intraday level. Intuitively, as market information is

gradually revealed, the intraday returns close to the end of each trading day should

be more informative than the earlier ones for daily estimates. Moreover, the shocks

in the cross products of intraday returns, which are used to calculate the realized

covariance, should be specifically weighted to avoid abnormal innovation values. In

addition, possible intraday patterns in correlation also call for different weighting

schemes for intraday observations. We show in this chapter that our assumption for

intraday correlation dynamics is able to accommodate these aspects, which further

serves the daily correlation estimation.

The question arises as to how can we incorporate intraday dynamics into a

2.1. Introduction 7

daily correlation estimate. The main contribution of this chapter is the daily aggre-

gation of intraday correlation dynamics so that the resulting recurrence equation

can be used to model daily correlation, a procedure which we call correlation aggre-

gation. To perform this procedure, the first step is deriving the link between the

intraday and the daily correlations, which is not explicit since correlation is not an

aggregated-type measure like variance or covariance. We show that the daily corre-

lation is a weighted sum of the intraday correlations, with weights determined by

the residual intraday variance pattern left after a pattern-filtering scheme is app-

lied. The second step assumes that the Fisher-transformed intraday conditional

correlation follows the GAS model recurrence equation in order to capture intra-

day dynamics. In the final step, we use the link derived in the first step together

with the intraday dynamics assumed in the second step to obtain the daily recur-

rence equation, which we refer to as the recurrence equation of the HFGAS model.

The aggregated daily recurrence equation of the HFGAS model is totally driven by

the realized score function, a particular nonlinear function of the intraday returns. We

show that this realized score function can be formulated as a time-varying linear

combination of realized variance and realized correlation, with the time-varying

coefficients determined by intraday conditional correlation. The Monte Carlo sim-

ulation results support the superior fit obtained when using the HFGAS model. By

comparing the mean absolute errors and the mean squared errors, we show that the

HFGAS model dominates the other candidates in a wide range of data generating

scenarios for intraday correlation.

In an empirical study, we apply the HFGAS model to 5-minute transaction price

quotes of the GBP/USD and EUR/USD exchange rates. By using the Nadaraya-

Watson regression, we first eliminate the well-known intraday volatility pattern

discussed in the literature, inter alia, by Wood et al. (1985) and Mclnish and Wood

(1992).1 The GJR-GARCH model is applied to the pattern-filtered return series to

obtain the standardized inputs for the copula function. To conduct a robust com-

1A reversed J-shape is found to be the typical intraday volatility pattern of the US equity market

by Wood et al. (1985) and Mclnish and Wood (1992).


parison with other models, we divide our sample period into two sub-samples. The

advantages of using the HFGAS model are illustrated by our estimation outcomes:

the maximized log-likelihoods and the information criteria consistently favor the

HFGAS model over other candidate models in both sub-samples and the whole

sample. A portfolio strategy using out-of-sample correlation forecasts based on the

HFGAS model performs best in terms of annual Sharpe ratio (1.17), when com-

pared with similar strategies based on either the daily GAS model (1.05) or the

GRAS model (1.08).

The remainder of this chapter develops as follows. In Section 2.2, we recap some

of the basic settings for copula GAS models. Section 2.3 presents the proposed

correlation aggregation procedure. Section 2.4 describes the estimation methods.

Section 2.5 shows the Monte Carlo evidence concerning the robustness of the corre-

lation aggregation and model performance. Section 2.6 provides the results of

empirical studies. Finally, Section 2.7 concludes this chapter.

2.2 The Gaussian Copula GAS Model

The main interest of this chapter concerns the modeling of correlation structures be-

tween two return series, therefore, we use the Gaussian copula to approximate the

correlation structure at both intraday and daily frequencies. Assume that the daily

bivariate returns (r1,t, r2,t)′ have conditional marginal distributions F1,t and F2,t

which are adapted to Ft−1, a suitably defined filtration at t− 1, i.e., rj,t|Ft−1 ∼ Fj,t,

j = 1, 2. Define u1,t := F1,t(r1,t) and u2,t := F2,t(r2,t) as the values of the cumulative

distribution function (CDF) evaluated at r1,t and r2,t. The Gaussian copula density

function c(.) with a time-varying correlation parameter ρt is constructed as

c(u1,t, u2,t; ρt) = −12

log(1− ρ2t )−

ρ2t(Φ−1(u1,t)

2 + Φ−1(u2,t)2)− 2ρtΦ−1(u1,t)Φ−1(u2,t)

2(1− ρ2t )

.

where ρt captures the time-varying linear dependence between r1,t and r2,t, and

Φ(.) is the normal pdf operator. Various models have been proposed to filter

this correlation parameter, such as the DCC model of Engle (2002), the copula

2.2. The Gaussian Copula GAS Model 9

GARCH model of Lee and Long (2009), and the conditional copula method of Pat-

ton (2006). In this chapter, we apply the GAS model of Creal et al. (2013) for the

reasons shown below. The GAS model, short for the generalized autoregressive

score model, suggests modeling the parameters of interest by using an intercept

term, an autoregressive term, and an innovation term which is the scaled score of

the log conditional joint density function of the observations. Specifically, under

the copula density function c(u1,t,u2,t;ρt), the recurrence equation of ρt suggested

by the GAS model is

ρt = ω(1− β) + βρt−1 + αst−1,ρ, (2.1)

where st,ρ = St,ρ∇t,ρ,

with ∇t,ρ =∂ log c(u1,t,u2,t;ρt)

∂ρt,

and St,ρ is a scaling factor

To see the advantage of using the scaled score function st,ρ, note that the score

function ∇t,ρ measures the steepest ascent direction in which the log-likelihood

can be improved given the current position of ρt, in which case ρt+1 is updated

given the information from the density instead of using observations only. To avoid

abnormal score function values, a scaling factor St,ρ is always needed to smooth the

score series. For the choice of the scaling factor St,ρ, we use

St,ρ = E[∇2

t,ρ|Ft−1

]−1,

which is Var(∇t,ρ|Ft−1

)−1 since E[∇t,ρ|Ft−1

]= 0. As ρt is a correlation coef-

ficient, its domain is (−1, 1). Applying (2.1) directly may lead to the problem

of unbounded correlation. A widely used approach to enforce this constraint is

to apply the Fisher transformation, where instead of modeling ρt, we model the

transformation f (ρt) s.t.

f (ρt) = ω(1− β) + β f (ρt−1) + αst−1,

with f (ρt) = log(1−ρt1+ρt

). f (ρt) can then be estimated with no bounding restrictions.

The scaled score function st will also change accordingly compared to st,ρ.


2.3 Correlation Aggregation under the GAS Model

The main focus of this chapter is on filtering the daily conditional correlation para-

meter using intraday information. One strand of the literature on this subject uses

realized measures such as the realized covariance matrix (the multivariate HEAVY

model by Noureldin et al. (2012)) or the realized correlation matrix (the GRAS

model by Salvatierra and Patton (2014)) to drive the conditional correlation. These

realized measures are calculated at the close of each trading day by using intraday

observations sampled at desirable frequencies, usually 5-minute or 10-minute so

as to avoid microstructure noise. The empirical results provide significant support

for the use of realized measures in lieu of innovations calculated from daily closing

returns. Both realized covariance and realized correlation are daily measures which

weigh intraday observations equally regardless of their recording time. Although

exploiting intraday information, the calculation of these methods could result in

some intraday dynamics being ignored.

We account for these missing intraday dynamics by proposing a new correlation

model in which the conditional correlation is updated at intraday frequencies. The

intraday conditional correlations in one trading day are then suitably aggregated

to a daily conditional correlation. The model possesses three possible advantages:

1) the model considers the evolution of the intraday correlation between two as-

sets by accommodating intraday correlation shocks, providing a clear view of how

the intraday correlations are aggregated to daily values; 2) both the intraday and

the daily conditional correlations can be estimated simultaneously, thus providing

correlation timing schemes for intraday portfolio strategy making; 3) the realized

measures which are used to update the daily correlation are derived in an intraday

dynamics setting instead of simply augmenting daily measures, thus responding

differently to the intraday arrival of information. To implement a model possess-

ing these advantages, in a first step, we derive the relation between correlations at

different frequencies, in both integrated and conditional forms. In a second step,

the dynamics of intraday conditional correlation are specified to follow the GAS

2.3. Correlation Aggregation under the GAS Model 11

model with an explicit expression for the scaled score function. Finally, the re-

currence equation for daily conditional correlation is obtained using the relation

derived in the first step and the intraday dynamics equation, a process which we

call correlation aggregation. We set t ∈Z and use the notation (t, j) to denote the j-th

intraday realization at trading day t and (i, t, j) to denote the same measure for a

specific asset i. We restrict our focus on the bivariate case.

2.3.1 Relation between daily and intraday correlation

The relationship between daily and intraday variances or covariances is straight-

forward to derive as these are measures which intrinsically aggregate over time.

However, the correlation parameter does not possess such a relationship as it is

a ratio between the latter measures. An intuitive way to illustrate this issue is

by assuming that the daily correlation equals the average value of the intraday

correlations. We will show that this equality only holds under very strong assump-

tions. In order to derive the relation under a general setting, we start by assuming

a semi-martingale form for the log-price process and the definition of integrated

correlation given by Barndorff-Nielsen and Shephard (2004).

Assumption 2.1. Denote by yt the bivariate log-price process and assume that it is a semi-

martingale of the form yt = at + κt, where at is an Ft-adapted finite-variation process

with a0 = 0 and κt is a local martingale of the form κt =∫ t

0 Θ(s)dW s with Θ(·) the

instantaneous or spot (co)volatility process and dW a vector with increments of independent

standard Wiener processes. The instantaneous covariance matrix V(·) = Θ(·)Θ(·)′ is

written as ⎛⎝ σ2

1 (·) σ12(·)σ12(·) σ2

2 (·)

⎞⎠ .

Denote by δt the daily integrated correlation from time (t− 1) to t, and δt,j the

integrated correlation from time(

t− 1 + j−1n

)to(

t− 1 + jn

)corresponding to an

intraday sampling scheme such that there are n equal-length intervals per trading


day. Under Assumption 2.1, the integrated correlations are defined as

δt :=

∫ tt−1 σ12(s)ds√∫ t

t−1 σ21 (s)ds

∫ tt−1 σ2

2 (s)ds, (2.2)

δt,j :=

∫ t−1+ jn

t−1+ j−1n

σ12(s)ds√∫ t−1+ jn

t−1+ j−1n

σ21 (s)ds

∫ t−1+ jn

t−1+ j−1n

σ22 (s)ds

. (2.3)

Proposition 2.1. Under Assumption 2.1, the integrated correlations in (2.2) and (2.3) are

linked by

δt =n

∑j=1

δt,jψt,j,

where

ψt,j =

⎛⎜⎜⎝∫ t−1+ j

n

t−1+ j−1n

σ21 (s)ds

∫ t−1+ jn

t−1+ j−1n

σ22 (s)ds∫ t

t−1 σ21 (s)ds

∫ tt−1 σ2

2 (s)ds

⎞⎟⎟⎠

1/2

denotes the square root of the product of the comparative integrated variances for asset 1

and 2 from time(

t− 1 + j−1n

)to(

t− 1 + jn

).

Proof. See Appendix.

Proposition 2.1 states that the integrated correlations for different frequencies

are connected by the terms ψt,j, each of which serves as the weight given to the

intraday integrated correlation δt,j, so that the daily integrated correlation δt is a

weighted sum of the intraday counterparts. Here we call ψt,j the square root of the

comparative variance product. By redefining ρt, the daily conditional correlation, and

ρt,j, the intraday conditional correlation, s.t.

ρt := E [δt|Ft−1] , (2.4)

ρt,j := E[δt,j|Ft,j−1

], (2.5)

we discretize the continuous-time assumption to a conditional form.2

2There are two ways of defining the conditional correlations in discrete time framework. The

first way is using the BEKK model (and its special cases such as the factor GARCH model) of Baba


Compared with ρt, ρt,j is constructed using the additional information between(t− 1, t− 1 + j−1

n

). Therefore, one would expect ρt,j to be more informative about

δt than ρt. The following proposition gives the conditional form of Proposition 2.1.

Proposition 2.2. Under Assumption 2.1, the conditional correlations defined in equations

(2.4) and (2.5) are linked by

ρt =n

∑j=1

λt,jE[ρt,j|Ft−1

]+ ct,

where λt,j = E[ψt,j|Ft−1

]and ct = ∑n

j=1 cov(δt,j, ψt,j|Ft−1).


The term λt,j denotes the conditional value of ψt,j at time(

t− 1 + j−1n

). We

adopt the following assumption to simplify the model:

Assumption 2.2. Under Assumption 2.1 and the results in Proposition 2.2, ψt,j is a ran-

dom variable with no autocorrelations and heterogeneity, the distribution of which admits a

specific mean λj, s.t.

ψt,j = λj + εψ,t,j, εψ,t,j ∼ i.i.d, E[εψ,t,j|Ft,j−1

]= 0,

for all j = 1, 2, ..., n.

This assumption suggests that the series{

ψt,j}T

t=1 fluctuates around a constant

mean value λj with no autocorrelations. In Section 2.5, we show that autocorrela-

tions of{

ψt,j}T

t=1 at all lags are insignificant based on an intraday 5-minute returns

et al. (1991) which first models conditional covariances and variances, then obtains conditional cor-

relations by taking the ratio between the two measures. The second way uses Dynamic Conditional

Correlation models of Engle (2002) or copula GARCH models of Lee and Long (2009) which es-

timate separately conditional variances and correlations, thus treating conditional correlations as

separate processes. Since our model is based on the general framework of copula GARCH mod-

els and separately estimates conditional correlations and variances, we adopt the second way of

defining conditional correlations as (2.4).


data sample. Assumption 2.2 also suggests that ct = ∑nj=1 cov(δt,j, ψt,j|Ft−1) =

∑nj=1 cov(δt,j, λj + εψ,t,j|Ft−1) = 0. As a result, the simplification leads to

ρt =n

∑j=1

E[ρt,j|Ft−1

]λj, (2.6)

such that the daily conditional correlation ρt is a weighted sum of the intraday con-

ditional correlations{

ρt,j}n

j=1 with constant weights{

λj}n

j=1, further conditioned

on Ft−1.

In an extreme case in which the instantaneous volatility is constant during one

trading day, i.e., σ1(q∗) = σ1(q) and σ2(q∗) = σ2(q), ∀q∗, q ∈ [t− 1, t), we have the

following proposition:

Proposition 2.3. Under Assumption 2.1 and σ1(q∗) = σ1(q) and σ2(q∗) = σ2(q), ∀q∗, q ∈[t− 1, t), (2.6) is further simplified to

ρt =1n

n

∑j=1

E[ρt,j|Ft−1

]. (2.7)


Proposition 2.3 shows that when the instantaneous volatility is constant within a

trading day, the daily conditional correlation ρt is the average value of the intraday

conditional correlations{

ρt,j}n

j=1 further conditioned on Ft−1. The assumption of

constant instantaneous volatility is certainly a very strong one, however, (2.7) serves

as a parsimonious choice for Monte Carlo simulation designs.

2.3.2 Dynamics of intraday conditional correlation

The conditional correlation is assumed to update at an intraday level according to

the GAS framework, s.t.

f (ρt,j) = ω(1− β) + β f (ρt,j−1) + αst,j−1, (2.8)

where f (·) is the Fisher transform operator. The transform maps ρt,j onto the whole

real line. The scaled score function st,j can then be calculated as:

st,j = ξ1,t,j(Φ−1(u1,t,j)2 + Φ−1(u2,t,j)

2) + ξ2,t,jΦ−1(u1,t,j)Φ−1(u2,t,j) + ξ3,t,j, (2.9)


where ξ1,t,j =2ρt,j

(1−ρ2t,j)(1+ρ2

t,j), ξ2,t,j = − 2

(1−ρ2t,j)

, and ξ3,t,j =2ρt,j

1+ρ2t,j

. It follows directly

that E[st,j|Ft,j−1] = 0 when the model is correctly specified. The term (Φ−1(u1,t,j)2 +

Φ−1(u2,t,j)2) measures the intraday variance contribution stemming from the stan-

dardized bivariate returns, while the product Φ−1(u1,t,j)Φ−1(u2,t,j) measures the

intraday covariance contribution (stemming from the standardized bivariate re-

turns). The advantage of using the GAS model can easily be seen as the scaled

score function contains the information from the intraday observations as well as

the information from the copula function. The calculation of the scaled score func-

tions under various choices of copulas can be found in Schepsmeier and Stober

(2014).

2.3.3 Correlation aggregation

An explicit daily recurrence equation for f (ρt) cannot be directly derived from (2.8)

as the linearity property implied by (2.6) no longer holds for f (ρt,j). To approximate

a linear relation between f (ρt) and f (ρt,j), we apply two Taylor expansions on f (ρt)

and f (ρt,j) around the previous daily estimate ρt−1, which gives

f (ρt) � f (ρt−1) + f ′(ρt−1)(ρt − ρt−1), (2.10)

f (ρt,j) � f (ρt−1) + f ′(ρt−1)(ρt,j − ρt−1). (2.11)

Solving for ρt and ρt,j and substituting back in (2.6), we have

f (ρt) �n

∑j=1

E[ f (ρt,j)|Ft−1]λj. (2.12)

The approximated linear relation between f (ρt) and f (ρt,j) resembles (2.6). Com-

bined with (2.8), the following proposition gives the implied recurrence equation

for f (ρt).

Proposition 2.4. Under Assumptions 2.1 and 2.2, and assuming the GAS-type recurrence

(2.8) for f (ρt,j), (2.12) implies the following recurrence equation for f (ρt):

f (ρt) � (1− βn)ωn

∑j=1

λj + βn f (ρt−1) + α(n

∑j=1

λjβj−1)

n

∑j=1

βn−jst−1,j︸��︷︷��︸Innovation term

. (2.13)



We call (2.13) the recurrence equation of the High-Frequency copula GAS model,

or in short, the HFGAS model. The innovation term contains the contributions

from residual intraday variance patterns{

λj}n

j=1 and the intraday score functions{st,j}n

j=1, both of which depend on the intraday observations. A weighting param-

eter βn−j is imposed on the intraday score function st,j, so that more weight is given

to the score function that is closer to the end of the trading day. We call the term

∑nj=1 βn−jst,j the realized score function. Note that from (2.9), we have

limβ→1

n

∑j=1

βn−jst,j =n

∑j=1

st,j

=n

∑j=1

{ξ1,t,j

(Φ−1(u1,t,j)

2 + Φ−1(u2,t,j)2)+ ξ2,t,jΦ−1(u1,t,j)Φ−1(u2,t,j) + ξ3,t,j

},

such that the realized score function can be approximated by the sum of the in-

traday score functions{

st,j}n

j=1 when the correlation process is very persistent. As

mentioned previously, the existing correlation modeling methods that use realized

correlations or realized variances as driving forces are constructed under daily dy-

namics assumptions, i.e., ρt,j = ρt, the result of which is that the realized score

function boils down to a linear combination of realized variances and realized co-

variances of the standardized intraday returns. This sheds light on the advantage of

assuming intraday correlation dynamics as it results in assigning intraday-varying

weights for the realized measures, thus treating intraday observations individually.

2.3.4 HFGAS vs. Intraday GAS

Another way to accommodate intraday correlation dynamics is by using the intra-

day GAS model specified in (2.8). However, the model itself cannot estimate daily

conditional correlation without using (2.13), rendering it incomparable with other

daily-frequency models. Moreover, further assumptions on the stochastic behaviors

of ct and λt,j could be incorporated in the HFGAS model, but not in the intraday

GAS model. In the HFGAS model, the relation between the intraday correlation

2.4. Estimation 17

and its daily counterpart depends on expectations formed from Ft−1, whereas the

estimated intraday correlation f (ρt,j) in the intraday GAS model is formed from

Ft−1,j, therefore not depending on ct and λt,j. Finally, the computational burden

will significantly increase for the intraday GAS model given the length of intraday

return series.

2.4 Estimation

As stated in Creal et al. (2013), estimating parameters by the standard maximum

likelihood (ML) method in the GAS model is very simple given the observed return

series {r1,t} and {r2,t}. The observable intraday return series make the HFGAS

model an observation-driven model in the spirit of Cox et al. (1981), and thus can

also be estimated through the standard ML method. The maximization problem

can be expressed as

θ = argmaxθ

T

∑t=1

�t,

where �t is the log conditional density function of the joint standardized return se-

ries at time t. We focus on the correlation structure and ignore the likelihood contri-

butions stemming from the marginal distributions of each return series. Therefore,

a standard quasi-ML (QML) method can be applied to the log-likelihood function

which is approximated by the log Gaussian copula density function.

2.4.1 Bivariate estimation

The conditional log-likelihood �t can be explicitly written as

�t = −12

log(1− ρ2t )−

ρ2t(Φ−1(u1,t)

2 + Φ−1(u2,t)2)− 2ρtΦ−1(u1,t)Φ−1(u2,t)

2(1− ρ2t )

,

where u1,t and u2,t are the empirical CDF values of the standardized return series;

ρt is constructed using (2.13) for which the intraday score functions{

st,j}n

j=1 were

calculated given (2.9).

The estimation procedure can be illustrated as follows: given an initial param-

eter set θ0 = (ω0, α0, β0)′, the initial daily conditional correlation ρt−1 and the


initial intraday conditional correlation ρt−1,0, the first step calculates the realized

score function ∑nj=1 βn−jst−1,j by iterating (2.8). Using the realized score function

value, the second step updates the daily conditional correlation ρt−1 to ρt using

(2.13) and the previously assumed value for ρt−1. The third step then sets t = t + 1

and repeats steps 1 and 2 for all trading days using the newly updated daily and

intraday conditional correlations. The final step maximizes the log-likelihood using

the standard QML method.

2.4.2 Multivariate generalization

In the cases where the dimension of the return vector rt, k := dim(rt), is larger

than 2, the aggregation and estimation procedure can still be applied. Denoting

the correlation matrix of the return vector rt by Rt, the log-likelihood function can

be written as

�t = c(ut; Rt) = −12

log(|Rt|)− 12

Φ−1(ut)T(R−1

t − I)Φ−1(ut), (2.14)

where Φ−1(ut) = (Φ−1(u1,t), Φ−1(u2,t), ..., Φ−1(uk,t))′. Denote by RV

t the vector

containing the elements of vech(Rt) obtained after eliminating all the rows with

element equal to 1, such that RVt solely contains all k(k− 1)/2 pairwise correlation

parameters. The time-varying correlation structure of the k assets depends entirely

on the time varying features of RVt . Denoting by ρk1,k2

t,j the conditional correlation

between the k1-th and the k2-th assets at time (t, j), where k1 = 2, ..., k, k2 < k1, k2 ∈N and assuming it follows (2.8), the intraday recurrence equation for f (RV

t,j), where

f (.) is element-wise Fisher transform, is then

f (RVt,j) = (I − B)ω + B f (RV

t,j−1) + Ast,j−1,

where ω is a k(k − 1)/2 vector containing the unconditional means of f (RVt,j); B

is a k(k − 1)/2 diagonal matrix with the autoregressive parameters on the diago-

nal; A is a k(k − 1)/2 diagonal matrix with the coefficients of the corresponding

scaled score functions on the diagonal, which are stacked in the k(k− 1)/2 vector

st,j−1. Denote by λj the k(k − 1)/2 vector containing λk1,k2j . Applying the same

2.4. Estimation 19

aggregation scheme as in the bivariate case, the daily recurrence equation can be

approximated by

f (RVt ) = diag(

n

∑j=1

λj)(I − Bn)ω + Bn f (RVt−1) + A(

n

∑j=1

Bj−1λj)n

∑j=1

Bn−jst−1,j. (2.15)

The score function st−1,j can be calculated following a suitable adjustment of the

results in Creal et al. (2011).

The separately modeled correlation parameters might fail to satisfy the positive

definitiveness restriction imposed on the correlation matrix. One promising adjust-

ment to enforce this restriction is provided by the hyperspherical decomposition

of the correlation matrix proposed by Jaeckel and Rebonato (2000), who instead of

modeling the correlation parameters directly, transform the correlation parameters

to time-varying angles expressed in radians. Aggregating correlations based on

time-varying angles is complicated and beyond the scope of this chapter, we will

investigate this procedure in future studies.

Creal et al. (2013) provide detailed steps for performing recursions on the gra-

dient functions of the log-likelihood function (2.14), which can also be applied for

the dynamics specified by (2.15) with parameter set θ = (ω′, diag(B)′, diag(A)′)′.

Applying the chain rule to (2.14), one gets

∂�t

∂θ′=

∂�t

∂ f (RVt )′ ·

∂ f (RVt )

∂θ′,

∂ f (RVt )

∂θ′=

∂(I − Bn)ω

∂θ′+ Bn ∂ f (RV

t−1)

∂θ′+ ( f (RV

t−1)⊗ I)∂vec(Bn)

∂θ′

+n

∑j=1

(s′t−1,j ⊗ I)∂A(∑n

j=1 Bj−1λj)Bn−j

∂θ′

+ A(n

∑j=1

Bj−1λj)n

∑j=1

Bn−j ∂st−1,j

∂θ′,

∂st−1,j

∂θ′= St−1,j

∂∇t−1,j

∂θ+ (∇′t−1,j ⊗ I)

∂vec(St−1,j)

∂θ′.

The gradient calculation can be very complicated, but computationally feasible in

the optimization procedures when k is large. Denote by θ0 the local maximizer of

�(θ) = ∑Tt=1 �t(θ). Under the assumption that the return process has finite fourth-


order moment and that it is covariance-stationary, we have

√T(θ− θ0)

d−→ N(0,J (θ0)),

with

J (θ0) = limT→∞

E[(∂�/∂θ0)(∂�/∂θ0)′]/T.

While the proof of the asymptotic normality is beyond the range of this chapter, we

investigate this property in simulations discussed in the next section. For analyses

of the asymptotic behavior of the GAS model estimators, we refer to Blasques et al.

(2014a) and Blasques et al. (2014b).

2.5 Monte Carlo Evidence

The HFGAS recurrence (2.13) was derived from the intraday recurrence (2.8) using

linear approximations and temporal aggregations. In this section, several Monte

Carlo simulation exercises are conducted in order to support our approximations

and the aggregation process. Firstly, we provide Monte Carlo evidence for the va-

lidity of the linear approximation showing that it does not lead to any significant

biases when compared to the true process across a range of parameter sets. Sec-

ondly, we justify the correlation aggregation process by testing whether parameter

estimates are robust across low and high frequencies. Finally, we compare the fits

of the HFGAS model, the daily GAS model and the GRAS model under correctly

specified and misspecified data generating processes (DGPs). We consider a wide

range of DGPs covering many possible scenarios for intraday correlation dynam-

ics. The benchmarks used to measure the performance are the mean absolute error

(MAE) and the mean squared error (MSE).

2.5.1 Data simulation

To simplify the simulation process, the bivariate data series are simulated under the

constant instantaneous volatility assumption mentioned in Proposition 2.3. We as-

sume, without loss of generality, that for each trading day consisting of 6.5 trading

2.5. Monte Carlo Evidence 21

hours, 30-minute bivariate return series are simulated from a time-varying Gaus-

sian copula. This assumption also ensures that the volatility of 30-minute returns

is 1. The synthetic daily return series are constructed by adding every 13 simulated

intraday returns.

2.5.2 The validity of the linearized Fisher transformation

The relation between f (ρt,j) and f (ρt) given in (2.12) is the result of a linear ap-

proximation after applying Taylor expansions to the Fisher-transformed variables

around ρt−1. We provide simulation evidence to support the approximation in the

sense that the filtered series based on the approximations match the true underly-

ing processes sufficiently close across the range of correlation time series patterns

considered. We start by assuming that the correlation structure of the bivariate

30-minute return series follows the Gaussian copula with conditional correlation

parameter ρt,j, s.t.

ρt,j = ω(1− β) + βρt,j−1 + αst,j−1.

Using the simplification in (2.6), with λt,j = 1/n, the true daily conditional correla-

tion can be calculated by

ρt =1n

n

∑j=1

[(1− βj−1)ω + βj−1ρt,1 + αβj−2st,1

]. (2.16)

Note that this calculation does not require any approximation since the equality

in (2.6) holds. For robustness considerations, four sets of parameters (ω, α, β) are

chosen to represent typical correlation patterns which are usually encountered em-

pirically. Some examples of the simulated correlation patterns are shown in Figure

2.1. The four sets of parameters generate respectively, a fluctuating pattern where

the correlations oscillate around zero but with periods of extreme values, and per-

sistent patterns with unconditional means 0.9 (high), 0.5 (medium) and 0 (low).

In each parameter set, 100 simulations are performed where each simulation con-

tains 65000 bivariate 30-minute returns, the non-overlapping sums of which lead

to 5000 bivariate daily returns. We therefore obtain 100 series of daily conditional

correlations {ρt}5000t=1 and 30-minute conditional correlations

{ρt,j}j=1, ..., 13

t=1, ..., 5000.


0 1000 2000 3000 4000 5000-1

-0.5

0

0.5

1ω = 0, α = 0.1, β = 0.999

0 1000 2000 3000 4000 5000-1

-0.5

0

0.5

1ω = 0.9, α = 0.01, β = 0.999

Simulated series Filtered series

0 1000 2000 3000 4000 5000-1

-0.5

0

0.5

1ω = 0.5, α = 0.01, β = 0.999

0 1000 2000 3000 4000 5000-1

-0.5

0

0.5

1ω = 0, α = 0.01, β = 0.999

Figure 2.1: Filtered series and simulated series

This figure shows the daily conditional correlation series (red thin lines) filtered according to (2.8)

and the series (grey thick lines) simulated according to (2.16). The sample size of each simulation is

65000 bivariate 30-minute returns. The graphs on the top left, top right, bottom left, bottom right,

present respectively the simulated series and the filtered series from the fluctuating case, and the

persistent cases with high, medium, and low levels.

The simulated bivariate series are then analyzed using (2.13) as the filter for

the daily conditional correlation ρt. Note that the filter is derived using a linear

approximation, therefore by testing the model fit, we are also checking the validity

of this approximation. As indicated in Figure 2.1, there are hardly any detectable

differences between the true correlation series and the filtered series in all four

cases. Table 2.1 presents the average of the 100 MAEs calculated for each simulated

correlation series given different parameter sets. The same measures from the GAS

model and the GRAS model are also provided for comparison since the two fil-

ters do not require linear approximation procedures. The intraday measure which

augments the GRAS model is realized correlation (RCorrt), s.t.

f (ρt) = ω + β f (ρt−1) + αst−1 + φRCorrt−1.

In all four cases, the MAEs from the HFGAS models are smaller than those from


Table 2.1: Mean absolute error under approximated linear relation

Model (0, 0.1, 0.999) (0.9, 0.01, 0.999) (0.5, 0.01, 0.999) (0, 0.01, 0.999)

HFGAS 0.0265 0.0024 0.0089 0.0200

GAS 0.1371 0.0164 0.0714 0.0966

GRAS 0.0597 0.0088 0.0245 0.0206

Note: This table presents the MAEs for the filtered daily conditional correlation parameters aggre-

gated based on the approximated linear relation (2.12). The corresponding MAEs calculated from

the GAS model and the GRAS model are also provided for comparison. 100 simulations based on

each parameter set generate 100 MAEs for each models. The reported values under each parameter

set are the averaged values across each of the 100 simulations. The model with the best performance

is marked in bold type. The reported numbers in each comparison are constructed using a sample

size of 65000.

the other candidates, indicating a better fit, while also suggesting that the linear

approximation does not cause any significant bias in the filtering process.

2.5.3 Parameter robustness and model fit check (correctly speci-

fied)

(2.13) is constructed through the temporal aggregation of (2.8), therefore, they share

the same set of parameters. This property can be checked by simulating intraday

bivariate returns with the intraday conditional correlation ρt,j generated by (2.8)

based on a specific parameter set (ω, α, β). These parameters can then be estimated

from the daily conditional log-likelihoods, constructed using the daily returns and

the daily conditional correlation ρt following (2.13). If the estimates (ω, α, β)

are sufficiently close to the true parameters, the aggregation procedure is justified,

further implying that the HFGAS model is locally the best model in terms of model

fit.

Table 2.2 shows the parameter estimates and the model fit measures for differ-

ent sets of true parameter values. The reported values are the averages across 100

simulations with sample size 65000. The daily GAS model is regarded as the bench-


Table2.2:Param

eterrobustness

andfinite-sam

plefitness

performance

(correctly-specified)

Trueparam

etersParam

eterestim

atesM

AE

MSE

ωα

βω

αβ

HFG

AS

GA

SG

RA

SH

FGA

SG

AS

GR

AS

-2.94440.1000

0.9990-1.5080

0.10160.9958

0.27431.0000

0.77760.0884

1.00000.7068

(0.7075)(0.0099)

(0.0015)

-2.94440.0100

0.9990-2.9456

0.01030.9991

0.23761.0000

0.60740.0612

1.00000.5133

(0.3782)(0.0023)

(0.0009)

-2.94440.1000

0.9900-2.5578

0.10220.9894

0.38051.0000

0.69370.1526

1.00000.5405

(0.2150)(0.0080)

(0.0021)

-2.94440.0100

0.9900-2.9457

0.01050.9896

0.47751.0000

0.76740.2380

1.00000.6124

(0.0655)(0.0082)

(0.0159)

-1.09860.1000

0.9990-0.5267

0.10160.9968

0.26961.0000

0.64710.0873

1.00000.5000

(0.5833)(0.0080)

(0.0012)

-1.09860.0100

0.9990-1.1078

0.00980.9991

0.22081.0000

0.39260.0517

1.00000.1586

(0.4669)(0.0025)

(0.0007)

-1.09860.1000

0.9900-0.9097

0.09820.9888

0.35491.0000

0.46720.1485

1.00000.2512

(0.1810)(0.0077)

(0.0019)

-1.09860.0100

0.9900-1.1025

0.00960.9906

0.45711.0000

0.62500.2175

1.00000.4048

(0.0563)(0.0045)

(0.0096)

0.00000.1000

0.9990-0.0691

0.10140.9970

0.26901.0000

0.61600.0882

1.00000.4552

(0.6834)(0.0072)

(0.0009)

0.00000.0100

0.9990-0.0711

0.01020.9989

0.18951.0000

0.26840.0368

1.00000.0731

(0.3223)(0.0019)

(0.0008)

0.00000.1000

0.99000.0421

0.09900.9884

0.35481.0000

0.44290.1452

1.00000.2234

(0.1752)(0.0078)

(0.0019)

0.00000.0100

0.9900-0.0051

0.01010.9901

0.45781.0000

0.52250.2221

1.00000.2848

(0.0564)(0.0078)

(0.0160)

Note:

Thefirst

sixcolum

nsof

thetable

presentthe

simulation

resultsregarding

theparam

eterrobustness,w

herethe

firstthree

columns

presentthe

truevalues

usedin

simulating

the

30-minute

conditionalcorrelation

series,andthe

lastthree

columns

presentthe

medians

ofthe

100sets

ofparam

eterestim

ates.The

standarddeviations

ofeach

parameter

acrossthe

100sim

ulationsare

listedbelow

inparentheses.

Thelast

sixcolum

nspresent

them

odelfit

performance

ofthe

HFG

AS

model,the

GA

Sm

odel,andthe

GR

AS

model.

The

MA

Esand

theM

SEsare

usedas

comparison

measures.

Eachparam

eterset

correspondsto

100sim

ulations,thereforegenerating

100values

forboth

MA

Esand

MSEs.

The

reportedvalues

under

eachparam

etersetare

theaverage

valuesacross

eachofthe

100sim

ulations.TheG

AS

modelis

settobe

thebenchm

arkm

odel.Them

odelwith

thebestperform

anceis

marked

inbold

type.Thereported

numbers

ineach

comparison

areconstructed

with

sample

size65000.


−5 0 50

20

40

60

80

100

120

140

−5 0 50

20

40

60

80

100

120

140

−5 0 50

20

40

60

80

100

120

140

ω Log(α) log(β/(1−β))

Figure 2.2: Parameters distribution in simulations.

This figure shows the estimated empirical distributions (in bars) of the standardized parameters

under sample size 65000. The number of simulation is 1000. The parameters α and β are respectively

transformed to log(α) and log(β/(1− β)) such that they do not have bounding restrictions. The

normal distribution is presented as comparison in solid lines.

mark model, the MAEs and MSEs of which are set to 1. The standard deviations

across the 100 simulations are reported in brackets. The true parameters (ω, α, β)

used to simulate the intraday conditional correlations are chosen respectively from

the following three sets: θω = (−2.9444, − 1.0986, 0), which measures three un-

conditional correlation levels;3 θα = (0.1, 0.01), which measures two fluctuation

levels; θβ = (0.999, 0.99), which measures two persistence levels. This setting leads

to 12 combinations of parameters. We can see that the estimated α’s and β’s in

all cases are very close to the true values. When α = 0.1, the intraday correlation

series exhibits a volatile pattern as shown in the upper left graph in Figure 2.1.

This distinct fluctuation poses difficulties for the capture of the true intercept term

ω(1 − β). However, the differences in most cases are insignificant. In terms of

model fit, as expected, the HFGAS model dominates the other two models.

The proof of the estimator asymptotic normality is beyond the scope of the

present chapter, however, the property can be checked for using empirical distribu-

tions of the parameter estimates. We perform 1000 simulations based on the param-

eter set (−1.0986, 0.01, 0.99), thus collecting 1000 parameter estimates (ω, α, β).

Figure 2.2 shows the empirical distributions of the suitably transformed param-

3The three values corresponds respectively to correlation measure 0.9, 0.5 and 0.


eters standardized by the true values and their obtained standard errors against

the normal distribution. The transformed parameters are no longer bounded by

stationarity conditions so that an unconstrained optimization tool can be used.

Asymptotic theory suggests that Gaussianity is preserved through transformations

provided the untransformed parameters are asymptotically normally distributed.

One can see that the distributions closely resemble the normal distribution.

2.5.4 Model fit comparison: misspecified

As further evidence, we compare model fit when the HFGAS model is misspecified.

Our first group of simulations follows the correlation simulation schemes used in

Creal et al. (2011) and Engle (2002) and simulates 65000 bivariate 30-minute returns

using the Gaussian copula with the following deterministic correlation patterns:

Sine wave : f (ρt,j) = ω +12

cos

(2π(t− 1 + j

13)

h/13

),

Constant : f (ρt,j) = ω,

Step : f (ρt,j) = ω−ω

(t− 1 +

jn> 2500

).

In the sine wave simulation, we choose h = 200 (fast sine), 1500 and 3000, which

approximately correspond to correlation cycle frequencies of 1/2 month, 1/2 year,

and one year. Out of consistency concerns, we restrict ω to lie in the low, medium

and high level ranges as in the previous subsection. Table 2.3 presents the relative

MAEs and MSEs between the estimated correlations and the simulated correlations

from the three models. The numbers reported are the averages of 100 simulations.

The GAS model is taken as the benchmark model, the MAEs and MSEs of which

are set to 1. The table shows that the HFGAS model and the GRAS model out-

perform the daily GAS model in all but the constant cases, which illustrates the

necessity of using intraday information when the correlation dynamics occur at

the intraday level. Moreover, the HFGAS model also outperforms the GRAS model

in all cases, which underlines the advantage of the HFGAS model since it is built

on the intraday dynamics assumption.


Table 2.3: Finite-sample fitness performance under deterministic correlation pattern

MAE MSE

HFGAS GAS GRAS HFGAS GAS GRAS

Sine Wave: h = 200; ω = −2.9444 0.7786 1.0000 0.8907 0.6741 1.0000 0.8490

Sine Wave: h = 1500; ω = −2.9444 0.5527 1.0000 0.7191 0.3368 1.0000 0.6267

Sine Wave: h = 3000; ω = −2.9444 0.5002 1.0000 0.6594 0.2723 1.0000 0.5063

Sine Wave: h = 200; ω = −1.9086 0.7844 1.0000 0.8573 0.7010 1.0000 0.8165

Sine Wave: h = 1500; ω = −1.9086 0.5469 1.0000 0.6539 0.3371 1.0000 0.4861

Sine Wave: h = 3000; ω = −1.9086 0.4868 1.0000 0.6081 0.2679 1.0000 0.4217

Sine Wave: h = 200; ω = 0 0.8216 1.0000 0.8258 0.7377 1.0000 0.7500

Sine Wave: h = 1500; ω = 0 0.5737 1.0000 0.5806 0.3728 1.0000 0.3817

Sine Wave: h = 3000; ω = 0 0.5321 1.0000 0.5500 0.3208 1.0000 0.3361

Constant: ω = −2.9444 1.0600 1.0000 1.1469 0.8839 1.0000 1.2116

Constant: ω = −1.9086 0.9152 1.0000 1.2386 0.8276 1.0000 1.3760

Constant: ω = 0 1.0388 1.0000 1.5403 1.1957 1.0000 2.1878

Step: ω = −2.9444; α = 2.9444 0.5638 1.0000 0.7456 0.3447 1.0000 0.6240

Note: This table presents the model fit performance of the HFGAS model, the GAS model, and the

GRAS model, when the correlations are simulated from the sine waves with h = 200, 1500 and 3000,

the constant case, and the step case. The MAEs and the MSEs are used as comparison measures.

Each parameter set corresponds to 100 simulations, therefore generating 100 values for both MAE

and MSE. The reported values under each parameter set are the average values across each of the

100 simulations. The GAS model is set to be the benchmark model. The model with the best

performance is marked in bold type. The reported numbers in each comparison are constructed

with sample size 65000.

In the second group of simulations, we simulate from the Factor Stochastic

Volatility (Factor SV) model of Jacquier et al. (1999), s.t.

rt = BFSV f t + εr,t, εr,t ∼ N(0, I),

fi,t = exp(hi,t/2)εi, f ,t, εi, f ,t ∼ N(0,1),

hi,t = 1 + φi(hi,t−1 − μi) + εi,h,t, εi,h,t ∼ N(0,1).

The setting suggests that the bivariate returns rt are generated by two factors f1,t

and f2,t, the log-variances of which follow two AR(1) processes with unit uncon-

ditional mean. BFSV is a 2 × 2 loading matrix whose first column contains two

elements b1 and b2 which measure the loadings of the two asset returns on the


first factor f1,t. The second column of BFSV is a vector (a, 0)′, such that only the

first asset has exposure to the second factor f2,t. For simplicity, all noise terms are

assumed i.i.d. and follow the standard normal distribution.

We choose 9 different forms of the matrix BFSV and 4 sets of the autoregressive

parameters ψ1 and ψ2. The simulation results are listed in Table 2.4. The reported

numbers are the averages of 100 simulations. Again, the MAEs and MSEs of the

GAS model are set to 1 as benchmarks. The table shows that the HFGAS model

dominates the other two when the autoregressive parameters are very close to 1.

The advantages start to vanish as the autoregressive parameters get smaller, i.e., the

log-variance series become less persistent. The two-factor variance model of Liu

and Xiao (2016) applied to daily equity returns shows that the first factor, which

represents the market variance factor, has an autoregressive parameter around 0.97.

The second factor is more persistent with an autoregressive parameter higher than

0.99. Following the temporal aggregation techniques for the GARCH process by

Dorst and Nijman (1993), we know that the corresponding autoregressive param-

eters of the two variance factors based on 30-minute returns should be around

0.971/13 and 0.991/13. Therefore, we would be more interested in the results when

ψ1 and ψ2 take the values 0.971/13 and 0.991/13 than others as they mimic empirical

estimates more closely. This setting corresponds to the first four columns in Table

2.4, according to which the HFGAS model outperforms the other two models by

pronounced margins.

2.6 Empirical Results

In this section, we apply the HFGAS model to intraday exchange rate data series,

and compare its performance with that of other daily correlation models.

2.6.1 Data description

The dataset contains 5-minute mid-price quotes of the EUR/USD and the GBP/USD

ranging from 6 March 2006 to 24 September 2010, a total of 1115 trading days.

2.6. Empirical Results 29

Tabl

e2.

4:Fi

nite

-sam

ple

fitne

sspe

rfor

man

ceun

der

the

Fact

orSV

mod

el

ψ1=

0.97

1/13

,ψ2=

0.99

1/13

ψ1=

0.99

1/13

,ψ2=

0.97

1/13

ψ1=

0.97

,ψ2=

0.99

ψ1=

0.99

,ψ2=

0.97

HFG

AS

GA

SG

RA

SH

FGA

SG

AS

GR

AS

HFG

AS

GA

SG

RA

SH

FGA

SG

AS

GR

AS

MA

E

b 1=

1;b 2

=1;

a=

00.

6500

1.00

000.

7395

0.57

631.

0000

0.91

080.

9851

1.00

000.

9749

0.85

681.

0000

0.83

24

b 1=

5;b 2

=5;

a=

00.

6903

1.00

000.

8644

0.66

151.

0000

0.94

510.

9942

1.00

000.

9744

0.86

671.

0000

0.87

61

b 1=

1;b 2

=5;

a=

00.

6598

1.00

000.

7656

0.56

351.

0000

0.91

850.

9848

1.00

000.

9731

0.86

101.

0000

0.84

82

b 1=

1;b 2

=1;

a=

10.

6048

1.00

000.

7218

0.54

901.

0000

0.81

420.

8878

1.00

000.

9000

0.84

281.

0000

0.81

38

b 1=

5;b 2

=5;

a=

10.

5952

1.00

000.

8608

0.58

081.

0000

0.89

600.

8650

1.00

000.

8903

0.84

861.

0000

0.86

56

b 1=

1;b 2

=5;

a=

10.

5798

1.00

000.

7480

0.54

841.

0000

0.83

570.

8549

1.00

000.

8805

0.85

501.

0000

0.84

03

b 1=

1;b 2

=1;

a=

50.

6281

1.00

000.

7061

0.55

921.

0000

0.71

850.

9986

1.00

000.

9911

0.94

921.

0000

0.95

62

b 1=

5;b 2

=5;

a=

50.

5627

1.00

000.

8589

0.54

591.

0000

0.84

130.

8051

1.00

000.

8561

0.83

461.

0000

0.82

80

b 1=

1;b 2

=5;

a=

50.

6068

1.00

000.

7110

0.60

211.

0000

0.71

580.

9720

1.00

000.

9967

0.96

031.

0000

0.96

29

MSE

b 1=

1;b 2

=1;

a=

00.

4301

1.00

000.

5694

0.32

481.

0000

0.85

650.

9726

1.00

000.

9526

0.75

061.

0000

0.70

11

b 1=

5;b 2

=5;

a=

00.

5250

1.00

000.

7949

0.44

951.

0000

0.95

490.

9786

1.00

000.

9540

0.80

211.

0000

0.77

81

b 1=

1;b 2

=5;

a=

00.

4569

1.00

000.

6343

0.31

251.

0000

0.90

720.

9721

1.00

000.

9483

0.76

631.

0000

0.72

92

b 1=

1;b 2

=1;

a=

10.

3777

1.00

000.

5390

0.30

651.

0000

0.68

120.

7998

1.00

000.

8192

0.71

851.

0000

0.66

74

b 1=

5;b 2

=5;

a=

10.

3380

1.00

000.

7793

0.33

301.

0000

0.74

810.

7567

1.00

000.

8129

0.77

041.

0000

0.76

42

b 1=

1;b 2

=5;

a=

10.

3419

1.00

000.

5860

0.30

241.

0000

0.72

720.

7386

1.00

000.

7876

0.74

241.

0000

0.71

13

b 1=

1;b 2

=1;

a=

50.

4092

1.00

000.

5019

0.33

581.

0000

0.53

970.

9814

1.00

000.

9742

0.90

301.

0000

0.91

92

b 1=

5;b 2

=5;

a=

50.

3142

1.00

000.

7871

0.29

471.

0000

0.74

870.

6702

1.00

000.

7658

0.71

211.

0000

0.69

20

b 1=

1;b 2

=5;

a=

50.

3829

1.00

000.

5125

0.37

201.

0000

0.51

420.

9082

1.00

000.

9607

0.91

861.

0000

0.92

94

Not

e:Th

ista

ble

pres

ents

the

mod

elfit

perf

orm

ance

ofth

eH

FGA

Sm

odel

,the

GA

Sm

odel

,and

the

GR

AS

mod

el,w

hen

the

corr

elat

ion

issi

mul

ated

from

the

Fact

orSt

ocha

stic

Vola

tilit

ym

odel

wit

hdi

ffer

ent

para

met

erse

ttin

gs.

The

MA

Esan

dth

eM

SEs

are

used

asco

mpa

riso

nm

easu

res.

Each

para

met

erse

tcor

resp

onds

to10

0si

mul

atio

ns,t

here

fore

gene

rati

ng10

0va

lues

for

both

MA

Esan

dM

SEs.

The

repo

rted

valu

esun

der

each

para

met

er

set

are

the

aver

age

valu

esac

ross

each

ofth

e10

0si

mul

atio

ns.

The

GA

Sm

odel

isse

tto

beth

ebe

nchm

ark

mod

el.

The

mod

elw

ith

the

best

perf

orm

ance

ism

arke

din

bold

type

.The

repo

rted

num

bers

inea

chco

mpa

riso

nar

eco

nstr

ucte

dw

ith

sam

ple

size

6500

0.


0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00

0.6

0.8

1

1.2

1.4

1.6

1.8

2

US MARKET

EUROPE MARKET

TOKYOMARKET

Figure 2.3: Intraday volatility pattern.

This figure shows the intraday volatility pattern of the GBP/USD returns filtered by the Nadaraya-

Watson regression with Gaussian kernel at the 5-minute frequency. The trading hours regarding

the three markets in our dataset are marked by arrows. The price quotes are collected during 00:00

to 16:00.

The price quotes are recorded from 00:00 to 16:00 every trading day during which

hours the two currencies are actively traded. Each series contains 192 5-minute

returns and 16 1-hour returns per trading day. Figure 2.3 shows the average in-

traday volatility pattern of GBP/USD.4 The figure shows that the typical trading

day starts with a relatively low volatility level as only the Tokyo market is open.

The volatility level increases after 2:00 when the European market opens and then

declines gradually until 7:00. The moderate level is followed by a distinguishable

peak around 8:00 when the US market opens and declines as the day proceeds. The

pronounced intraday volatility pattern calls for cyclical adjustments since it cannot

be well captured by a GARCH-type filtration.

We follow Section 2.2 and denote by yi,t,j the intraday log-price of asset i, i =

1, 2. The intraday return and the daily return are calculated by ri,t,j = yi,t,j − yi,t,j−1

and ri,t = ∑nj=1 ri,t,j. We adopt the assumption in Andersen and Bollerslev (1998)

4The pattern is the average of five weekday-specific patterns. The intraday volatilities of

EUR/USD show a somewhat similar pattern.


for the (demeaned) intraday and daily returns, s.t.

ri,t,j = di,jσi,t,jεi,t,j, εi,t,j ∼ i.i.d. (2.17)

ri,t = σi,tεi,t, εi,t ∼ i.i.d.. (2.18)

Both σ2i,t,j and σ2

i,t are assumed to follow a GJR-GARCH(1,1,1) process. We use

the skewed t distribution of Hansen (1994) to model the innovation terms εi,t,j

and εi,t. The intraday volatility pattern is captured by di,j, which is filtered by the

Nadaraya-Watson regression with Gaussian kernel with the restriction ∑nj=1 di,j = 1.

Figure 2.4 shows the autocorrelations of the absolute intraday returns before and

after the intraday volatility patterns were removed for the 5-minute and the 1-

hour returns. Significant moderation of the autocorrelation patterns can be seen

in both frequencies. Given the estimated conditional variances σ2i,t and σ2

i,t,j, and

the estimated intraday volatility pattern di,j, the standardized innovation terms are

calculated by

εi,t =ri,t

σi,t, εi,t,j =

ri,t,j

di,jσi,t,j.

The input values in the copula functions are the empirical CDF values of the series{εi,t,j

}j=1, ..., nt=1, ..., T and {εi,t}T

t=1 for both currencies.

To construct the likelihood function and the recurrence (2.13), the parameter λj

has to be pre-computed. We assume for simplicity that λj is time-invariant, and that

ψt,j = λj + εψ,t,j has no serial correlation. We test this conjecture by approximating

ψt,j by its realized counterpart Rψt,j, s.t.

Rψt,j =

(RV1,t,jRV2,t,j

RV1,tRV2,t

)1/2

,

where RVi,t,j stands for the realized variance during(

t− 1 + jn , t− 1 + j+1

n

)of asset

i, i = 1, 2. As in Barndorff-Nielsen and Shephard (2004) and by applying the delta

method, we know that Rψt,jp−→ ψt,j. We construct Rψt,j on an hourly basis using

5-minute returns, therefore j = 1, ..., 16 as we are considering 16 trading hours

per trading day. As expected, the autocorrelations of the{

Rψt,j}T

t=1 series are not

statistically significant at any lags for j = 1, ..., 16, making it redundant to model


2 4 6 8 10 12 14 16 18 200.1

0.15

0.2

0.25

0.3

0.355-Minute Absolute Return Autocorrelation

Pattern adjusted absolute return autocorrelationRaw absolute return autocorrelation

2 4 6 8 10 12 14 16 18 200.1

0.15

0.2

0.25

0.31-Hour Absolute Return Autocorrelation

Figure 2.4: Autocorrelations in intraday absolute returns.

This figure shows the autocorrelation of the absolute returns of GBP/USD up to 20 days. The upper

and bottom graphs present respectively the autocorrelations for the 5-minute and 1-hour absolute

returns. The red dashed lines in both graphs represent the autocorrelations of the unadjusted

absolute returns (raw returns); the black lines represent the autocorrelations of the pattern-adjusted

absolute returns.

the conditional expectation of Rψt,j as a time-varying process. Therefore, we take

λj =1T ∑T

t=1 Rψt,j as the proxy for λj. Empirically,{

λ}16

j=1 are around 0.0489 with

a standard deviation of 0.0040, and ∑16j=1 λj = 0.7825. With λj, the empirical CDF

values of εi,t and εi,t,j, and the initial values for the parameters in (2.13), we can

construct the log-likelihood function and perform the estimation.

2.6.2 In-sample performance

To measure the in-sample performance in a robust way, we divide the sample into

two sub-sample periods. The first period ranges from 6 March 2006 to 9 February

2009, a total of 715 trading days. The second period contains a total of 400 trad-

ing days from 11 February 2009 to 24 September 2010, which also serves as the

out-of-sample period. The estimation results for the whole sample period are also

provided. The models to be compared are the daily GAS model of Creal et al. (2013),


the GRAS model of Salvatierra and Patton (2014), and the HFGAS model. We apply

the same volatility filtering method in all three models, so the performance differ-

ence only comes from each model specification of the recurrence equation of the

correlation parameter. The correlation parameter is Fisher transformed to remove

the boundedness constraints. The recurrence equations of the daily conditional

correlation ρt for each of the three models are:

HFGAS : f (ρt) = (1− βn)ωn

∑j=1


∑j=1

λjβj−1)

n

∑j=1

βn−jst−1,j,

GAS : f (ρt) = (1− β)ω + β f (ρt−1) + αst−1,

GRAS : f (ρt) = ω + β f (ρt−1) + αst−1 + φRCorrt−1.

where RCorrt−1 is the realized correlation calculated at the end of trading day

(t− 1) by using all the intraday 5-minute realizations during the same day. Note

that because E[RCorrt−1] � E[ f (ρt−1)], the interpretation of the parameter ω in

the GRAS model is different from the one it has in the GAS model or the HFGAS

model, since ω no longer serves as the target unconditional mean.5 Also note that

the parameter β in the HFGAS model is the autoregressive parameter of the hourly

conditional correlation, which is not comparable with the β’s in the GAS model

and the GRAS model, which in turn are the autoregressive parameters of the daily

conditional correlation.

Table 2.5 reports the in-sample estimation results for the two sub-sample peri-

ods and the whole sample period. In all three periods, the HFGAS model is favored

in terms of model fit and information criteria. The parameters of the realized score

functions are all statistically significant at the 5% significance level. The HFGAS

model shows high persistence in the correlation parameters as the estimated β’s

are close to 1, even for the daily persistence parameter (β)16.

One feature of the HFGAS model is that it estimates daily and intraday condi-

tional correlations simultaneously, which allows us to examine the intraday corre-

lation patterns. Note that the intraday correlation pattern cannot be eliminated by

5We also estimate another form of the GRAS model using f (RCorrt−1) instead of RCorrt−1, but

the results are no better than using RCorrt−1 directly. We omit reporting them.


Table 2.5: Parameters, maximized log-likelihoods and information criteria

Model ω β α φ Log-lik. AIC BIC

Period 1

HFGAS -3.5514∗∗∗ 0.99771/16 ∗∗∗ 0.0119∗∗ – 248.8938 -491.7876 -478.0708

(0.6785) (0.0001) (0.0058)

GAS -2.1946∗∗∗ 1.0000∗∗∗ 0.0168∗∗∗ – 240.5794 -475.1588 -461.4420

(0.0626) (0.0000) (0.0031)

GRAS 0.0317 0.9713∗∗∗ 0.0200∗∗∗ -0.1304 241.4011 -474.8022 -456.5131

(0.0251) (0.0185) (0.0052) (0.0888)

Period 2

HFGAS -1.6514∗∗∗ 0.99981/16 ∗∗∗ 0.0048∗∗∗ – 101.1982 -196.3964 -184.4220

(0.5208) (0.0001) (0.0018)

GAS -1.5077∗∗∗ 0.9553∗∗∗ 0.0189∗∗ – 98.1714 -190.3428 -178.3684

(0.0083) (0.0180) (0.0085)

GRAS 0.1065 0.8404∗∗∗ -0.0087 -0.5612∗∗∗ 99.5082 -191.0164 -175.0505

(0.0883) (0.0504) (0.0175) (0.2137)

Full sample

HFGAS -3.0134∗∗∗ 0.99871/16 ∗∗∗ 0.0081∗∗∗ – 347.1549 -688.3098 -673.2600

(0.7510) (0.0000) (0.0037)

GAS -1.6775∗∗∗ 0.9893∗∗∗ 0.0236∗∗∗ – 336.5351 -667.0702 -652.0204

(0.0313) (0.0036) (0.0043)

GRAS 0.0393 0.9559∗∗∗ 0.0216∗∗∗ -0.1804∗ 338.4323 -668.8646 -648.7982

(0.0291) (0.0218) (0.0048) (0.0996)

Note: This table presents the parameter estimates, the maximized log-likelihoods, and the informa-

tion criteria from the HFGAS model, the GAS model and the GRAS model. The test statistics that

are significant at the 1%, 5%, and 10% levels are denoted with 3, 2, and 1 asterisks respectively. The

best performance is marked as bold-type. Sample period: 6 March 2006 – 9 February 2009 (period

1), 11 February 2009 – 24 September 2010 (period 2), and 6 March 2006 – 24 September 2010 (full

sample).


0 5 10 150.68

0.682

0.684

0.686

0.688Period 1

0 5 10 150.639

0.64

0.641

0.642

0.643Period 2

0 5 10 150.665

0.666

0.667

0.668

0.669

0.67Whole Period

01/08/2008 15/08/2008 02/09/2008 15/09/2008 30/09/20080.6

0.65

0.7

0.75

0.8Daily versus intraday conditional correlation

Daily conditional correlationIntraday conditional correlation

Figure 2.5: Intraday Correlation Pattern.

The upper three graphs show the intraday correlation patterns of GBP/USD and EUR/USD during

the three in-sample periods: 6 March 2006 – 9 February 2009 (period 1), 11 February 2009 – 24

September 2010 (period 2), and 6 March 2006 – 24 September 2010 (full sample). The trading starts

at 00:00 and ends at 16:00. The bottom graph shows the comparison between the daily and intraday

conditional correlations during August - September 2008. The daily conditional correlations (black)

are repeated 16 times to be comparable with the intraday conditional correlations (red dashed).

previously removing the intraday volatility pattern, as the correlation parameter

governs the joint distribution of the standardized innovation terms. In all three

in-sample periods, the intraday correlation patterns{

dρ,j}

are determined by tak-

ing the average of the intraday conditional correlations, such that dρ,j =1T ∑T

t=1 ρt,j.

The upper graph in Figure 2.5 exhibits robust U-shape patterns in all three peri-

ods with the lowest intraday correlation value reached around 8:00 when the US

market opens. On the other hand, the small scale used on the y-axis suggests that

the pattern’s scale is notably small, indicating that the average level of intraday

correlations does not alter much given different times of the trading day. However,


this does not necessarily imply persistent intraday correlation series, as can be seen

in the bottom graph of Figure 2.5, where changes in the intraday correlation are

dominated by small intraday shocks, which are not detected by daily dynamics

models.

2.6.3 Out-of-sample performance

In this subsection, we compare the portfolio performance based on the correlation

series estimated by the HFGAS model, the GAS model and the GRAS models. The

portfolio is constructed using the two exchange rates and is rebalanced on a daily

basis. The out-of-sample period ranges from 11 February 2009 to 24 September

2010, a total of 400 trading days (period 2). The parameters used to forecast the

daily conditional correlations are pre-estimated from the in-sample period ranges

from 6 March 2006 to 9 February 2009 (period 1).

The portfolio selection problem we consider is based on the mean-variance

framework of Markowitz (1952). We set the initial wealth of the mean-variance

manager to be 1 at the beginning of every trading day. Also, we assume a zero in-

terest rate which is reasonable since the portfolio is rebalanced on a daily basis. The

risk free asset can be regarded as cash holdings. At the end of each trading day t,

the mean-variance manager allocates his wealth following the optimal normalized

portfolio weights wt+1 which solve

minwt+1

w′t+1Σt+1wt+1,

s.t. w′t+1μt+1 = μ∗t+1,

where μt+1 is the daily conditional expected return vector; μ∗t+1 is the target return;

wt+1 and Σt+1 separately represent the vector of conditional portfolio weights and

the daily conditional covariance matrix of the two return series estimated based on

the information set Ft.

Specifically, μt+1 and Σt+1 are estimated by

μt+1 =

⎛⎝ ∑t

t∗=t−t∗+1 r1,t∗/t∗

∑tt∗=t−t∗+1 r2,t∗/t∗

⎞⎠ , Σt+1 =

⎡⎣ σ2

1,t+1 ρt+1σ1,t+1σ2,t+1

ρt+1σ1,t+1σ2,t+1 σ22,t+1

⎤⎦ .


09/2006 09/2007 09/2008 09/2009 09/20100.3

0.4

0.5

0.6

0.7

0.8

0.9

HFGAS GAS GRAS

In-sample estimated Out-of-sample forecast

Figure 2.6: In-sample and out-of-sample conditional correlations.

This figure shows in-sample estimated and out-of-sample forecasted conditional correlations from

the HFGAS model (black), the GAS model (grey circled), and the GRAS model (red dashed). In-

sample: 6 March 2006 – 9 February 2009; Out-of-sample: 11 February 2009 – 24 September 2010.

The conditional volatilities σ1,t+1 and σ2,t+1 for t = 1, ..., T are modeled using a GJR-

GARCH(1,1,1) specification in (2.18) on a rolling-window basis. The conditional

expected returns for the two currency rates are calculated using a rolling-window

average of length t∗. We choose t∗ to be 200 (trading days), which corresponds to

the traditional look-back period of most trend-following strategies, such as Asness

et al. (2013) and Moskowitz et al. (2012). The target return μ∗t+1 is assumed to be the

conditional expected return on a benchmark strategy, where the portfolio weights

of both assets are 0.5, i.e., μ∗t+1 = μ′t+1ι/2, where ι is a 2× 1 vector of ones. We call

this benchmark strategy the naive strategy.

The assumptions above ensure that the portfolio performance differences orig-

inate solely from the correlation forecasts of different models. Figure 2.6 shows

the in-sample estimated and the out-of-sample forecasted conditional correlations

from the three models featured in our comparison. One can see that the forecasts

from the GAS model and the GRAS model share a great level of similarity. Still,

the forecasts from the HFGAS model fluctuate more strongly.


Table 2.6: Portfolio performance

Characteristics

Strategy Return Volatility Sharpe Ratio Cum. Return

GBP/USD 11.05% 11.01% 1.0034 17.54%

EUR/USD 3.38% 10.95% 0.3089 5.37%

Naive 7.22% 9.91% 0.7280 11.45%

Short selling allowed

HFGAS 9.99% 8.50% 1.1748 15.85%

GAS 8.98% 8.52% 1.0536 14.25%

GRAS 9.27% 8.54% 1.0859 14.72%

Short selling prohibited

HFGAS 8.80% 8.69% 1.0135 13.97%

GAS 8.26% 8.72% 0.9468 13.11%

GRAS 8.14% 8.74% 0.9319 12.92%

Note: This table presents the performance of three portfolio strategies using the correlation fore-

casted by the HFGAS model, the GAS model, and the GRAS model. The first three rows show

the strategies that invest, respectively, 100% in GBP/USD, 100% in EUR/USD, and 50% in each

currency. The return, volatility, and Sharpe ratio are annualized measures. The cumulative returns

are calculated by cumulatively adding all the daily log-returns of the portfolios. The out-of-sample

period ranges from 11 February 2009 to 24 September 2010.

In Table 2.6 we present portfolio optimization results based on three strategies

using correlation forecasts from each of the HFGAS model, the GAS model, and

the GRAS model. The naive strategy which equally invests in the two currencies

is also included in the comparison. We consider the scenarios with and without

short selling. In either scenario, the strategies based on the assumption of time-

varying correlation outperform the naive strategy, which emphasizes the value of

correlation timing. Among the three model-based strategies, the HFGAS strategy

generates the highest annualized average return of 9.99% (8.80%) and a annual-

ized Sharpe ratio of 1.1748 (1.0135) in the scenario with(out) short selling allowed.

Moreover, the HFGAS model also delivers the lowest annualized volatility and the

highest cumulative return.

2.7. Conclusion 39

2.7 Conclusion

The dynamics of correlation at intraday levels are often omitted by daily condi-

tional correlation models. To capture the intraday dynamics, we derive an explicit

link between the correlations at different frequencies and a temporal aggregation

procedure which allows for intraday dynamics to be accommodated into a daily

recurrence equation. Simulation studies show that the resulting HFGAS model pro-

vides parameter estimates which are robust over aggregation at various frequencies

and is superior in fitting the DGP when compared to other GAS-type models. Em-

pirical studies based on two currency rates also favor the HFGAS model in terms of

model fitness. The portfolio selection application further demonstrates the practical

advantage of using the HFGAS model. One possible extension to the current work

is using a Student’s t copula to capture the dependence structure, in which case the

aggregation theory for the copula’s degrees of freedom parameter is needed.


Appendix

Proof of Proposition 2.1

Proof. From (2.2), we have

δt =

∫ tt−1 σ12(s)ds√∫ t

t−1 σ21 (s)ds

∫ tt−1 σ2

2 (s)ds=

n

∑j=1

∫ t−1+ jn

t−1+ j−1n

σ12(s)ds√∫ tt−1 σ2

1 (s)ds∫ t

t−1 σ22 (s)ds

=n

∑j=1

∫ t−1+ jn

t−1+ j−1n

σ12(s)ds√∫ t−1+ in

t−1+ j−1n

σ21 (s)ds

∫ t−1+ jn

t−1+ j−1n

σ22 (s)ds

⎛⎜⎜⎝∫ t−1+ j

n

t−1+ j−1n

σ21 (s)ds

∫ t−1+ jn

t−1+ j−1n

σ22 (s)ds∫ t

t−1 σ21 (s)ds

∫ tt−1 σ2

2 (s)ds

⎞⎟⎟⎠

1/2

=n

∑j=1

δt−1+ jnψt−1+ j

n

=n

∑j=1

δt,jψt,j,

where

δt,j =

∫ t−1+ jn

t−1+ j−1n

σ12(s)ds√∫ t−1+ jn

t−1+ j−1n

σ21 (s)ds

∫ t−1+ jn

t−1+ j−1n

σ22 (s)ds

and

ψt,j =

⎛⎜⎜⎝∫ t−1+ j

nt−1+ i−1

nσ2

1 (s)ds∫ t−1+ j

n

t−1+ j−1n

σ22 (s)ds∫ t

t−1 σ21 (s)ds

∫ tt−1 σ2

2 (s)ds

⎞⎟⎟⎠

1/2

.


Proof. By the law of iterated expectations, (2.4) and (2.2), we have

E[δt,j|Ft−1

]= E

[E[δt,j|Ft−1+ j−1

n

]|Ft−1

]= E

[ρt,j|Ft−1

].

2.7. Conclusion 41

Therefore,

ρt =n

∑j=1

E[δt,jψt,j|Ft−1

]=

n

∑j=1

E[ρt,j|Ft−1

]E[ψt,j|Ft−1

]+ cov(δt,j, ψt,j|Ft−1).

Define λt,j = E[ψt,j|Ft−1

]and ct = ∑n

j=1 cov(δt,j, ψt,j|Ft−1), then

ρt =n

∑j=1

λt,jE[ρt,j|Ft−1

]+ ct.


Proof. From (2.2), we rewrite the daily conditional correlation as

ρt = E

⎡⎣ ∫ t

t−1 σ1(s)σ2(s)κ(s)ds√∫ tt−1 σ2

1 (s)ds∫ t

t−1 σ22 (s)ds

|Ft−1

⎤⎦ ,

where κ(s) stands for the instantaneous correlation between asset 1 and 2. Since

the instantaneous volatilities are constants in a trading day, ρt can be written as

ρt = E

[σ1(t−)σ2(t−)

∫ tt−1 κ(s)ds

σ1(t−)σ2(t−)|Ft−1

]= E

[∫ t

t−1κ(s)ds|Ft−1

].

Applying a similar analysis on ρt,j, we have

ρt,j = nE

[∫ t−1+ jn

t−1+ j−1n

κ(s)ds|Ft−1+ j−1n

].

Therefore, by the law of iterated expectations and the relation can be derived as

ρt =1n

n

∑j=1

E[ρt,j|Ft−1

].

Compared with the previous case, we see that λj =1n .



Proof. The proof is based on (2.12), where

f (ρt) �n

∑j=1

E[

f (ρt,j)|Ft−1]

λj.

From the recurrence (2.8),

f (ρt,j) = ω(1− β) + β f (ρt,j−1) + αst,j−1, (2.19)

the relation between f (ρt,j) and f (ρt−1,j) can be expressed as

f (ρt,j) = (1− βn)ω + βn f (ρt−1,j) + αn

∑i=1

βi−1st,j−i,

the intraday sum of which isn

∑j=1

f (ρt,j)λj = (1− βn)ωn

∑j=1

λj +n

∑j=1

λjβn f (ρt,j) + α

n

∑j=1

λj

n

∑i=1

βi−1st,j−i. (2.20)

Using relation (2.12), we have

f (ρt) � (1− βn)ωn

∑j=1

λj +n

∑j=1

λjβn f (ρt,j) + α

n

∑j=1

λj

n

∑i=j

βi−1st,j−i. (2.21)

Subtracting (2.21) from (2.20) leads to

n

∑j=1

f (ρt,j)λj − f (ρt) � αn

∑j=1

λj

j−1

∑i=1

βi−1st,j−i.

Changing t to t− 1, we have

n

∑j=1

f (ρt,j)λj � f (ρt−1) + αn

∑j=1

λj

j−1

∑i=1

βi−1st−1,j−i. (2.22)

Using the relation that

α

(n

∑j=1

λjβj−1

)n

∑j=1

βn−jst−1,j = βnαn

∑j=1

λj

j−1

∑i=1

βi−1st−1,j−i + αn

∑j=1

λj

n

∑i=j

βi−1st,j−i,

and substituting (2.22) to (2.21), the recurrence equation of the daily conditional

correlation is

f (ρt) � (1− βn)ωn

∑j=1


∑j=1

λjβj−1)

n

∑j=1

βn−jst−1,j.

3 | Score-Driven Variance-Factor Mod-

els

3.1 Introduction

It has been a major topic of interest in asset pricing and risk management to es-

timate the conditional covariance structure of asset returns in a portfolio. When

the observed multiple series display temporal or contemporaneous dependencies,

the class of multivariate GARCH models is usually applied to jointly analyze the

second-order moment features such as volatility comovement and time-varying

correlation.

A number of multivariate GARCH models have been introduced in the last

decades. The VEC model of Bollerslev et al. (1988) directly generalizes the univari-

ate GARCH model of Bollerslev (1986) to a multivariate case where the conditional

covariance matrix is modeled as a function of the lagged conditional covariance

matrix estimate and the lagged cross product of the observed demeaned return

vector. Built on the VEC model, the BEKK model parameterized by Baba et al.

(1991) and Engle and Kroner (1995) takes a step further by ensuring the positive

definiteness of conditional covariance matrix. The constant conditional correlation

(CCC) model of Bollerslev (1990) decomposes the covariance matrix to a volatility

matrix and a correlation matrix thus separately estimating conditional variances

and correlations. The conditional variance of each asset return in the system is

assumed to follow a univariate GARCH process, and the correlation parameters

43

44 Chapter 3. Score-Driven Variance-Factor Models

are assumed to be constant. This constant correlation assumption is relaxed in the

Dynamic Conditional Correlation (DCC) model of Engle (2002) and the Varying-

Correlation (VC) model of Tse and Tsui (2002).

Another class of multivariate GARCH models is the class of factor GARCH

models.1 First introduced by Engle et al. (1990), the factor GARCH model assumes

that a number of independent latent components summarize the second-order mo-

ment features of returns through a nonsingular linear combination of the vector of

returns. Following the work of Engle et al. (1990), Alexander (2001) proposes the or-

thogonal GARCH model, i.e., the OGARCH model, which assumes that observed

returns are generated by an orthogonal transformation of a group of independent

univariate GARCH processes. The assumption of the orthogonal factor loading

matrix is relaxed in the generalized OGARCH model, i.e., the GOGARCH model,

by van der Weide (2002). Furthermore, Lanne and Saikkonen (2007) introduce the

reduced-factor orthogonal GARCH model that allows for a reduced number of uni-

variate GARCH factors. Such models were first suggested as a special case of the

multivariate GAS model by Creal et al. (2011).

Previous work in factor GARCH modeling assumes that the components driv-

ing the multivariate return series affects the level of the returns. By a linear com-

bination of these components which are modeled as univariate GARCH processes,

one can mimic a factor structure in the conditional variances of the multivariate

return series. A straightforward but yet to be explored approach to model the

volatility comovement is to assume a factor structure directly in the conditional

variances instead of the returns. In this case, the dynamics of the multiple condi-

tional variance series are governed by a linear combination of the variance factors.

We call this class of models variance-factor models.

In this chapter, we develop and extend the approach of Creal et al. (2011) to

the class of variance-factor models, resulting in variance-factor GAS models. These

models distinguish themselves from the other factor models in the sense that the

dynamics in the multiple conditional variance series are governed by a linear com-

1Engle et al. (1990) show that the factor GARCH model is a special form of the BEKK model.

3.2. Model Formulation 45

bination of several observation-driven factors. Empirical results show that this

setting significantly increases model fit. Moreover, the good performance cannot

be attributed to over-fitting the data given that the out-of-sample VaR coverage rate

comparisons still favor variance-factor GAS models. In a further application, we

show that variance-factor GAS models encompass a special formulation that fol-

lows the conditional CAPM assumption of Jagannathan and Wang (1996) exactly.

Fama-Macbeth regression results show that the R-squares obtained by this special

form variance-factor GAS models are the highest when compared to those obtained

by the DCC and the GOGARCH models, in- and out-of-sample, at both the daily

and the monthly frequencies.

The chapter develops as follows: Section 3.2 describes the formulation of vari-

ance-factor GAS models, the conditions for parameter identification, comparisons

with factor GARCH models, and a test for the number of common variance fac-

tors. Monte Carlo evidence on parameter estimation is presented in Section 3.3.

A number of empirical studies are conducted in Section 3.4. Finally, Section 3.5

concludes.

3.2 Model Formulation

In this section, we introduce the formulation of variance-factor GAS models fol-

lowed by an analysis of the conditions for parameter identification and a compar-

ison with different classes of multivariate GARCH models. For ease of implemen-

tation and simplicity, all GARCH-type recurrence equations are assumed to follow

a GARCH(1,1) process. We also provide an estimation method and a test for the

number of common variance factors.

3.2.1 Variance-factor GAS models

For a demeaned daily log-return vector rt, assume

rt = Σ1/2t εt, rt ∈RN, t ∈Z,


where εt is i.i.d. with mean 0 and an identity covariance matrix. The components

of εt are assumed to have finite fourth-order moments. This assumption on εt also

implies that rt is a covariance-stationary process with mean 0 and conditional co-

variance matrix Σt which is adapted to Ft−1, a suitably defined filtration at t− 1.

To model the heteroskedasticity in Σt, we follow the covariance matrix decomposi-

tion approach in the DCC model of Engle (2002), s.t. Σt = DtRtDt, where Rt is the

correlation matrix and Dt is an N × N diagonal matrix whose diagonal elements

are the conditional volatilities (square root of the conditional variances) of each

element in rt.

Unlike factor-GARCH models in which factor structures are imposed in rt, in

variance-factor GAS models we assume that a factor structure exists in the con-

ditional variances with a number of M (M ≤ N) factors driving the conditional

variances. The reason for this assumption is to accommodate the feature of volatil-

ity comovement empirically estimated in equity and index returns. To guarantee

positive volatilities and to avoid identification problems in the correlation estima-

tion, following Creal et al. (2011) we use log-transformed conditional variances and

a hyperspherically decomposed correlation matrix to model the time-varying co-

variance matrix Σt, i.e.

⎛⎝ log(diag(D2

t ))

φt

⎞⎠ =

⎛⎝ αD

αφ

⎞⎠+

⎡⎢⎣ βD 0

N× N(N−1)2

0 N(N−1)2 ×M

βφ

⎤⎥⎦⎛⎝ f Dt

f φt

⎞⎠ . (3.1)

Here βD is an N ×M loading matrix of the variance factors f Dtand βφ is an N∗ ×

N∗ loading matrix of the correlation angle factors f φt, where N∗ = N(N − 1)/2.

α = (α′D, α′φ)′ is a vector of intercepts. f Dtand f φt

are the observation-driven

factors which determine the dynamics of log(diag(D2t )) and φt, respectively.2 The

angle vector φt originates from the setup in which the correlation matrix Rt is

hyperspherically decomposed as X′tXt, where Xt = Xt(φt) is an upper-triangular

2We assume that there are no explicit interactions between the conditional log-variances and the

conditional correlation angles, so that f Dtonly impact log(diag(D2

t )) and f φtonly impact φt.


matrix (Jaeckel and Rebonato (2000)) of the form

Xt(φt) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 c12t c13t ... c1Nt

0 s12t c23ts13t ... c2Nts1Nt

0 0 s23ts13t ... c3Nts2Nts1Nt...

...... . . . ...

0 0 0 ... cN−1,Nt ∏N−2l=1 slNt

0 0 0 ... ∏N−1l=1 slNt

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

with cijt = cos(φijt) and sijt = sin(φijt), where φijt is the time-varying angle mea-

sured in radians. The vector φt contains N∗ = N(N − 1)/2 angles.

The recurrence equations for the factors f Dtand f φt

are assumed to follow the

GAS framework of Creal et al. (2013), s.t.⎛⎝ f Dt

f φt

⎞⎠ = f t = (I − B)ω + Ast−1 + B f t−1, (3.2)

where A and B are diagonal matrices of dimension N∗ + M. The vector ω mea-

sures the unconditional levels of the factors f t. The dynamics of the factors are

determined by their lagged values f t−1 and the scaled score function st−1. The

scaled score function st is calculated by

st = St · ∇t, ∇t =∂ log p(rt| f t,Ft−1;θ)

∂ f t.

The function p(rt| f t,Ft−1;θ) is the conditional density function of the observations

rt given the estimated conditional factors f t, the filtration Ft−1 and the parameters

θ. ∇t is the score function that measures the steepest ascent direction towards

which the log-likelihood changes according to the factor changes. We also need a

scaling matrix St of dimension N∗ + M to smooth the path of the score function.

Throughout this chapter, we choose St = I−1/2t|t−1 , where It|t−1 = E [∇t∇′t|Ft−1] is

the information matrix. This setting ensures var(st) = I. To calculate the score

function under the assumption of Gaussianity, i.e., that rt follows a joint normal

distribution, we first define

Ψt = Ψ( f t) =∂vech(Σt)

∂ f ′tfor Σt = Σ( f t).


Then following Creal et al. (2011) and Hamilton (1994), the analytical form of Ψt

given (3.1) is

Ψt = BN(I ⊗ DtRt + DtRt ⊗ I)WDtD2t [βD 0N×N∗ ]

+ BN(Dt ⊗ Dt)[(I ⊗ X′t) + (X′t ⊗ I)CN ]Zt[0N∗×M βφ].

Subsequently, the analytical forms for the score function and the information ma-

trix are

∇t =12

Ψ′tD′N(Σt ⊗ Σt)−1(rt ⊗ rt − vec(Σt)),

It|t−1 =14

Ψ′tDN(Σ′Jt ⊗ Σ′Jt)[G− vec(I)vec(I)′](ΣJt ⊗ ΣJt)DNΨt,

where ⊗ is the Kronecker product sign; Zt = ∂vec(Xt)/∂φ′t; the matrices DN, BN

and CN are defined respectively as the duplication matrix, the elimination matrix

and the commutation matrix; the matrix ΣJt is obtained by any proper matrix de-

composition procedures such that Σ−1t = Σ′JtΣJt; the matrix WDt is constructed from

the N2 × N2 diagonal matrix with diagonal elements vec(D−1t )/2 after dropping

the columns containing only 0s; and the matrix G is defined as G = E[(zz′ ⊗ zz′)]

with z ∼ N(0, IN).

By changing the number of factors in conditional variances and correlations,

variance-factor GAS models encompass a wide range of asset pricing models.

When M = 1, N ≥ 2, and rNt is the market portfolio return, the variance-factor

structure represents that of the conditional CAPM model of Jagannathan and Wang

(1996). Such a model will be discussed in detail in Section 3.4.2. By restricting the

number of correlation factors to 1 when N ≥ 3, the equicorrelation model of Engle

and Kelly (2012) can be replicated.

3.2.2 Conditions for parameter identification

Our main interest in this chapter is to study the factor structure in conditional

log-variances, therefore, we can set αφ = 0 and βφ = I so that each correlation

angle serves as its own factor. The model formulations under this setting may lead


to various identification problems for the parameters of the conditional variance

structure, both in the local levels and in the scales of the conditional log-variances.

Denote by Λt the vector of conditional log-variances at time t, s.t.

Λt = log(diag(D2t )) = αD + βD f Dt

, Λt ∈RN, t ∈Z, (3.3)

As stated in the previous section, αD is an N-vector of the intercept terms and βD

is an N ×M factor loading matrix with N ≥ M.

We assume the recurrence equation of the variance factor f Dtas follows

f Dt= (I − B2

D)1/2sDt−1 + BD f Dt−1

. (3.4)

This assumption restricts the unconditional mean of f Dtto be zero, thus leaving αD

to be the only determinant of the local level. Moreover, with the diagonal elements

of Bd less than 1 in absolute value, (3.4) restricts the scale of the factors to be 1

given that var( f Dt) = var(sDt) = I, which makes the loading matrix βD the only

determinant of Λt’s scale with var(Λt) = βDβ′D.

With no restrictions on the loading matrix βD, a different stationary process{f Dt

}with E

[f Dt

]= 0 and var( f Dt

) = I may result in the same Λt series. Assum-

ing that we can find another process Λt = αD + βD f Dtand Λt = Λt, then taking the

unconditional expectations of Λt and Λt suggests that αD = αD. The equality in the

unconditional variances of Λt and Λt leads to βDβ′D = βD β′D. However, this equal-

ity does not suggest that βD = βD since as long as βD equals βD MD, with MD an

orthogonal matrix of dimension M, we always have βD β′D = βD MD M′Dβ′D = βDβ′D.

Therefore, without further restrictions, we cannot identify the factor loading matrix

βD.

Proposition 3.1. Assume that {Λt} is a stationary process with finite covariance matrix,

then the the uniqueness of the loading matrix βD holds up to column sign changes, if the

columns of βD can be arranged so that for s = 1, 2, ..., M, the s-th column contains at

least s− 1 zeros. Let βsD be any submatrix of βD consisting of the s− 1 left-most elements

of any s− 1 rows of βD which have zeros in the s-th column, then for all s = 2, 3, ..., M,

there exists |βsD| � 0.


Proof. See Dunn (1973).

Following Proposition 3.1, when the elements in one of the rows are assumed

to be non-negative, βD can be uniquely determined by the covariance matrix of Λt.

Therefore, if Λt = Λt, we have αD = αD and βD = βD.

Proposition 3.2. Assuming that the factor loading matrix βD and the unconditional level

vector αD in (3.3) can be uniquely identified, the persistence matrix BD of the variance

factors f Dtin (3.4) can be uniquely identified.

Proof. Given Λt = Λt and αD = αD, we have βD f Dt= βD f Dt

, which from Proposi-

tion 3.1, implies that f Dt= f Dt

. From (3.4), we get

(I − B2D)

1/2sDt−1 + BD f Dt−1= (I − B2

D)1/2sDt−1 + BD f Dt−1

.

Since f Dt−1= f Dt−1

, we have

(BD − BD) f Dt−1= (I − B2

D)1/2sDt−1 − (I − B2

D)1/2sDt−1 .

Note that the right-hand side of the equation is an Ft−1-measurable function,

whereas the left-hand side is an Ft−2-measurable function. Therefore, we must

have (BD − BD) f Dt−1= 0, for t ∈Z. This result shows that BD = BD.

An example setting for βD in a 3× 2 case is⎛⎜⎜⎜⎝

β11 β12

β21 β22

β31 0

⎞⎟⎟⎟⎠

such that there is one zero term in the second column. To make sure the parameters

can be uniquely estimated without column sign changes, we can restrict β11 and

β12 to be positive.

To conclude, assuming the covariance matrix of Λt, the conditional log-variances,

exists, all model parameters featured in equations (3.3) and (3.4) can be uniquely

identified provided the loading matrix βD satisfies the restriction imposed by Propo-

sition 3.1 and all the elements in one of its rows are restricted to be positive.


3.2.3 Comparison with the DCC model

Note that variance-factor GAS models use the same covariance matrix decomposi-

tion approach as in the DCC model. The difference between the two models origi-

nates from the approaches they use to model conditional (log-) variances. Variance-

factor GAS models assume a factor structure in the conditional (log-) variances,

while the DCC model assumes that conditional (log-) variances follow univariate

GARCH models. In fact, the variance-factor GAS model resembles the DCC model

when M = N, and βD = I.

Proposition 3.3. Denote by G(.) a twice differentiable increasing function and set βD is

an identity matrix of dimension N. Then, the variance-factor GAS model defined as

G(diag(D2t )) = αD + βD f Dt

, (3.5)

and

f Dt= BD f Dt−1

+ ADsDt−1 , (3.6)

represents the DCC model, where the innovation terms in the univariate GARCH process

are replaced by sDt.

Proof. The proof follows immediately by substituting the recurrence (3.6) into (3.5),

after which one obtains

G(diag(D2t )) = αD + BD f Dt−1

+ ADsDt−1

= αD + BD(G(diag(D2t−1)))− αD) + ADsDt−1

= (I − BD)αD + BDG(diag(D2t−1)) + ADsDt−1 , (3.7)

which can be regarded as the recurrence equation for G(diag(Dt)2). Set Λt equal

to diag(D2t ), the standard DCC model assumes

Λt = (I − BD)αD + BDΛt−1 + AD(rt−1 ◦ rt−1 −Λt−1),

where ◦ is the Hadamard product operator. The above equation is thus a special


form of (3.7) with G(.) being an identity function and sDt−1 = rt−1 ◦ rt−1 − Λt−1.3

3.2.4 Comparison with factor GARCH models

As mentioned in Section 3.2.1, the core difference between variance-factor GAS

models and factor GARCH models is in the location of the factor structures. Variance-

factor GAS models assume factor structures in the conditional variances, while

factor GARCH models assume factor structures in the level of returns. The com-

parison is conducted for two cases, namely Λt = diag(D2t ) and Λt = log(diag(D2

t )),

where, same as in Section 3.2.1, Dt stands for the N × N diagonal matrix of condi-

tional volatilities. For ease of comparison, only full-factor models are considered,

in which case the factor loading matrix βD in variance-factor GAS models is an

N × N full-rank matrix.

The full-factor GARCH models can be broadly defined as

rt = Wxt,

where W is an N × N full-rank matrix and xt is an N-vector representing the in-

dependent factors of the returns, with xt = H1/2t ξt, where Ht = diag(ht), ht =

(h1t, h2t, ..., hNt)′, and ξt a sequence of i.i.d. random vectors with zero mean and

identity covariance matrix. This model is labeled ‘full-factor’ since the number of

factors is the same as the number of returns and there is no idiosyncratic noise

term. The conditional variances of xt are assumed to follow GARCH(1,1) recur-

rence equations, i.e.

ht = (I − BFD)α

FD + AF

DsFD,t−1 + BF

Dht−1, (3.8)

where BFD and AF

D are diagonal matrices of dimension N, and αFD is an N-vector

representing the unconditional variances of xt. Note that we introduce the term

sFD,t−1 to accommodate the score function so that its i-th component equals the

martingale difference (x2i,t−1 − h2

i,t−1).

3Creal et al. (2013) show that under proper scaling, the equality holds under normally distributed

returns.


3.2.4.1 Case 1: Λt = diag(D2t )

When modeling conditional variances instead of conditional log-variances, variance-

factor GAS models assume that

Λt = diag(D2t ) = αD + βD f Dt

.

Denoting by ΛFt the conditional variances of rt in factor GARCH models, we have

ΛFt = (W ◦W)ht.

By denoting f FDt

= ht − αFD and from (3.8) we get that f F

Dt= AF

DsFD,t−1 + BF

D f FDt−1

with E[ f FDt] = 0. Set βF

D = W ◦W, then

ΛFt = βF

DαFD + βF

D f FDt

,

where f FDt

can be treated as the vector containing the variance factors with βFD the

corresponding factor loading matrix. Compared with (3.3), a first difference is that

the loading matrix βFD appears in both the local level and the scale estimations

in factor GARCH models, as opposed to variance-factor GAS models in which

the two characteristics are separately estimated. Note that in a full-factor model,

this feature does not cause any difference since one can always find αFD such that

βFDαF

D = αD. In a reduced-factor model, i.e., N assets and M common factors,

however, this feature of variance-factor GAS models is advantageous since they use

the N parameters in αD to model the levels of unconditional variances, while factor

GARCH models only use the M parameters in αFD together with some degrees of

freedom in βFD. Moreover, in a setting with time-varying loading matrix βDt

, this

feature of separately estimating levels and scales can be also very advantageous,

since it enables the estimation of the variances with a time-varying factor loading

structure but with static level terms.

A second difference is that in variance-factor GAS models we separately esti-

mate the correlation structure and the variance structure so that the loading matrix

βD has no involvement in the correlation estimation. On the other hand, in factor

GARCH models, the correlation matrix Rt is determined by

Rt = D−1t WHtW ′D−1

t = diag(βFDht)

−1WHtW ′diag(βFDht)

−1, (3.9)


where one can easily see that βFD does have an impact on the correlation struc-

ture. Therefore, variance-factor GAS models have less restrictions on the correla-

tion structure than factor GARCH models. We consider this fact to be the second

advantage of variance-factor GAS models. Note that in factor GARCH models, a re-

duction in the number of factors not only imposes more restrictions on the variance

dynamics, but also more restrictions on the correlation dynamics. Therefore, the

second advantage will be even stronger in reduced-factor models. Summing up,

the two advantages of using variance-factor GAS models against factor GARCH

models under Λt = diag(D2t ) are: first, variance-factor GAS models estimate the

local levels separately from the scales of conditional variances; second, variance-

factor GAS models separate the estimation of conditional correlations from the

estimation of conditional variances.

3.2.4.2 Case 2: Λt = log(diag(D2t ))

We now turn to the case when Λt = log(diag(D2t )). Note that we still have Λt =

αD + βD f Dt, with the vector f Dt

containing the factors of the conditional log-

variances, βD the corresponding loading matrix for f Dt, and αD the vector of the

unconditional log-variances. The loading matrix βD is different than in Case 1,

since the variance factors f Dtare driving the conditional log-variances instead of

the conditional variances. Denoting by ΛFt the conditional log-variances in factor

GARCH models, then we have

ΛFt = log(βF

Dht). (3.10)

Note that βFD no longer serves as the factor loading matrix for the conditional log-

variance factors. Therefore, we need to find the loading matrix of the conditional

log-variances in factor GARCH models which would be comparable to βD. To do

so, we use our assumption that var( f Dt) = I, as mentioned in (3.4).

For a diagonal matrix ΞF of dimension N, (3.10) can be rewritten as

ΛFt = log(diag(ΞF)) + log((ΞF)−1βF

Dht).


Let ΥP and ΥΛ denote the matrices with, respectively, the orthonormal eigenvec-

tors and the eigenvalues of the matrix var(

log((ΞF)−1βFDht)

), and set Υ equal to

ΥPΥ1/2Λ . ΛF

t can be rewritten as

ΛFt = log(diag(ΞF)) + Υ

(Υ−1 log((ΞF)−1βF

Dht))

.

Set Υ−1 log((ΞF)−1βFDht) = f F

Dt, then f F

Dtcan serve as the factors of the conditional

log-variances in factor GARCH models since var( f FDt) = I. Assuming E[ f F

Dt] = μF

Df,

then

ΛFt = log(diag(ΞF)) + ΥμF

Df︸��︷︷��︸local level

+Υ( f FDt− μF

Df)︸��︷︷��︸

scale

.

Setting ΞF a diagonal matrix with diagonal elements exp(E[Λt]), we have μFDf

= 0,

and

ΛFt = log(diag(ΞF)) + Υ f F

Dt.

We can see that the factor loading matrix Υ only appears in the scale estimation.

Therefore, the first advantage of variance-factor GAS models in the previous case

is no longer applicable here. Note that for any chosen ΞF, βFD impacts both the

loading matrix Υ and the correlation matrix Rt as shown in (3.9). The estimation of

the correlation structure thus cannot be separated from that of the variance, which

is another advantage in the previous case of using variance-factor GAS models.

3.2.5 Estimation and diagnostic tests

Under the assumption that

rt|Ft−1 ∼ N(0, Σt), rt ∈RN, t = 1, ..., T,

the log-likelihood can be calculated by

� =T

∑t=1

�t =T

∑t=1

{−N

2log(2π)− 1

2log(|Σt|)− 1

2r′tΣ−1

t rt

}.

The dynamics of Σt are determined by the factors f Dtand f φt

, the recurrence equa-

tions of which are specified in (3.2). Since the factors are driven by the score func-

tions which are functions of the observed return series, variance-factor GAS mod-

els can be classified as observation-driven models as opposed to parameter-driven


models, see Cox et al. (1981). Creal et al. (2011) show that it is easy to implement

a standard quasi-maximum likelihood (QML) method for an observation-driven

model. Define θ as the full parameter vector, and the maximum likelihood estima-

tor

θ = argmaxθ

T

∑t=1

�t,

Assume by θ0 the unique maximizer of �, such that E[�] = E[∑T

t=1 �t

]. Following

the standard QML method and using the GAS model filtering mechanism, we have√

T(θ− θ0)d−→ N(0,J (θ0)

−1), (3.11)

with

J (θ0) = limT→∞

E[(∂�/∂θ0)(∂�/∂θ0)′]/T.

While the proof of estimator asymptotic normality is beyond the scope of this chap-

ter, we investigate this property via simulations in the next section. For the analysis

of the asymptotic behavior of parameter estimation in the GAS model, we refer to

Blasques et al. (2014a) and Blasques et al. (2014b).

The integer M, i.e., the number of common factors driving the multivariate

conditional (log-) variance series, is unknown. Here, we follow the approach of

Ling and Li (1997) and test whether a pre-specified number of common factors are

able to explain the heteroskedasticity in the conditional (log-) variances.

Denote by Σt the model-based estimator of the conditional covariance matrix

Σt. Define the lag-l autocorrelation in the sum of squared standardized residuals

as

Γl =T

∑t=l+1

(r′tΣ−1t rt − N)(r′t−lΣ

−1t−lrt−l − N)/

T

∑t=1

(r′tΣ−1t rt − N)2.

Set Γ = (Γ1, Γ2, ..., ΓK)′ for some integer K. Under the null hypothesis that the

model is correctly specified, Ling and Li (1997) show that

TΓ′Ω−1Γ ∼ χ2(K),

and under the assumption that the returns are multivariate normally distributed,

we have Ω = IK − ΔJ −1Δ′/(4N2), with

Δ = (Δ1, Δ2, ..., ΔK)′,

3.3. Monte Carlo Experiment 57

and

Δl = E

[∂vec(Σt)′

∂vec(θ)vec

{Σ−1

t (r′t−lΣ−1t−lrt−l − N)

}].

In practice, for a consistent maximum likelihood estimator θ of θ, Δ and J can be

replaced by their sample averages as in Li and Mak (1994), i.e.

J =1T(∂�(θ)/∂θ

)(∂�(θ)/∂θ

)′,

and

Δl =1T

T

∑t=l+1

∂vec(Σt)′

∂vec(θ)vec

{Σ−1

t (r′t−lΣ−1t−lrt−l − N)

}.

where the term ∂vec(Σt)′

∂vec(θ)can be calculated numerically.

3.3 Monte Carlo Experiment

In this section, we investigate the distributions of the finite-sample parameter esti-

mates for variance-factor GAS models under our proposed conditions of parameter

identification. We restrict our focus to the case in which N = 3 and M = 2, i.e., a tri-

variate series with corresponding conditional log-variances driven by two factors.

We name the class of variance-factor GAS models with M = N − 1 the reduced-

factor GAS models, in short, RF-GAS models. Analogously, the class of models

with the same number of variance factors as the return series are named the full-

factor GAS models, in short, FF-GAS models. Our main interest is to investigate

whether the estimated loading matrix βD is sufficiently close to the values used in

the data generating process, and whether the QML estimates approximate normal

distribution in finite-sample settings. Throughout this chapter, we only focus on

modeling factor structures in the conditional (log-) variances, therefore a constant

correlation assumption is imposed in all the simulations.

For a loading matrix βD of the form

βD =

⎛⎜⎜⎜⎝

β11 β12

β21 β22

β31 β32

⎞⎟⎟⎟⎠ ,


the conditions for parameter identification suggest that at least one of the elements

in the second column should be zero. Without loss of generality, we restrict β32

to be zero, so that the conditional log-variances of the third asset return series

are driven by only one factor. Therefore, the parameters to be estimated are θ =

(β11, β12, β21, β22, β31, B1, B2, φ12, φ13, φ23, αD1, αD2, αD2)′, where B1 and B2 are

the diagonal elements of BD. The constant correlations are determined by the

constant angles assumed in the hyperspherically decomposed correlation matrix.

The number of angles is N(N − 1)/2 = 3. We set the true angle values to φ12 =

−1.2, φ13 = −1.1 and φ23 = −1. The corresponding correlations for each pair of

simulated series are ρ12 = 0.3624, ρ13 = 0.4535, and ρ32 = 0.6132. The value for

αD is (0.9, 1.0, 1.1)′. To avoid column sign changes, we restrict β11 and β12 to be

positive.

We simulate 1000 tri-variate series under two different sample sizes, 500 and

2000. The parameter values used in the simulations are

βD =

⎛⎜⎜⎜⎝

β11 β12

β21 β22

β31 0

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎜⎝

0.60 0.20

0.55 0.23

0.7 0

⎞⎟⎟⎟⎠ , BD =

⎛⎝ B1 0

0 B2

⎞⎠ =

⎛⎝ 0.98 0

0 0.93

⎞⎠ .

The chosen parameter values determine a setting with two variance factors driving

the system. The first factor, with an autoregressive parameter 0.98, represents the

persistent factor that impacts all three conditional log-variance series. The second

factor, with an autoregressive parameter 0.93, represents the factor with less mem-

ory and serves as an extra source of dynamics for the conditional log-variances of

the first two assets. When the number of assets and factors are large, for simplicity

one can assume the persistent matrix BD to have equal values along the diagonal.

This setting does not cause identification problems on the factors since the loading

matrix specifies the roles of each factor by imposing zero elements.

Table 3.1 presents the parameter estimates obtained from 1000 simulations with

sample sizes of either 500 or 2000. The numbers in brackets indicate sample stan-

dard deviations for each parameter across 1000 simulations. The parameters are

estimated using numerical maximization of the log-likelihood function. The re-

3.3. Monte Carlo Experiment 59

Table 3.1: Monte Carlo simulation results for the RF-GAS model

True Est.(500) Std.(500) Est.(2000) Std.(2000)

β11 0.6000 0.5724 (0.2231) 0.5845 (0.0815)

β12 0.2000 0.1830 (0.0740) 0.1950 (0.0305)

β21 0.5500 0.5285 (0.2110) 0.5380 (0.0777)

β22 0.2300 0.2088 (0.0894) 0.2258 (0.0336)

β31 0.7000 0.6667 (0.2475) 0.6845 (0.0939)

B1 0.9800 0.9717 (0.0160) 0.9780 (0.0054)

B2 0.9300 0.8980 (0.0613) 0.9236 (0.0193)

φ1 -1.2000 -1.2011 (0.0428) -1.2009 (0.0207)

φ2 -1.1000 -1.1000 (0.0403) -1.1002 (0.0192)

φ3 -1.0000 -1.0066 (0.0319) -1.0036 (0.0161)

αD1 0.9000 0.8902 (0.3664) 0.8990 (0.1904)

αD2 1.0000 0.9900 (0.3434) 0.9989 (0.1771)

αD3 1.1000 1.1013 (0.4219) 1.1034 (0.2232)

Note: This table presents the means and standard deviations (in brackets) of 1000 sets of estimates

from 1000 Monte Carlo replications of the estimates. Data is generated under the RF-GAS model

with sample size 500 and 2000. The loading matrix contains five parameters, the persistence param-

eter matrix BD contains 2 parameters.

sults show that the estimates closely approximate the true parameter values in the

sense that the biases are very small in magnitude compared with the standard devi-

ations. Asymptotic normality of the QML estimators for correctly specified models

also suggests that the distribution of(θ− θ0

)/std(θ) approximates the standard

normal distribution, where the std(θ) is obtained from the information matrix in

(3.11). Figure 3.1 presents empirical density of the suitably transformed parameter

estimates. The transformations for β11, β12, B1 and B2 embed the restrictions that

β11 > 0, β12 > 0, 0 < B1 < 1, and 0 < B2 < 1. Provided the estimated transformed

parameters are asymptotically normally distributed, so should the untransformed

ones be. One can see that the empirical densities of the parameters become closer

to the standard normal density as the sample size increases from 500 to 2000.


-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

log(β11)

Dist (500)Dist (2000)Standard Normal

-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

log(β12)

-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

β21

-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

β22

-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

β31

-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

log(B1/(1−B1))

-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

log(B2/(1−B2))

Figure 3.1: Parameter distribution in simulations.

The figure shows the empirical densities of the centered t-statistics of all parameter estimates in

1000 simulations under the parameter setting. Each sample tri-variate series contains 500 (light-

grey) or 2000 (dark-grey) tri-variate observations. The standard normal distribution is indicated in

each graph by the red line.

3.4 Empirical Applications

In this section, we apply variance-factor GAS models for the prediction of Value-at-

Risk (VaR) and for the estimation of time-varying market betas. The stock return

data is from the CRSP database.

3.4.1 Model fit and VaR coverage

We apply the models developed in Section 3.2 to a panel of daily equity log-returns

between January 3rd 1998 and November 7th 2012. The dataset consists of 3836

time series observations for three equities with ticker symbols IBM (International

Business Machines Corporation), DD (E I Du Pont De Nemours And Co) and BA

(The Boeing Company). Throughout our empirical studies, we assume conditional

joint-normality for the daily log-returns with potential non-zero mean subtracted.

3.4. Empirical Applications 61

3.4.1.1 In-sample fit comparison

Our in-sample study considers three cases: the 1F-GAS model (one factor), the 2F-

GAS model (two factors) and the 3F-GAS model (three factors). In all cases, we set

Λt to be the vector of the conditional log-variances follows from (3.3), where f Dt

is the vector of variance factors and βD is the corresponding factor loading matrix.

The in-sample period ranges from January 3rd 1998 to 12th November 2010, a total

of 3236 trading days.

We stack the returns in the vector rt as follows

rt = (rIBM,t, rDD,t, rBA,t)′ .

The unconditional mean of the tri-variate conditional log-variances are determined

by the vector αD = (α1, α2, α3)′, and the factor structures with model-specific di-

mensions all follow (3.4). Moreover, since we do not consider a reduction in the

number of correlation factors and assume no interactions between the correlation

factors, the correlation factor loading matrix βφ is an identity matrix of dimension

3.4 The parameters governing the dynamics of the correlation angle vector φt are(Bφ,1, Bφ,2, Bφ,3, Aφ,1, Aφ,2, Aφ,3, ωφ,1, ωφ,2, ωφ,3

)′, s.t.

φt =

⎛⎜⎜⎜⎝

ωφ,1(1− Bφ,1)

ωφ,2(1− Bφ,2)

ωφ,3(1− Bφ,3)

⎞⎟⎟⎟⎠+

⎛⎜⎜⎜⎝

Bφ,1 0 0

0 Bφ,2 0

0 0 Bφ,3

⎞⎟⎟⎟⎠φt−1 +

⎛⎜⎜⎜⎝

Aφ,1 0 0

0 Aφ,2 0

0 0 Aφ,3

⎞⎟⎟⎟⎠ sφ,t−1

(3.12)

For ease of estimation, we restrict Bφ,1 = Bφ,2 = Bφ,3 and Aφ,1 = Aφ,2 = Aφ,3.5

4Given that we use the time series of daily returns for three different assets, we have three

conditional correlation series to be estimated.5The sample correlations between the three equity return series are 0.3794, 0.3137, and 0.4560,

and as the differences are not too pronounced, one can assume ωφ,1 = ωφ,2 = ωφ,3. In addition,

this simplification reduces the optimization time significantly while only causing some very small

differences in the estimated correlation series that cannot be detected from the graphs. One can

also see this from Table 3.2, where the bottom three estimates are very similar. We also apply this

assumption in the out-of-sample study to speed up the estimation since we are using a rolling-

window estimation.


In the 1F-GAS model, the tri-variate conditional log-variance series are assumed

to be driven by a single variance factor. The factor loading matrix βD is a 3× 1 vec-

tor. The persistence matrix BD of the single factor in (3.4) is a scalar B1. The param-

eters of the factor loading matrix and the factor dynamics are (β11, β21, β31, B1)′.


to be driven by two variance factors. The corresponding factor loading matrix βD is

a 3× 2 matrix. Following the conditions for parameter identification, we set β32 = 0,

β11 > 0, and β12 > 0. This setting suggests that the conditional log-variances for

IBM returns and DD returns are driven by two factors. By placing BA as the third

asset, the model assumes that the conditional log-variances of BA returns only load

on the first factor.6 The parameters in the βD matrix can be interpreted to be the

unscaled weights given to the two variance factors by each of the return series.

The persistence matrix in (3.4) in this case is a 2-dimensional diagonal matrix, with

elements B1 and B2. The parameters of the factor loading matrix and the factor

dynamics are (β11, β12, β21, β22, β31, B1, B2)′.


to be driven by three variance factors. The factor loading matrix βD is a 3× 3 ma-

trix. Following the conditions for parameter identification, we set β13 = β23 =

β32 = 0, β11 > 0, β12 > 0, and β33 > 0. Compared with the 2F-GAS model,

the 3F-GAS model suggests that a single factor assumed on the conditional log-

variances of BA returns is not enough to explain its variance dynamics. The per-

sistence matrix in (3.4) is a 3-dimensional diagonal matrix with elements B1, B2

and B3. The parameters of the factor loading matrix and the factor dynamics are

(β11, β12, β21, β22, β31, β33, B1, B2, B3)′.

Table 3.2 reports the parameter estimates and White (1982) standard errors as

well as the maximized log-likelihood values, the Akaike information criteria (AIC)

6The results of a principal component analysis on the squared returns of all three stocks show

that BA returns have the highest loading on the first principle component, which suggests that if all

three stocks are driven by one single factor, BA returns have a conditional variance series that most

resembles that of the single factor. This gives us a reason to assume the conditional variances of BA

returns are single-factor driven in the 2F-GAS model setting.


Table 3.2: Parameter estimates, log-likelihoods and information criteria

Parameters 1F-GAS 2F-GAS 3F-GAS

Aφ,1 0.0069*** (0.0016) 0.0074*** (0.0020) 0.0068*** (0.0014)

B1 0.9923*** (0.0038) 0.9891*** (0.0057) 0.9917*** (0.0035)

B2 – 0.9990*** (0.0029) 0.9991*** (0.0016)

B3 – - 0.7123*** (0.0823)

Bφ,1 0.9976*** (0.0018) 0.9975*** (0.0020) 0.9974*** (0.0018)

β11 0.3719*** (0.0547) 0.3719*** (0.0604) 0.3373*** (0.0464)

β12 0.2379*** (0.0290) 0.2291*** (0.0254)

β21 0.3501*** (0.0474) 0.3376*** (0.0574) 0.3523*** (0.0400)

β22 -0.0215 (0.0984) -0.0092 (0.0362)

β31 0.2804*** (0.0402) 0.2736*** (0.0505) 0.2709*** (0.0303)

β33 - 0.0945*** (0.0253)

α1 0.9975*** (0.1983) 0.9774*** (0.3007) 0.9905*** (0.2640)

α2 1.1460*** (0.1764) 1.1121*** (0.1547) 1.1327*** (0.1423)

α3 1.3919*** (0.1521) 1.3710*** (0.1338) 1.3608*** (0.1108)

ωφ,1/(1− Bφ,1) -1.2275*** (0.0889) -1.2187*** (0.0891) -1.2186*** (0.0810)

ωφ,2/(1− Bφ,1) -1.2909*** (0.0652) -1.2921*** (0.0667) -1.2924*** (0.0635)

ωφ,3/(1− Bφ,1) -1.2623*** (0.0742) -1.2707*** (0.0762) -1.2768*** (0.0694)

Num.Pars 12 15 17

Log-Lik. -18866.1 -18807.0 -18786.5

AIC 37756.1 37643.9 37607.1

BIC 37829.1 37735.2 37710.5

Note: The table presents the parameter estimates, the maximized log-likelihoods, and the infor-

mation criteria for the 1F-GAS model, the 2F-GAS model and the 3F-GAS model. The best log-

likelihood and information criteria are indicated in bold type. The test statistics that are significant

at the 1%, 5%, and 10% level are denoted with 3, 2, 1 asterisks respectively. The in-sample period

ranges from January 1998 to November 2010.


Table 3.3: Likelihood and information criterion comparisons of different models

Model Num.Pars Log-lik. AIC BIC

Unit-GARCH 9 -19477.5 38972.9 39027.7

BEKK (Diagonal) 12 -18853.4 37730.9 37803.9

DCC 14 -18828.7 37685.3 37770.5

GOGARCH 9 -18874.5 37767.1 37821.8

R-GOGARCH 16 -19078.9 38189.8 38287.1

1F-GAS 12 -18866.1 37756.1 37829.1

2F-GAS 15 -18807.0 37643.9 37735.2

3F-GAS 17 -18786.5 37607.1 37710.5

Note: The table presents the maximized log-likelihoods for different class of multivariate

GARCH(1,1) models together with the 1F-GAS model, the 2F-GAS model, and the 3F-GAS model.

The prefix ‘R’ on R-GOGARCH denotes the reduced-factor class of the GOGARCH models (Lanne

and Saikkonen (2007)). The second column presents the number of parameters to be estimated

for each model. The third column presents the maximized log-likelihoods. The last two columns

present the Akaike information criteria (AIC) and the Schwartz Bayesian information criteria (BIC).

The best two log-likelihoods and information criteria are indicated in bold type. The in-sample

period ranges from January 1998 to November 2010.

and the Schwartz Bayesian information criteria (BIC) for the three different mod-

els. From left to right, the columns present the estimates for the 1F-GAS model,

the 2F-GAS model, and the 3F-GAS model. In all three models, the loadings on

the first factor are always statistically significant and positive. The 2F-GAS model

estimates show that the added second factor, although of little importance for DD

returns, does explain a significant amount of variance dynamics for IBM returns.

The 3F-GAS model estimates show that the third factor imposed on the system,

i.e., the second factor imposed on the conditional log-variances of BA returns, sig-

nificantly improves the sample fit by more than 20 points when compared to the

2F-GAS model. In a comparison, Table 3.3 presents the estimated log-likelihoods

and information criteria for some of the classic multivariate GARCH models. We

can see that even with a reduced-factor structure, the 2F-GAS model achieves a

better in-sample fit than other multivariate GARCH models.

For a comprehensive overview of the factors, Figure 3.2 presents the estimated


1998 2000 2002 2004 2006 2008 2010-5

0

5

10Fac-1 (1F) Fac-1 (2F) Fac-1 (3F)

1998 2000 2002 2004 2006 2008 2010-4

-1

2

5Fac-2 (2F) Fac-2 (3F)

1998 2000 2002 2004 2006 2008 2010-10

0

10

20Fac-3 (3F)

Figure 3.2: Factor comparisons.

The figure shows the factor comparisons of the 1F-GAS model, the 2F-GAS model and the 3F-GAS

model from 1998 to 2010. The top graph shows the first factor in all three models; the middle graph

shows the second factor in the 2F-GAS model and the 3F-GAS model; the bottom graph shows the

one extra factor in the 3F-GAS model compared with the 2F-GAS model.

factors from the 1F-GAS model, the 2F-GAS model and the 3F-GAS model. Since

the first factor is the only factor on which all three conditional log-variance series

load on, one would be tempted to assume that the first factor mimics the global

log-variance factor. This conjecture is supported by the top graph, where the first

factor estimated by each of the three models follows a similar time series pattern

which mimics the conditional log-variances of the S&P 500 Index. The middle

graph shows the second factors estimated in the 2F-GAS model and the 3F-GAS

model. Since the first factor is the main driving force as indicated by the large

factor loadings, the second factor could only provide limited information about the

variance dynamics, which potentially is the reason why it has a much smoother

pattern than the first factor. The bottom graph shows the extra factor in the 3F-GAS

model that drives the third series, i.e., the factor ignored by the 2F-GAS model. One

can see that this factor fluctuates strongly. Table 3.2 shows that while B1 and B2 are

all close to 0.99, the autoregressive parameter B3 is only 0.7123, drawing attention

to a possible over-fitting, i.e., the high likelihood of the 3F-GAS model is reached


by fitting the noise in the squared BA returns. Another possible explanation for

the high likelihood is that the third factor captures the jump component in the

volatility.

1998 2000 2002 2004 2006 2008 2010

102030 2F-GAS

1998 2000 2002 2004 2006 2008 2010

102030 3F-GAS

1998 2000 2002 2004 2006 2008 2010

102030 GOGARCH

1998 2000 2002 2004 2006 2008 2010

102030 DCC

IBM DD BA

1998 2000 2002 2004 2006 2008 2010

102030 1F-GAS

Figure 3.3: Estimated conditional variances.

The figure shows the estimated conditional variances from 1998 to 2010 under the 1F-GAS model,

the 2F-GAS model, the 3F-GAS model, the GOGARCH model and the DCC model. The blue

lines represent the condition variances of the daily returns of IBM; the green dotted lines represent

the condition variances of the daily returns of DD; the red dashed lines represent the conditional

variances of the daily returns of BA.

Figure 3.3 presents the conditional variances estimated by the three models.

We also show the conditional variances estimated by the GOGARCH model and

the DCC model for comparison. As can be seen from the graph, the factor-based

models are more capable of describing the the conditional variance comovement

than the DCC model. One such example occurs during the ‘9/11’ event, when

the Boeing company suffered a large price drop. The DCC model responds to

this event by showing a variance peak for BA returns, while the variances of IBM

returns and DD returns remain unaffected. However, the 2F-GAS model reacts in a

way that increases the level of the conditional variances for IBM and DD returns to

a similar level as that of BA returns. We regard this feature of variance-factor GAS

models as an advantage since it reflects the variance spillover between different


industries.

1998 2000 2002 2004 2006 2008 20100

0.5

12F-GAS

IBM vs DD IBM vs BA DD vs BA

1998 2000 2002 2004 2006 2008 2010-0.5

0

0.5

1GOGARCH

1998 2000 2002 2004 2006 2008 20100

0.5

1DCC

Figure 3.4: Estimated conditional correlations.

This figure shows the estimated conditional correlations from 1998 to 2010 under the 2F-GAS model,

the GOGARCH model and the DCC model. The blue lines represent the conditional correlations

between the daily returns of IBM and DD; the green dotted lines represent the conditional correla-

tions between the daily returns of IBM and BA; the red lines represent the conditional correlations

between the daily returns of DD and BA.

Figure 3.4 presents the conditional correlation series estimated by the 2F-GAS

model, the GOGARCH model and the DCC models. In Section 3.2.3, we demon-

strate that one advantage of using variance-factor GAS models is that the load-

ing matrix does not feature in the correlation estimation, while it does so in the

GOGARCH model. Therefore, as expected, the estimated correlations from the

GOGARCH model deviate significantly from those of the 2F-GAS model and the

DCC model. From the graph, the conditional correlation series estimated by the

GOGARCH model exhibits strong fluctuations, which result in multiple periods

of negative correlations among the three stock return series. Since negative cor-

relations between individual stocks are usually considered unrealistic, we regard

this as an undesirable feature of the GOGARCH model. We know that unrealistic

correlation levels occur even more often when simulating multivariate return series


from the GOGARCH model. Note that the DCC model separates the correlation

estimation from the variance estimation as variance-factor models do. Therefore,

its estimated correlation patterns are close to those of the 2F-GAS model.

To test the number of factors, we use the approach detailed in Section 3.3.2 and

test for the first lag autocorrelation, since all of our results are implemented in a

GARCH(1,1)-type framework. We start by testing the residuals from the 1F-GAS

model in order to check whether a single factor is enough to explain the variance

dynamics. The test statistic for the 1F-GAS model is 3.9668, with a p-value of

0.0465, which indicates a rejection at 95% confidence level of the hypothesis that

one common variance factor is enough to explain the variance dynamics of the

three stocks. By extending the 1F-GAS model to the 2F-GAS model, the statistic

reduces to only 2.4277, with a p-value of 0.12, which indicates that we cannot reject

the hypothesis that the two variance factors estimated by the 2F-GAS model are

enough to explain the heteroskedasticity in the conditional log-variances of the

three return series. A further test on the 3F-GAS model provides a test statistic

of 1.8452, with a p-value of 0.17, which again suggests the inadequacy of the 1F-

GAS model. Out of parsimony concerns, we conclude that the 2F-GAS model is

sufficient in our example to explain the common structure in variance dynamics,

since the extra factor in the 3F-GAS model might be the result of over-fitting.

3.4.1.2 Out-of-sample performance: simulations and VaR coverage

One reason for implementing the class of variance-factor models is that it pro-

vides a new approach to simulate multivariate returns with volatility comovement.

Variance-factor models such as the 2F-GAS model and the 3F-GAS model sepa-

rate the dynamics of variances and correlations so that the variance factors do not

influence the correlation estimation. The models we choose to compare with are

the 2F-GAS model, the 3F-GAS model, the GOGARCH model, the reduced-factor

GOGARCH model, and the DCC model. We use the parameters estimated in the

last subsection as the simulation input, and set the sample size to 5000. The noise


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

5

102F-GAS

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

5

103F-GAS

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20GOGARCH

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20R-GOGARCH

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

25

50DCC

Figure 3.5: Simulated conditional variances.

This figure shows the simulated conditional variances from the 2F-GAS model, the 3F-GAS model,

the GOGARCH model, the reduced-factor GOGARCH model and the DCC model. Sample size is

5000. The estimates from Table 3.3 are used as parameter values to perform the simulations.

terms are assumed to follow a conditionally normal distribution.7

Figure 3.5 and Figure 3.6 present the simulated series of the conditional vari-

ances and correlations from the five models. From top to bottom, the graphs show

the results from the 2F-GAS model, the 3F-GAS model, the GOGARCH model,

the reduced-factor GOGARCH model and the DCC model. In Figure 3.5, clear

comovement features exist in the first four models, which are factor-type models.

The DCC model fails to capture this feature since the variance series are modeled

independently. As previously noted, even though the GOGARCH-type models

are capable of producing multivariate return series with volatility comovement, in

simulations they might produce strongly fluctuating correlations with periods of

negative values as shown in Figure 3.6. The only two models that are able to simu-

late multivariate return series with volatility comovement and realistic correlation

series are the 2F-GAS model and the 3F-GAS model.

7Simulated series for each model are under the seed 156 in MATLAB R2013a.


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.5

12F-GAS

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.5

13F-GAS

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1

0

1GOGARCH

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1

0

1R-GOGARCH

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1

0

1DCC

Figure 3.6: Simulated conditional correlations.

The figure shows the simulated conditional correlations from the 2F-GAS model, the 3F-GAS model,

the GOGARCH model, the reduced-factor GOGARCH model and the DCC model. Sample size is

5000. The estimates from Table 3.3 are used as parameter values to perform the simulations.

To empirically test the advantages of using variance-factor GAS models to simu-

late multivariate return series, we conduct an out-of-sample study closely following

the procedure of Creal et al. (2011) who use the Value at Risk (VaR) coverage rate as

performance benchmark. The out-of-sample performance is judged by comparing

the forecasting ability for 1% and 5% VaR at 1-day horizon for different portfolios

constructed with the three stocks. Denote the 3× 1 vector wi as the weight vector

for portfolio i. We choose four weight vectors which are w1 = (0.25, 0.25, 0.25),

w2 = (−0.5, 0.25, 0.25), w3 = (0.25, − 0.5, 0.25), and w4 = (0.25, 0.25, − 0.5). The

strategy w1 takes long positions on all three stocks, and the other three strategies

wi, i = 2,3,4, take short positions on each of the three stocks respectively. The

out-of-sample window ranges from November 15th 2010 to November 7th 2012,

a total of 500 trading days. We use a rolling-window of 1500 days for estimating

model parameters. We set the start of the window at t = 1 and the end at t = 500.

For every t, the conditional covariance matrix Σt is calculated using the 2F-GAS


Table 3.4: VaR exceedance rates and p-values for the Kupiec test of correct coverage

ω1 ω2 ω3 ω4

Model 1% 5% 1% 5% 1% 5% 1% 5%

2F-GAS 2.00 4.00 1.20 5.00 1.20 4.40 1.20 4.80

(0.05) (0.29) (0.66) (1.00) (0.66) (0.53) (0.66) (0.84)

DCC 2.00 4.40 1.20 3.80 0.60 3.40 0.80 3.80

(0.05) (0.53) (0.66) (0.20) (0.33) (0.08) (0.64) (0.20)

GOGARCH 2.40 4.60 1.20 3.80 1.40 3.20 1.80 4.00

(0.01) (0.68) (0.66) (0.20) (0.40) (0.05) (0.11) (0.29)

Note: The table presents the VaR exceedance rates in percentage of the 2F-GAS model, the DCC

model, and the GOGARCH model. The p-values for the Kupiec test are presented in parentheses.

The best models in each case are in bold type. The out-of-sample period ranges from November

2010 to November 2012.

model, the GOGARCH model, and the DCC model with the in-sample estimation

period ranging from t− 1500 to t− 1. The VaR estimates at time t are calculated

by simulating rst , such that rs

t ∼ N(0,Σt) for s = 1, ..., Np, where Np is sample path

number. We choose Np = 10000 and calculate the simulated portfolio returns at

time t by psi,t = wirs

t , i = 1, ..., 4, and s = 1, ..., 10000. The VaR estimates of the i-th

portfolio at time t with coverage 1% and 5% are calculated based on the empirical

distribution of{

psi,t

}10000

s=1. We repeat this procedure for each t = 1, ..., 500, then

for each portfolio i, we obtain the VaR series VaRit(a) for t = 1, ..., 500 and a = 1%

and 5%. Next we compare the VaR estimates with the real portfolio return pi,t,

for all four portfolios and all t. For portfolio i, we calculate the proportion of VaR

exceedances, that is, the number of t such that pit < VaRit(a), divided by 500, after

which we collect eight VaR exceedance rates (four portfolios × two coverage rates:

1% and 5%) for each of the three multivariate models. Denote by ai,a and γi,a the

exceedance rate and total number of exceedances for portfolio i at coverage rate a,

i = 1, ..., 4 and a = 1%, 5%.

To compare the performance of the three models, we use the test statistic of

Kupiec (1995) to test the null hypothesis H0 : ai,a = a against the alternative hy-


pothesis H1 : ai,a � a for each portfolio i = 1,2,3,4. The test statistic of Kupiec

is

LRi,a = 2{

log[aγi,ai,a (1− ai,a)

500−γi,a ]− log[aγi,a(1− a)500−γi,a ]}

,

which is asymptotically χ2(1) distributed. Table 3.4 presents the proportion of

VaR exceedances and the p-values of the corresponding likelihood ratio test of

Kupiec. We can see that in 7 out of 8 cases, the 2F-GAS model is the best choice,

providing the most accurate coverage rates. These results underline the advantages

of using variance-factor GAS models for simulations since they better approximate

real return processes.

3.4.2 Fitting the CAPM with time-varying beta

As mentioned in Section 3.2.3, variance-factor GAS models are closely related to

the conditional CAPM of Jagannathan and Wang (1996), in the sense that the linear

relation between stock and the market returns is inherited by their variance struc-

tures. In this subsection, we show that by adopting a slightly modified variance-

factor GAS model, we can estimate time-varying market betas, or in short, betas,

following the exact structure of the conditional CAPM. The estimated betas from

such an approach should provide the best fit in explaining stock return variation

in the conditional CAPM setting.

3.4.2.1 Model setting and Fama-Macbeth regression results

Assume the univariate daily excess log return rit of stock i follows the conditional

CAPM for the market portfolio excess return, denoted as rmt, s.t.

rit = βitrmt + σIitεit = σitηit, ηit ∼ N(0,1),

rmt = σmtηmt, ηmt ∼ N(0,1),

where εit, ηit, ηmt are i.i.d. standard normally distributed variables which sepa-

rately represent the idiosyncratic return shock to stock i, the total return shock to

stock i, and the systematic return shock. The conditional idiosyncratic variance of


rit is denoted by σ2Iit, where the subscript I stands for idiosyncratic. The conditional

market beta for stock i is then calculated by

βit =cov(rit,rmt|Ft−1)

var(rmt|Ft−1)=

ρitσitσmt

σ2mt

= ρitσit

σmt,

where ρit denotes the conditional correlation between rit and rmt. Taking the con-

ditional variance of rit leads to

σ2it = β2

itσ2mt + σ2

Iit, (3.13)

which simply states that the conditional variances σ2it has two components: the first

one, measured by β2itσ

2mt, is due to the variance spillover σ2

mt from the market port-

folio return scaled by β2it, and the second one is due to the idiosyncratic variance

σ2Iit which according to the theory is independent from the variance of the market

portfolio return. Directly applying variance-factor GAS models on (3.13) is cum-

bersome since it requires time-varying factor loadings βit for the market variance.

However, the time-varying loadings can easily be accommodated by assuming a

time-varying conditional correlation between rit and rmt. Note that the conditional

CAPM assumes that the conditional correlation between rit and rmt originates solely

due to the fact that βit is nonzero, which can be seen from

ρit =βitσ

2mt√

(β2itσ

2mt + σ2

Iit)(σ2mt)

.

The conditional variance of rit can also be written as a function of the conditional

correlation ρit and the idiosyncratic variance σ2Iit only:

σ2it =

11− ρ2

itσ2

Iit. (3.14)

(3.14) shows that we can write the conditional variance of rit as a single factor

model, where this single factor is the idiosyncratic variance σ2Iit, with time-varying

factor loadings determined by ρit. In this way, a variance-factor GAS model can

easily be implemented as⎛⎝ σ2

mt

σ2it

⎞⎠ = SLt

⎛⎝ σ2

mt

σ2Iit

⎞⎠ = SLt

(αD + βD f Dt

), (3.15)


where

αD =

⎛⎝ α1

α2

⎞⎠ , βD =

⎛⎝ β11 0

0 β22

⎞⎠ , SLt =

⎛⎝ 1 0

0 1/(1− ρ2it)

⎞⎠ .

The structure of (3.15) is a special bivariate case of the variance-factor models,

where the factor loading of the second factor, i.e., the factor governing the dynam-

ics of idiosyncratic variance, depends on the level of the conditional correlation ρit.

Therefore, (3.15) represents a class of variance-factor models with time-varying fac-

tor loadings. Following the variance-factor GAS model framework, we can model

the dynamics of the variance factors using a GAS-type recurrence equation, i.e.

fDt = BD fDt−1 + (I − B2D)

1/2sDt−1

where we update the factor levels by the scaled score function sDt . Note that the

CAPM restricts σ2Iit and σ2

mt to be uncorrelated, which can be imposed by assuming

the scaling matrix to be I−1/2t|t−1 such that var( f Dt

) = I. The recurrence equation for

f Dtis then precisely (3.4). The time-varying angle parameter φt can be estimated

following the recurrence equation of fφ given in (3.2). The correlation ρit can then

be calculated as cos(φt).

To test the model empirically, we take daily log-returns for 100 stocks in the

S&P 500 Index, and compare the estimated betas from the variance-factor GAS

model, the DCC model, the GOGARCH model, and the constant CAPM model.

The data window ranges from January 2001 to December 2010, a total of 2515

trading days. We apply Fama-Macbeth regressions of Fama and MacBeth (1973)

using the estimated betas from the 4 models for each trading day in the sample

period. The regression equation reads

rit − r f t = γ0t + γ1t βit + εit, i = 1, ..., 100.

A reasonable beta estimate should produce a positive and significant market risk

premium, and also gives the best model fit by producing the highest R-square

compared with other competing models. We also compare the betas at the monthly

frequency, for which we use the betas estimated in the last day of the previous


Table 3.5: Fama-Macbeth regression results

Daily frequency Monthly frequency

Model γ0 γ1 R2 γ0 γ1 R2

In-sample:

Var-GAS 0.0065 0.0499 0.0886 0.0146 1.1421** 0.0821

0.3306 1.4414 0.0348 2.0102

DCC -0.0019 0.0564* 0.0827 -0.1002 1.2099** 0.0740

-0.0973 1.7981 -0.2421 2.3188

GOGARCH 0.0200 0.0322 0.0714 0.3441 0.7492 0.0753

0.9655 0.9659 0.8682 1.2211

Const 0.0192 0.0333 0.0650 0.4000 0.7000 0.0790

0.8060 0.8763 0.8174 0.8897

Out-of-sample:

Var-GAS 0.0589 -0.0101 0.1177 1.4109 -0.4820 0.1520

1.9565 -0.1740 1.7059 -0.3201

DCC 0.0578 -0.0133 0.1160 0.9343 0.0126 0.1361

1.8586 -0.2290 1.1798 0.0087

GOGARCH 0.0648 -0.0202 0.1057 1.2778 -0.2985 0.1351

1.9378 -0.3039 1.3756 -0.1736

Const 0.0686 -0.0240 0.1033 1.4463 -0.5148 0.1376

1.8581 -0.3323 1.3084 -0.2696

Note: This table presents the Fama-Macbeth regression results on four different models and for both

the daily and the monthly frequencies. The in-sample results are based on the daily log-returns of

100 stocks over the data window from January 2001 to December 2010; the out-of-sample window

ranges from January 2011 to December 2012. The t-statistics are presented in the lower rows. The

significance levels of 10%, 5%, and 1% are labeled as 1, 2, and 3 asterisks.

month as a proxy for the predicted monthly beta. Table 3.5 shows that when

the regression is conducted at the daily frequency, none of the four models can

predict significant market risk premia. The R-squares favor the variance-factor GAS

model, which is reasonable as the variance-factor GAS model exactly follows the

conditional CAPM assumption. The regression results at the monthly frequency

show that the DCC model and the variance-factor GAS model are able to predict

significant market risk premia at 5% significance level. The R-squares again suggest

that the variance-factor GAS model provides the best model fit.

The out-of-sample analysis is based on data spanning the period from January


2011 to December 2012, with a total of 502 trading days. We use a fixed window

approach so that the parameters are fixed at the in-sample estimated values. Ta-

ble 3.5 shows that none of the market risk premia estimated from the models are

significant. The R-squares again suggest that at both the daily and the monthly

frequencies, the variance-factor GAS model provides the best model fit.8

3.4.2.2 Portfolio sorting

To further demonstrate the superiority of the variance-factor GAS model in fitting

the conditional CAPM, we implement a portfolio sorting strategy according to the

level of the estimated betas. Again, we implement this strategy at both the daily

and the monthly frequencies. Since we are more interested in fitting the model and

because the out-of-sample analysis is based on fixed parameters, we only study the

performance during the in-sample period. The idea of the sorting scheme follows

the conditional CAPM assumption that the market return is the only risk factor

that is priced in a cross-section of stock returns. The portfolio with a high beta

implies a large exposure to the only existing risk factor, generating a higher return

than the portfolio with a low beta. Therefore, reasonable beta estimates should

exhibit a strong positive relation with portfolio returns. To check this relation for

each trading day, we first sort the 100 stocks according to their estimated betas.

Then we divide the 100 stocks into 5 groups so that Group 1 contains the 20 stocks

with the lowest betas and Group 5 contains the 20 stocks with the highest betas.

The betas of the stocks increase from Group 1 to 5.

Table 3.6 shows the results of portfolio sorting obtained using the beta esti-

mated from the variance-factor GAS model, the DCC model and the GOGARCH

model. In the daily sorting scheme, daily portfolio returns (scaled to monthly re-

turns by multiplying with 22) and annualized volatility levels sorted according to

their beta levels estimated by the variance-factor GAS model increase monotoni-

8To test the robustness of these results, we also perform year-by-year regressions. In an un-

reported result, the R-squares of the variance-factor GAS model are the highest for 11 out of 12

years.

3.5. Conclusion 77

cally from the lowest to the highest group. This monotonic relation in the daily

returns cannot however be found in the portfolios sorted according to the DCC

and the GO-GARCH model estimated betas. Similar patterns can be found for

the monthly sorting scheme, where the portfolio with the highest betas estimated

by the variance-factor GAS model generates an average monthly return of 1.42%,

which is 1% higher than the return of the portfolio with the lowest betas. The

monthly return again monotonically increases with beta, and this feature cannot

be found in any of the other two models.

3.5 Conclusion

We have introduced a class of variance-factor GAS models in which the factor

structure is imposed in the (log-) conditional variances. Given the identification

adjustment for the loading matrix, the number of factors that can be specified in

the model is flexible. In empirical applications, the number of factors can be deter-

mined using a test on residual autocorrelations. By comparing with factor GARCH

models, several advantages are found in favor of variance-factor GAS models. Em-

pirical results show the superiority of using variance-factor GAS models when

compared against a range of multivariate GARCH models, both in- and out-of-

sample. Another refinement proposed in our model can be obtained by assuming

a factor structure in the conditional correlations while simultaneously assuming a

factor structure in the (log-) conditional variances. The inclusion of high frequency

data would likely improve model performance significantly. Finally, an extension

for the purpose of option pricing can be considered by assuming a variance struc-

ture which contains a risk-neutral variance factor and a variance risk premium

factor.


Table3.6:Portfolio

sortingresults

accordingto

betas

Daily

frequencyM

onthlyfrequency

Model

Lowbeta

23

4H

ighbeta

Lowbeta

23

4H

ighbeta

Var-G

AS

Return

0.46%0.81%

1.24%1.39%

1.49%0.43%

0.77%1.18%

1.32%1.42%

Vol14.09%

17.28%20.37%

24.51%32.35%

12.32%12.82%

16.40%18.83%

24.95%

SharpeR

atio0.1093

0.15780.2055

0.19180.1556

0.12200.2072

0.24890.2434

0.1967

DC

C

Return

0.51%0.58%

1.22%1.57%

1.50%0.49%

0.55%1.16%

1.50%1.43%

Vol14.33%

17.13%20.25%

24.39%32.04%

12.62%13.73%

15.43%20.14%

24.34%

SharpeR

atio0.1216

0.11460.2033

0.21840.1584

0.13470.1394

0.26000.2580

0.2034

GO

GA

RC

H

Return

0.68%0.87%

1.11%1.50%

1.17%0.65%

0.83%1.06%

1.43%1.11%

Vol13.83%

17.39%21.07%

24.26%31.12%

10.79%13.83%

15.46%20.06%

25.44%

SharpeR

atio0.1673

0.16850.1782

0.20960.1273

0.20930.2066

0.23690.2471

0.1517

Note:

Thistable

presentsthe

portfolioperform

ancein

boththe

dailyand

them

onthlyfrequencies,

sortedby

thesize

ofbetas.

Thereturns

and

SharpeR

atiosare

reportedatthe

monthly

frequency.Thevolatilities

arereported

inannualized

values.Theportfolio

sortingis

implem

entedduring

thesam

pleperiod

fromJanuary

2001to

Decem

ber2010,a

totalof2515

tradingdays.T

henum

berof

stocksselected

is100.

4 | Factor Premia in Variance Risk

4.1 Introduction

The evolution of individual stock return variances exhibits strong commonalities

which risk modeling should account for. The market embedded pricing of this risk

feature can only partially be explained by linear factor-based asset pricing mod-

els. A comprehensive and systematic analysis of variance comovement requires

imposing a factor model directly on the second moment of returns, such that the

dynamics of the factors determine the uncertainty of the variances, thus represent-

ing the sources of variance risk.

This is the first work to analyze the properties of the systematic variance-factors

driving individual stock variances and their implications for the variance risk pre-

mia. The first contribution of this chapter is the identification of a particular vari-

ance factor which coexists with the market variance factor. This factor, depicted by

the grey line in Figure 4.1, displays a time series pattern rarely seen in the exist-

ing financial econometrics literature and plays a nontrivial role in explaining the

common dynamics of individual stock variances. The dynamic nature of these two

factors suggests their potential significance for the pricing of the variance risk of in-

dividual stocks. Demonstrating the pricing implication of the variance factors and

providing econometric evidence to support it constitutes the second contribution

of this chapter. The quantified factor risk premia together with the estimated factor

memory features jointly predict the variance risk premia of individual stocks, lead-

ing to the third contribution of this chapter, which involves the implementation of

an option portfolio strategy using the term structure of straddle returns.

79

80 Chapter 4. Factor Premia in Variance Risk

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-4

-2

0

2

4

6

8

10

Figure 4.1: Two factors filtered from individual stock variances.

This figure shows the time series plots of the two factors filtered from multivariate series of condi-

tional variances of individual stock returns. The first factor illustrated by the black line represents

the market variance factor, and the second factor by the grey line represents the second source of

variance dynamics.

The chapter starts by presenting the intuition behind variance-factor models,

particularly for the two-factor case. Firstly, important intuition stems from a re-

cently documented attribute of individual stock variances: idiosyncratic variance

comovement.

Linear factor-based asset pricing models such as the CAPM of Merton (1973)

and the APT of Ross (1976) suggest that equity returns can be explained by the

market return or a linear combination of factors, leaving the residuals to be id-

iosyncratic, or firm specific. The factors represent the sources of systematic risk

that are priced in a cross-section of stock returns. The unexplained residuals from

the linear equations are regarded as idiosyncratic returns, the risk of which, usu-

ally quantified in the form of idiosyncratic variance, is not priced under perfect

diversification. However, empirical tests of the pricing implications of idiosyncratic

variances suggest otherwise, e.g., the well-known idiosyncratic volatility puzzle of

Ang et al. (2009), who find that high residual variances from a Fama-French 3-factor

model (Fama and French (1993)) predict low future returns. Although many stud-

ies have focused on explaining this puzzle such as in Chen and Petkova (2012), Fu

(2009) and Cao and Han (2013), who regard missing factors as the main reason for

pricing idiosyncratic risk, it still draws the question: why, as indicated in Herskovic

4.1. Introduction 81

et al. (2014), do the idiosyncratic variances of individual stocks still exhibit a high

level of comovement, even though the co-variation in individual stock returns is

almost totally extracted by the return systematic factors.

I conduct a similar analysis using the CAPM for 105 stocks listed in the S&P 500

Index. The CAPM represents the class of one-factor models for the return struc-

ture, in which the single factor that governs the common structure of stock returns

is represented by the market return. The CAPM also implies a factor structure in

stock return variances, according to which the single factor structure driving in-

dividual stock variances is represented by the market return variance. By treating

the S&P 500 Index return as the benchmark market return, the results based on

the CAPM show that the market return explains over 94% of the average pair-wise

correlations between the individual stock returns. However, the market return

variance only explains 26% of the average pair-wise correlations between the in-

dividual stock variances. The strong common structure embedded in the stock

idiosyncratic variances suggests the use of variance-factor models which impose

factors directly on the stock variances. I show that the one-factor variance-factor

model enables the index variances to extract 62% of the average pairwise correla-

tion in the stock variances, thus being a better choice than return-factor models in

analyzing the common structure of stock variances. Moreover, the remaining 38%

correlation should not be overlooked. Therefore, the one-factor model is extended

to a two-factor model. The choice of the number of factors is further supported

by the results of a principal component analysis conducted on the individual stock

variances.

Secondly, intuition in support of the two-factor model can be gained by consid-

ering variance risk. Using a one-factor model, Christoffersen et al. (2013) show that

the variance risk premia of individual stocks should be linearly related to that of

the index. However, this linear relation does not hold empirically as a one-factor

variance model always overestimates the variance risk premia of individual stocks

compared with those implied from option prices. I show that this overestimation

is the result of excess risk premium spillover from the market variance to individ-


ual stock variances. Since in extant literature the index variance risk premium1 is

reconciled to be positive, e.g., in Bakshi and Kapadia (2003), Bakshi and Madan

(2006), and Carr and Wu (2009), a second factor which contains a negative factor

premium would be needed in order to offset the aforementioned spillovers.

Motivated by these intuitions, a two-factor model is constructed by imposing

two factors on a multivariate series of individual stock variances. The first fac-

tor is the market variance (MV) factor which captures the variance spillover from

the market portfolio to individual stocks. The MV factor is thus a standardized

market portfolio variance and its factor premium represents the market variance

risk premium. The second factor, should it exist, captures the remaining variance

dynamics in individual stock variances that the market portfolio fails to explain.

The factor is therefore called the variance residual (VR) factor and the pricing of

individual stock variances requires that it carries a negative premium.

The second part of this chapter first aims at filtering the two factors to justify

their existence and then at quantifying the factor premia to reveal their pricing im-

plications. In order to provide evidence that both factors are necessary in explain-

ing the common structure in individual stock variances, I adapt the variance-factor

GAS model in Chapter 3 by tailoring it to a two-factor case. The model is later

referred to as the physical measure model as it uses stock log-returns observable

only under the physical measure. The empirical performance of the model is ro-

bustly assessed using 8 different groups representing different sectors, each group

containing 10 stocks. I show that the model is able to identify the MV factor, in

the sense that the filtered MV factors from different groups are similar and closely

mimic the time-pattern of the index variance. The filtered VR factors from different

groups also share a high level of similarity, their existence being justified as most

stocks have positive, statistically significant loadings on them. The model perfor-

mance is also investigated using a larger group with 25 stocks listed in the Dow

Jones Industrial Average Index, for which similar results are found.

1The variance risk premium is documented as the difference between the risk-neutral implied

variances and the realized variances (Q−P).

4.1. Introduction 83

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-5

0

5

10

15

20

25Average Factor Premium Comparison

Average MV Factor PremiumAverage VR Factor Premium

Figure 4.2: Factor premia filtered from the implied variances of individual stocks.

This figure shows the time series plots of the two factor premia estimated by the Kalman filter

approach. The black bar represents the premium of the MV factor, which resembles the scaled

index variance risk premium. The grey bar represents the premium of the VR factor, which serves

to offset the excess risk premium spillover from the market variance to individual stock variances.

When the common structure of stock variances is assumed to be captured by the

MV and the VR factors, the systematic variation of stock variances is governed by

the variation of the two factors. Accordingly, the variance risk premia of individual

stocks, which are the results of the variation of variances, are jointly determined

by the two factor premia. Note that the variance risk premia are usually calculated

as the difference between the variances under the risk-neutral and the physical

measure respectively, therefore, quantifying the factor premia thus also requires

taking the difference between the factor estimates obtained under each measure.

To calculate factor premia, the factors filtered under the risk-neutral measure are

needed. Since the physical measure model requires return series data, it cannot

be estimated under the risk-neutral measure. Therefore, I apply a Kalman filter

approach directly on the risk-neutral measure variance series of individual stocks.

The risk-neutral measure filter is able to consistently estimate the conditional and

unconditional factor premia for the MV and the VR factors. I calculate the model-

free implied variance following the methodology of Britten-Jones and Neuberger

(2000), Jiang and Tian (2005) and Bakshi et al. (2003) for all the stocks in differ-

ent asset groups. The Kalman filter is implemented on the multivariate series of

the implied variances with the same factor loadings and persistence parameters


suggested by the physical measure model. The main parameters of interest to be

estimated are the ones governing the two factor premia. These parameters are iden-

tified in such a way that they jointly determine the level changes in variances from

the physical to the risk-neutral measure. The estimation results across 9 different

asset groups suggest: 1) the MV factor carries a positive premium, which is in

agreement with the empirical evidence of a positive market variance risk premium

and 2) in line with the pricing implication conjecture, the VR factor carries a signif-

icant negative premium, serving to compensate the excess risk premium spillover

from the market variance to individual stock variances. Figure 4.2 shows the aver-

age time patterns of the factor premia across 9 groups, the black bar representing

the MV factor premium which resembles the market variance risk premium, and

the grey bar representing the VR factor premium which offsets the market variance

risk premium by displaying an opposite pattern.

The empirical analyses also show that the VR factor has a longer memory than

the MV factor. Specifically, the VR factor has a half-life of 192 days, which is longer

than the 62 days half-life of the MV factor. In the short-term, both factors and

their premia are important determinants of variance and variance risk premium

predictions. In the long-term, however, the dominant determinant is the VR factor.

The model predicted variance risk premium is further used to design an option

portfolio strategy. The idea of the strategy is to take advantage of the difference in

the factor memories and the factor premia, especially as the long-run variance risk

premium is dominated by the VR factor which carries a negative premium. The

portfolio follows these model insights and takes long positions on long-term indi-

vidual stock straddles which have the lowest predicted variance risk premia. Also

note that in order to collect the positive MV factor premium, taking short positions

on short-term options is optimal given its short memory. The strategy then requires

shorting the short-term index straddles with the highest predicted variance risk

premia to collect the market variance risk premia. As a result, the portfolio strat-

egy leads to negative exposure to the VR factor premium and positive exposure to

the MV factor premium, thus collecting both factor premia in a profitable way. I

4.2. Variance-Factor Model and Variance Risk Premium 85

show that this model-based strategy outperforms competing model-free strategies

in most asset groups, especially in the large portfolio with 25 Dow Jones stocks.

Particularly, when investing equally in different asset groups, the strategy gener-

ates an in- and out-of-sample monthly return of 4.30% and 7.07%, both of which

are higher than 3.33% and 6.46% achieved by the best model-free strategies. The

CAPM regression results further demonstrate the superiority of the strategy which

generates the highest alphas in most asset groups. In a follow-up analysis, the

VR factor premium collection strategy is shown to qualify as a dispersion trading

procedure. The strategy only requires investing in a small number of individual

stock straddles, which is different from the correlation trading strategy designed

by Driessen et al. (2009) whose strategy invests in the individual straddles for all

the index components.

This chapter develops as follows: Section 4.2 provides the variance risk pre-

mium analysis and the motivations for using two-factor variance-factor models;

Section 4.3 studies the methodologies for filtering factors and estimating param-

eters; Section 4.4 shows the empirical results for the estimated factors and pa-

rameters; Section 4.5 implements the portfolio strategy; Section 4.6 concludes the

chapter.

4.2 Variance-Factor Model and Variance Risk Premium

This section introduces the basic setting of variance-factor models and the intuition

behind its two-factor version.

4.2.1 The variance-factor model

The class of variance-factor models assumes that a limited number of factors ex-

plain the common dynamic structure of multivariate conditional variance series.

Let Ft be a suitably defined filtration and denote by Λt−s,t, an Ft−s-measurable

vector, the vector of conditional variances over the time period [t − s, t]. The


variance-factor model with M common factors (M < N) is written as

Λt−s,t = sα + β f t,

where α is the vector of unit-period unconditional variances under the physical

measure; β is the N × M factor loading matrix; f t is the M× 1 vector of common

factors that are mutually statistically independent and have mean zero. The factors

capture systematic variation of individual stock variances.

4.2.2 The two-factor case

As the number of factors increases, the number of factor loading parameters in-

creases at a rate of N. A parsimonious approach is the one-factor model, in which

a single factor is assumed capable of capturing the common dynamic structure of

the conditional variance series of N individual stocks. The model thus identifies

the factor as the only factor that prices variance risk. In the spirit of the CAPM,

I name this factor the Market Variance factor, or the MV factor, since it can be

best reconciled as a measure of the market variance spillovers to individual stock

variances.

The one-factor model imposes a strong assumption on individual stock vari-

ances which is usually rejected by empirical tests. Specifically, it assumes the time

series patterns of different individual stock variances are proportional. This draw-

back is evident in Figure 4.3 where the time series pattern of an individual stock

variance estimated from the one-factor model is proportional to that of the market

variance. However, it deviates significantly from the GJR-GARCH model (Glosten

et al. (1993)) estimates (the grey area), as the GJR-GARCH model is a typically used

in univariate variance modeling.

To relax the strong variance restriction imposed by the one-factor model and

yet preserve parsimony, a class of two-factor models is considered in this chapter.

A two-factor model is an extension of the one-factor model in which the first factor

can still be identified as the MV factor, while the additional second factor aims

to capture remaining variance dynamics. The two-factor model defined on the


01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

20

40

60

80

100

120

GJR-GARCH stock varianceOne-Factor model fitted stock varianceThe market variance

Figure 4.3: Biased estimates from the one-factor model.

This figure shows the estimated conditional variance time pattern of the stock with ticker ‘DLTR’.

The GJR-GARCH(1,1,1) model is first applied on the daily log-returns of the S&P 500 Index (s=1),

which is regarded as the market variance σ2mt. The conditional variance σ2

it of the stock is then

estimated by the method of quasi-maximum likelihood assuming σ2it = αi + βmiσ

2mt, where αi and βmi

are to be estimated. The black line represents the filtered series of an individual stock conditional

variance from the one-factor model; the red dotted line stands for the GJR-GARCH model estimates

for the index conditional variance; the grey area is the GJR-GARCH model estimate for the stock

conditional variance.

conditional variance of stock i over the time period [t− s, t] can be written as

Λi,t−s,t = sαi + βmi fmt + βxi fxt, (4.1)

where βmi is the factor loading on the MV factor fmt; βxi is the factor loading on

the second factor fxt. Both loadings are stock-specific. As mentioned previously, αi

measures the unconditional level of unit-time variance under the physical measure.

The unconditional level of Λi,t−s,t under the physical measure is then measured

by sαi. However, a shift in the variance level is expected when moving from the

physical to the risk-neutral measure due to the variance risk premium. Denote

by μm and μx any possible nonzero mean for fmt and fxt under the risk-neutral

measure. Note that (4.1) holds for i = 1, ..., N, therefore the common structure in

individual stock variances is governed entirely by the MV factor fmt and the added


second factor fxt with stock-specific loadings βmi and βxi. Also note that fmt is

assumed to measure the dynamics of the market variance, which suggests that the

market conditional variance Λm,t−s,t can be modeled by a one-factor structure, i.e.

Λm,t−s,t = sαm + βmm fmt, (4.2)

This equation becomes equivalent to (4.1) after setting i = m and βxm = 0. Here the

abbreviation m is used to denote the market portfolio. Throughout this chapter,

the S&P 500 Index is treated as the market portfolio, and the S&P 500 Index option

variance risk serves as the market variance risk.

4.2.3 Reasons for two-factor model

In addition to the market variance factor, the second factor is imposed to model

the remaining dynamics in individual stock variances which the market variance

factor typically fails to capture. I show that the existence of the second factor is

not only justified by its explanatory power on variance dynamics, but also by its

variance risk pricing implications.

4.2.3.1 Reason 1: Idiosyncratic volatility comovement

Linear factor-based asset pricing models such as the CAPM of Merton (1973) and

the APT of Ross (1976) suggest that individual stock returns can be explained by

the market return or by a linear combination of common factors, leaving the resid-

uals to be idiosyncratic, or firm specific. The factors are assumed to capture all

systematic variation in individual stock returns, therefore, the residuals from such

regressions are regarded as idiosyncratic returns. The term ‘idiosyncratic’ simply

suggests that all kinds of comovement in the statistical characteristics of the resid-

uals can be made insignificant after a certain level of diversification. However, the

idiosyncratic volatility puzzle described in Ang et al. (2009) shows the pricing im-

plications of idiosyncratic volatilities, which is in contradiction to what financial

theories would suggest. A further study by Herskovic et al. (2014) shows that firm-

specific idiosyncratic variances preserve high comovement levels which cannot be


explained by missing return-factors. This comovement feature suggests the exis-

tence of another systematic variance factor(s) that linear asset-pricing models fail

to capture.

I conduct a similar analysis as in Herskovic et al. (2014) on the conditional vari-

ances of the daily log-returns for 105 stocks listed in the S&P 500 Index. The S&P

500 Index return is treated as the market return. The results resemble many of

the results presented in Herskovic et al. (2014) under the CAPM setting, in which

the market daily log-return rmt is regarded as the single common factor that drives

individual stock log-returns. The equation is addressed as

rit = βirmt + eit, eit ∼ N(0,σ2id,t), (4.3)

where rit is the daily log-return; βi is the static market beta; eit measures the id-

iosyncratic return with time-varying idiosyncratic variance σ2id,t. All four measures

are stock-specific. The average pair-wise correlation of rit, i = 1, ..., 105 is 0.37,

meaning that in general the individual stock returns are positively correlated. Af-

ter considering their exposure to the market return rmt, the residuals eit, i.e., the

idiosyncratic returns of each stock, have an average pair-wise correlation of 0.02,

indicating that the market return factor explains 94% of the correlation between the

individual stock returns.2 A conclusion which can be drawn from the results is that

the market return factor is essential, or even adequate in explaining the common

variation in stock returns.

Taking the conditional variance on both sides, the CAPM (4.3) also implies a

variance-factor model with a single factor, s.t.

Λi,t−1,t = β2i Λm,t−1,t + σ2

id,t, (4.4)

The single factor is no longer the market return, but the market conditional vari-

ance, which can be regarded as the MV factor discussed in Section 4.2.2. I apply the

GJR-GARCH(1,1,1) model on daily returns to estimate the conditional variances of

2The significant reduction in correlations are robustly checked in 9 small groups, where similar

results are obtained.


the individual stocks and the S&P 500 Index. The average pair-wise correlation of

the conditional variances {Λi,t−s,t}105i=1 calculated by

1105× (105− 1) ∑

i=2, ..., 105, j<icorr(Λi,t−s,t,Λj,t−s,t)

is 0.61, which confirms the existence of a strong common structure in firm-level

volatilities. Subtracting the exposure to the MV factor from the individual stock

variances, the idiosyncratic variances{

σ2id,t

}105

i=1, not surprisingly, are still highly

correlated with an average pair-wise correlation of 0.45, meaning that the market

variance only explains 27% of the correlation between the individual stock vari-

ances, and that the idiosyncratic variances of the individual stocks still carry a

strong common structure.

Note that β2i is not the optimal estimator to fit the market variance to individual

stock variances, since it serves to minimize the variance of the pricing error eit. As

a comparison, the one-factor model discussed in Section 4.2.2 extracts the market

variance from the individual stock variances by running the regression

Λi,t−1,t = βviΛm,t−1,t + εi,t. (4.5)

The parameter βvi is the so-called variance-beta as in Carr and Wu (2009), and

as suggested by (4.5), is a better choice to use than β2i in order to fit the market

variance to individual stock variances.

An analysis to test this conjecture is conducted under both the physical and

the risk-neutral measure. Under the physical measure, the static beta is calcu-

lated by cov(rit,rmt)/var(rmt). The variance-beta is calculated by performing OLS

regressions on the conditional variances of all the stocks and the index returns.

These conditional variances are estimated by the GJR-GARCH(1,1,1) model. Under

the risk-neutral measure, the implied static beta is calculated by first deriving the

implied conditional beta series following Buss and Vilkov (2012) and then taking

its time-series average. The variance-beta is calculated in the same way as under

the physical measure using the model-free implied variance (MFIV) of Bakshi and

Kapadia (2003).


Squared Return Beta (P)0 0.5 1 1.5 2 2.5

Var

ianc

e B

eta

(P)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Beta Compare (P)

Squared Return Beta (Q)0 0.5 1 1.5 2 2.5

Var

ianc

e B

eta

(Q)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Beta Compare (Q)

Figure 4.4: Squared return-beta and variance-beta.

This figure shows the scatter plots of the variance-betas βvi versus the squared return-betas β2i

under both the physical and the risk-neutral measure. The y = x line is drawn in dark grey for

comparison. The return-beta in the physical measure is calculated by cov(rit,rmt)/var(rmt), where

cov(rit,rmt) is the sample covariance between the daily log-returns of stock i, i = 1, ..., 105, and the

index, and var(rmt) is the sample variance of the daily log-returns of the index. The return-beta

under the risk-neutral measure is first calculated by the implied beta of Buss and Vilkov (2012) and

then taken the time-series average. In both graphs, all stocks have higher variance-betas than the

squared return-betas.

Figure 4.4 shows that the estimated variance-betas βiv of all the stocks are higher

than the corresponding squared return-betas β2i under both the physical and the

risk-neutral measure. For simplicity and following customary notation, the for-

mulas in this chapter use P to denote the physical measure, and Q to denote the

risk-neutral measure. To confirm the conjecture that the variance-betas extract more

market variance than the squared return-betas, the average correlation of the resid-

uals {εi,t}105i=1 in (4.5) across different stocks is 0.24 (62% explained), a decrease when

compared to 0.45 which is the correlation of{

σ2id,t

}105

i=1in (4.4). Even though the

decrease in correlations suggests that the one-factor model already substantially

mitigates the comovement feature in the variance residual, the remaining 38% cor-

relation cannot be ignored and calls for another factor(s) independent from the MV

factor to explain the systematic variation. For the purpose of parsimony, I extend


the one-factor case to a two-factor case.3

4.2.3.2 Reason 2: Variance risk premia of individual stocks

When the systematic dynamics of individual stock variances are explained by two

factors, the variance risk premia of individual stocks, which are due to the sys-

tematic variation of variances, are jointly determined by the dynamic nature of the

two factors. Specifically, the variance risk premium (VRP) of stock i or of the index

(i = m) over a time interval [t− s, t] is defined by

VRPit,s = ΛQi,t−s,t −ΛP

i,t−s,t, (4.6)

which measures the difference between the forward-looking variances under the

risk-neutral measure and the physical measure. VRP reflects the premium investors

demand for bearing the risk of randomness in future variance. In general, the vari-

ances implied from option prices are higher than the realized variances calculated

using observed returns. The empirical evidence of VRP has been addressed exten-

sively in the literature, such as in Bakshi and Kapadia (2003), Bakshi and Madan

(2006), and Carr and Wu (2009).

By substituting (4.1) into (4.6), the VRP process inherits the linear relation be-

tween the conditional variances, i.e.

VRPit,s = βmi( f Qmt − f P

mt) + βxi( f Qxt − f P

xt). (4.7)

Therefore, the VRP of individual stock i has two components. The first component

is the difference between the MV factor fmt under the risk-neutral measure and the

physical measure, which can be interpreted as the MV factor premium conditioned

at time t− s. The second component is the difference between fxt in the risk-neutral

measure and the physical measure, i.e., the factor premium of fxt conditioned at

3A principal component analysis on all individual stock conditional variances shows that the

first factor, which resembles the market variance time-pattern, explains 75% of the total variation,

while the second factor explains 10% variation. The individual explanatory powers of all the other

factors are no higher than 2%.


time t − s. The market variance risk premium VRPmt,s can then be obtained by

restricting βxi to 0, so

VRPmt,s = βmm( f Qmt − f P

mt),

which suggests that the market VRP is proportional to the MV factor premium

with scaling parameter βmm. Taking both the unconditional expectation and the

cross-sectional average of (4.7), the average individual stock variance risk premium

VRPs over a time period of length s is a linear combination of the unconditional

factor premia μm and μx:

VRPs = βmμm + βxμx, (4.8)

with μm = EQ[ f Qmt], μx = EQ[ f Q

xt ], βm = 1N ∑N

i=1 βmi and βx = 1N ∑N

i=1 βxi. The

same method implies the time-series average of the index variance risk premium

VRPm,s = βmmμm. Therefore, (4.8) can be written as

VRPs =βm

βmmVRPm,s + βxμx. (4.9)

From (4.9), the time-series and cross-sectional average of the variance risk premia

of individual stocks has two components: 1) the premium spillover VRPm,s of the

market variance scaled by βmβmm

, and 2) the premium of the second factor μx scaled

by βx. Empirically, VRPm,s is found to be statistically significant and positive, such

as in Bakshi and Kapadia (2003) and Carr and Wu (2009). However, VRPs are

empirically much smaller than VRPm,s. It suggests that if the average loadings

βm and βx are positive and if βm/βmm is on average larger than 1, then the factor

premium parameter μx is negative in the sense that it compensates the excess risk

premium spillover from the market variance to individual stock variances.

I adopt the VRP calculation of Carr and Wu (2009), who show that the variance

risk premium of stock i can be calculated empirically as the time-series average of

the difference between the variance swap rate and the realized variance over a time

interval [t− s, t]:

SWi,t−s,t − RVi,t−s,t,

where RVi,t−s,t denotes the annualized realized variance between t− s and t; SWi,t−s,t

denotes the variance swap rate pre-fixed at time t− s which can be consistently es-


timated by the MFIV. The annualized realized variance is calculated as

RVi,t−s,t =252s∗

s

∑j=1

r2i,t−s+j,

where s∗ denotes the number of trading days between t − s and t; if t − s + j

represents a trading day, ri,t−s+j is the daily log-return, or else, ri,t−s+j = 0 . The

variance risk premia of individual stocks are calculated by taking the difference

between the MFIV and the realized variance with s = 30. For days during which

either there are no call options or no put options, or only one option is available

for each type so that there are insufficient entries to deliver an accurate estimate,

the interpolated implied variances are used.

The estimated annual variance risk premium for the S&P 500 Index is on av-

erage 6.53%, with a t-statistic (Newey and West (1987) with 22 lags) of 13.0465.

The equally weighted averages of individual variance risk premia in the stock pool

are on average 3.73% with an averaged t-statistic of 5.0353. Note that the average

variance risk premium of individual stocks is about half the size of the index vari-

ance risk premium, and yet taking the average of the variance-beta of all the stocks

gives βm/βmm = 1.3190, meaning that the average size of the index variance risk

premium spillover to individual stocks is 1.3 times the size of the index variance

risk premium. From (4.8), the one-factor model implied average variance risk pre-

mium of individual stocks is βmβmm

VRPm = 1.3190× 6.53% = 8.61%, which is much

larger than the empirical level 3.73%. To fix this mismatch, a second variance factor

should be included to offset the spillover.

4.2.4 The second factor: the VR factor

Since the role of the second variance factor is to capture the remaining variance

residual after the MV factor is extracted, I name the second factor the variance

residual (VR) factor. The corresponding factor premium is called the VR factor

premium, which serves to offset the excess risk premium spillover from the market

variance to individual stock variances. With positive loadings βx, it is reasonable

to conjecture that this premium is negative (μx < 0). Empirically collecting this


Market VRP

Stock 1 VRP

Stock i VRP

Stock N VRP

Market VRP+ VR premium

βm1

βmi

βmN

wm1

wmi

wmN

+ Corr premium

...

...

Figure 4.5: Relation between the variance risk premia of the market portfolio and individual

stocks.

This figure shows the relation between the market variance risk premium and the variance risk

premia of its individual components. The variance risk premia of individual stocks are composed

of the market variance risk premium and the VR factor premium, so that the VR factor premium

exists in individual stocks. The market variance risk premium is composed of the individual stock

variance risk premia and the correlation risk premium, thus, the correlation risk premium exists in

the market portfolio, not in individual stocks.

negative premium is feasible following (4.9):

−βxμx =βm

βmmVRPm,s −VRPs.

This corresponds to the expected return of a strategy which shorts delta-hedged

market portfolio straddles and buys delta-hedged individual stock straddles. This

trading strategy resembles a dispersion trade, which is often used to collect the

correlation risk premium of Driessen et al. (2009). However, the VR factor premium

is different from the correlation risk premium in terms of definition and feasibility.

Figure 4.5 describes the relation between the variance risk premia of the market

portfolio and individual stocks. The figure shows two differences between the

two premia: 1) the VR factor premium exists in the variances of individual stocks,

while the correlation risk premium exists in their combination which is the market

portfolio; 2) the VR factor premium can be collected by trading options on a fairly

small number of individual stocks together with options on the market portfolio,


while in order to collect the correlation risk premium, in theory,4 one would have to

invest in options on each of the market portfolio component. This second difference

highlights the advantage of using the variance-factor model for the management of

low dimension portfolios.

4.3 Methodology

The previous section shows that the VR factor, should it exist, would carry a neg-

ative factor premium given that individual stocks have positive factor loadings on

it. The MV factor, instead, should have a positive premium to match empirical

findings. In this section, I propose estimation methods for the MV and the VR fac-

tors under both the physical and the risk-neutral measure. In the physical measure

estimation, the parameters to be estimated are the factor loadings βmi and βxi, and

the unconditional variances αi. These parameter estimates are further used in the

filtering conducted under the risk-neutral measure to obtain the factor premium

parameters μm and μx which by assumption only exist under the risk-neutral mea-

sure.

4.3.1 Filtering under the physical measure

The observed daily return series is used to construct a multivariate model. Un-

like the multivariate GARCH models where the outer product of the return vector

serves as the driving force of the conditional covariance matrix, the two-factor vari-

ance model requires modeling the MV and the VR factors directly, after imposing

the condition that the two factors are mutually independent.

Denote by rit the daily excess return of stock i, i = 1, ..., N, and rmt the daily

excess return of the index. Also denote by rt an (N + 1)-vector containing all the

individual stock daily returns in the portfolio and with the index return rmt stacked

as the last entry. The vector rt is assumed to follow a conditional multivariate Stu-

4Empirically, at least 30%-40% of the stocks in S&P 500 Index are included in a dispersion trading

strategy to collect the correlation risk premium

4.3. Methodology 97

dent’s t distribution, with conditional covariance matrix Vt and degrees of freedom

ν based on the information set Ft−1. The basic model reads

rt|Ft−1 ∼ t(0,Vt,ν) and Vt = DtRtDt.

Following the Dynamic Conditional Correlation model of Engle (2002), the covari-

ance matrix Vt is decomposed into Dt and Rt, where Dt = diag(ΛPt−1,t)

1/2 is a di-

agonal matrix of the conditional volatilities and Rt is the correlation matrix. ΛPt−1,t

again stands for the conditional variance vector under the physical measure, which

for ease of notation, is denoted by ΛPt . From (4.1), the process of ΛP

t can be written

as

ΛPt =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

α1

α2...

αN

αm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

︸��︷︷��︸unconditional level α

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

βm1 βx1

βm2 βx2...

...

βmN βxN

βm 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

︸��︷︷��︸factor loading matrix β

⎛⎝ f P

mt

f Pxt

⎞⎠ = α + β f P

t .

This setting identifies f Pmt as the MV factor and f P

xt as the VR factor under the phys-

ical measure. One reasonable expectation is that f Pmt is proportional to the market

variance while f Pxt is somewhat different from f P

mt if the two factors are assumed in-

dependent. To ensure a clear distinction between f Pmt and f P

xt, the recursive equation

of the two factors is assumed to follow:

f Pt+1 =

⎛⎝ f P

m,t+1

f Px,t+1

⎞⎠ = (I − B2)1/2

⎛⎝ smt

sxt

⎞⎠+ B

⎛⎝ f P

mt

f Pxt

⎞⎠

= (I − B2)1/2st + B f Pt , (4.10)

where B is a persistence matrix with diagonal elements bm and bx and zero other-

wise, and smt and sxt are the driving forces for the MV and the VR factors.

If both driving forces smt and sxt are martingale difference series mutually inde-

pendent w.r.t. Ft−1, then taking the unconditional variance on both sides of (4.10)

leads to

var( f Pt+1) = (I − B2)var(st) + Bvar( f P

t )B′.


Setting var( f Pt+1) equal to var( f P

t ) results in⎛⎝ (1− b2

m)var( f Pmt) bmbxcov( f P

mt, f Pxt)

bmbxcov( f Pmt, f P

xt) (1− b2x)var( f P

xt)

⎞⎠ =

⎛⎝ (1− b2

m)var(smt) 0

0 (1− b2x)var(sxt)

⎞⎠ .

This leads to cov( f Pmt, f P

xt) = 0, and var( f Pmt) = var(smt), var( f P

xt) = var(sxt). There-

fore, the factor independence property is proved. Moreover, the unconditional co-

variance matrix of the factors equals the unconditional variance of st, i.e., var( f Pt ) =

var(st).

The question then arises: how to find driving forces for the factors? In the

univariate GARCH models, the driving forces are the squared returns, while in the

multivariate GARCH models the driving forces are the outer products of the return

vectors. To find a counterpart for the observation-driven factors, one could resort

to using the scaled score function.

Proposition 4.1. The variance of the independent factors f Pmt and f P

xt can be restricted to 1

if the observation-driven vector st is modeled as

st =

⎛⎝ sP

mt

sPxt

⎞⎠ = St · ∇t, ∇t =

∂ log�t(rt|Vt;ν)∂ f P

t

St = I−1/2t , It = Et−1

[∇t∇′t]

.

Proof. ∇t stands for the score function. Under a correctly specified model, the

conditional expectation of ∇t should be zero under the local maximum. There-

fore, Et−1[∇t] = 0. The conditional covariance matrix of the score function is then

Et−1[∇t∇′t]. Scaling the score function using the inverse square root of the condi-

tional covariance matrix leads to

st = Et−1[∇t∇′t

]−1/2∇t.

Taking the conditional covariance matrix of st leads to

vart−1(st) = Et−1[∇t∇′t

]−1/2Et−1

[∇t∇′t]

Et−1[∇t∇′t

]−1/2= I,

which again confirms that the unconditional covariance matrix of the scaled score

function is an identity matrix. Since the covariance matrix of the factors and of the

4.3. Methodology 99

scaled score function are the same, the unconditional covariance matrix of the fac-

tors is an identity matrix. Therefore, the variances of the two factors are restricted

to 1.

As mentioned in Creal et al. (2013), the scaled score function thus measures

the steepest ascent direction in which the log-likelihood can be improved, i.e., the

scaled score function provides a local best measure to fit the model to the data.

When the number of asset increases, the number of correlation parameters in-

crease by a rate of N/2. Therefore, a single factor is assumed to drive all the

cross-sectional pairwise correlations. This assumption is resemblant of the equicor-

relation model by Engle and Kelly (2012). One way to adopt this assumption is

to assume a portfolio-wide correlation factor f Pρt

such that this factor measures the

average correlation level of all the pairwise correlations in the portfolio. f Pρt

can also

be modeled as an observation-driven process, where the driving force sρt can be

derived following the procedure in Proposition 4.1. Consequently, the scaled score

function st can be generalized to s∗t such that it includes time-varying correlation

features, i.e., s∗t =(smt, sxt , sρt

)′ and f P∗t =

(f Pmt, f P

xt , f Pρt

)′.

Proposition 4.2. Under the assumption that the return series follow a multivariate Stu-

dent’s t distribution, the scaled score function of the two variance factors and the portfolio-

wise correlation factor can be analytically derived as

s∗t = I∗−1/2t · ∇∗t ,

where

Ψt = BN(I ⊗ DtRt + DtRt ⊗ I)WDtDt[β 0N×1] + BN(Dt ⊗ Dt)DN [0 N(N+1)2 ×2

vech(U)],

∇∗t =12

Ψ′tD′N(Vt ⊗Vt)−1(wtrt ⊗ rt − vec(Vt)),

I∗t =14

Ψ′tDN(J′t ⊗ J′t)[gG− vec(I)vec(I)′](Jt ⊗ Jt)DNΨt.

The matrix Ψt is derived from

Ψt = Ψ( f ∗t ) =∂vech(Vt)

∂( f P∗t )′

for Vt = V( f P∗t ), where f P∗

t = ( f Pmt, f P

xt, f Pρt)′,


where U is a N-dimensional square unit matrix, with diagonal elements 0; ⊗ is the Kro-

necker product sign; the matrices DN, BN and CN are defined as the duplication matrix,

the elimination matrix and the commutation matrix; the matrix Jt can be obtained by any

proper matrix decomposition procedure such that V−1t = J′t Jt; the matrix WDt is constructed

from the (N + 1)2 × (N + 1)2 diagonal matrix with diagonal elements vec(D−1t )/2 after

dropping the columns containing only 0s; the matrix G is defined as G = E[(zz′ ⊗ zz′)]

with z ∼ N(0, IN+1); wt =ν+N

ν−2+rt ′V−1t rt

; g = ν+Nν+2+N . The model reduces to a multivariate

Gaussian model as ν→∞.

Proof. The proof of the closed-form solution can be found in Creal et al. (2011).

The scaled score function, though complicated in terms of formulation, can be

calculated easily given its analytical form. Note that the scaled score function is

an observation-driven function, that is, it is measurable given the observed return

vector rt and the conditional covariance matrix Vt. The question arises as to how the

parameters can be identified. The main parameters that may suffer possible finite-

sample identification issues are the persistence matrix B and the factor loading

matrix β. The following proposition shows that by calculating the driving forces as

illustrated in Proposition 4.2, the parameters can be well-identified.

Proposition 4.3. Under the definition of st in Proposition 4.1 and following the calculation

procedure in Proposition 4.2, the persistence matrix B and the unconditional level α can be

uniquely identified, and the exposure β of the conditional variances Λt to the two systematic

factors can be identified up to column sign change.

Proof. The detailed proof can be found in Section 3.2.2.

The parameters and the conditional factor series can be estimated by quasi-

maximum likelihood methods. Under the assumption that the return series follow

a conditional multivariate Student’s t distribution, the log-likelihood contribution

for observation rt is

�t(rt|Vt;θ) = log(

Γ(

ν + N2

))− log

(Γ(ν

2

))− N

2[(ν− 2)π]

4.3. Methodology 101

−12

log (|Vt|)− ν + N2

log

(1 +

rt′V−1

t rt

ν− 2

),

where the parameter set θ = (α′, vec(β), diag(B), θρ, ν), with θρ denoting all

correlation related parameters.

4.3.2 Filtering under the risk-neutral measure

Given that daily return series are only observed under the physical measure, the

model as developed so far cannot be applied under the risk-neutral measure. The

filtering under the risk-neutral measure is then directly performed on forward-

looking integrated variance series implied under the risk-neutral measure. The

estimated factor loadings and the unconditional variance parameters are preserved

from the physical measure estimation. The focus is to estimate the two factor

premium parameters μm and μx, which by assumption only exist under the risk-

neutral measure.

From (4.1) and following the definition of ΛPt , the process for the 1-day forward-

looking integrated variance vector implied under the risk-neutral measure can be

written as

ΛQt = α + βm f Q

mt + βx f Qxt , (4.11)

where βm = [βm1, βm2, ..., βmN, βmm]′ and βx = [βx1, βx2, ..., βxN, 0]′. The fac-

tor loadings are equal to their physical measure values, an approach following

Christoffersen et al. (2013), Duan and Wei (2009) and Serban et al. (2008). The con-

ditional factors under the risk-neutral measure are assumed to follow the same

processes assumed under the physical measure. From (4.10), the factors are driven

by their own lagged estimates and the scaled score function, which by definition

should have mean zero and identity covariance matrix. Following this formulation,

a state equation can be written for the processes of f Qmt and f Q

xt , where the innova-

tion terms are replaced by the disturbance terms ηt. The values estimated under

the physical measure are retained for the persistence matrix B, so that the factor


memories are assumed to be the same under different probability measures.⎛⎝ f Q

m,t+1

f Qx,t+1

⎞⎠ = (I − B)

⎛⎝ μm

μx

⎞⎠+ B

⎛⎝ f Q

mt

f Qxt

⎞⎠+ ηt+1, ηt+1 ∼ N(0,Σ). (4.12)

Note that there are no intercept terms in (4.10), corresponding to the assumption

that the factors have zero mean in the physical measure. When the two factors

are assumed to be the only two systematic risk sources for the individual stock

variances, the premia these factors carry jointly determine the variance risk premia

of the individual stocks. The setting in (4.12) ensures that the unconditional MV

factor and the VR factor premia are μm and μx.

To construct an observation equation, note that the model-free implied variance

(MFIV) of Britten-Jones and Neuberger (2000), Jiang and Tian (2005) and Bakshi et

al. (2003) can serve as a consistent estimator of ΛQt−s,t. The MFIV for stock i can be

calculated by

MFIVi,t−s,t =∫ ∞

Si,t−s

2(

1− log[

KiSi,t−s

])K2

iC(i, t− s, s;Ki)dKi

+∫ Si,t−s

0

2(

1 + log[Si,t−s

Ki

])K2

iP(i, t− s, s;Ki)dKi,

where Si,t−s is the spot price of stock i, Ki is the strike price of its corresponding

options, and C(i, t− s, s;Ki) and P(i, t− s, s;Ki) are the call and put option prices. A

discrete version of this formula given finite numbers Nic and Nip of available calls

and puts, respectively, is

MFIVi,t−s,t =Nic

∑j=1

2(

1− log[

Kic,jSi,t−s

])K2

ic,jC(i, t− s, s;Kic,j)ΔKic

+

Nip

∑j=1

2(

1 + log[Si,t−sKip,j

])K2

ip,jP(i, t− s, s;Kip,j)ΔKip.

The MFIV can be used to construct the observation equation of the state-space

model:

MFIVt−s,t = ΛQt−s,t + εt−s, εt−s ∼ N(0, H), (4.13)

4.3. Methodology 103

where MFIVt−s,t = [MFIV1,t−s,t, ..., MFIVN,t−s,t, MFIVm,t−s,t]′ and εt−s denotes the

vector of estimation errors, or the observation errors.

(4.13) and (4.12) together form a state space model, in which the observations

are the MFIVs calculated from observed option prices, and the dynamics of the

observations are assumed to be captured by two common factors. Note that the

factors are estimated at the daily frequency so that they determine the daily vari-

ances. The MFIVs, however, usually represent forward-looking variances over a

period of s trading days, where s is much larger than 1. Therefore, in order to

apply a filter which produces estimates comparable with the factors filtered under

the physical measure, the MFIVs should be tailored to represent daily-frequency

measures.

Proposition 4.4. Under (4.12), the s-trading-day ahead conditional variances can be writ-

ten using the 1-day ahead conditional variances in the form of

ΛQt−s,t = μ + δ f Q

t−s+1, (4.14)

where

f Qt−s+1 = [ f Q

m,t−s+1, f Qx,t−s+1]

′, δ[i,:] = [βmi

s

∑i=1

bi−1m , βxi

s

∑i=1

bi−1x ],

μ = sα + βmμm(1− bm)s

∑h=1

h−1

∑i=0

bim + βxμx(1− bx)

s

∑h=1

h−1

∑i=0

bix.

Proof. To aggregate the 1-trading-day ahead conditional variance to an s-trading-

day ahead conditional variance, the following aggregation procedure is necessary

to implement a filter:

ΛQt−s,t =

s

∑i=1

EQt−sΛ

Qt−s+i−1,t−s+i,

using (4.11):

ΛQt−s,t = sα + βm

s

∑i=1

EQt−s

[f Qm,t−s+i

]+ βx

s

∑i=1

EQt−s

[f Qx,t−s+i

]. (4.15)

From (4.12), the h-step forward conditional factors are

EQt−s

[f Qm,t−s+h

]= μm(1− bm)

h−1

∑j=0

bjm + bh

m f Qm,t−s+1, (4.16)


and

EQt−s

[f Qx,t−s+h

]= μx(1− bx)

h−1

∑j=0

bjx + bh

x f Qx,t−s+1. (4.17)

Substituting (4.16) and (4.17) into (4.15) leads to Proposition 4.4.

(4.13), (4.14), and (4.12) form a state-space model for the same variance fre-

quencies, where the factor updating mechanisms can be approximated through a

Kalman filter. The detailed filtering procedures are illustrated as follows, assuming

Pt−1 = varQt−1( f Q

t+1):

vt−1 = MFIVt−1,t − μ− δB f Qt ,

Ft−1 = βPt−1β′ + H,

f Qt+1 = (I − B)[μm, μx]

′ + B f Qt + BPt−1β′F−1

t−1vt−1,

Pt = B(Pt−1 − Pt−1β′F−1t−1βPt−1)B′ + Σ.

The parameters to be estimated are the two unconditional factor premium param-

eters μm and μx, the covariance matrix H of the error terms in the observation

(4.13), and the covariance matrix Σ of the error terms in the state (4.12). To avoid

finite-sample identification problems, Σ is restricted to be equal to the physical

measure factor variances. The parameters can then be easily estimated by the

quasi-maximum likelihood method.

4.4 Empirical Results

In this section, empirical data on stock returns and option prices is used to filter

the MV and the VR factors assumed in the model setting. The first part of this

empirical study uses the stock return data to obtain parameter estimates under the

physical measure, then the second part uses the option data to estimate the factor

premia under the risk-neutral measure. A trading strategy based on the model-

predicted variance risk premia is implemented to show the economic value of the

empirical findings.


4.4.1 Data description

The stocks are selected from those listed in the S&P 500 Index composites. The 500

stocks are then grouped into eight sectors according to the GICS sector classifica-

tion rule. The eight sector classifications used represent: consumer discretionary,

consumer staples, energy, health care, industrials, information technology, mate-

rials, and utilities. Within each sector, a 10-stock portfolio is chosen to form a

sector-portfolio, where the ten stocks cover a wide range of variance levels. Specif-

ically, the selection procedure goes as follows: all the stocks which belong to one

sector are first ranked by their unconditional variance levels and included in the

sector-portfolio if their unconditional variance levels lie closest to either of the 5%,

15%, 25%, ..., 95% quantiles. To account for the leverage effect in volatility, a GJR-

GARCH(1,1,1) model is applied to estimate the unconditional variance levels. The

data window ranges from 2nd January 2001 to 31 December 2010, a total of 2515

trading days, during which the 2008 financial crisis started, intensified, and un-

folded. As a robustness check, the same procedure is applied to a large portfolio

consisting of 25 constituents in the Dow Jones Industrial Average Index during the

same time periods. The S&P 500 Index return is taken as a proxy for the market

return. The data source is the CRSP database. The out-of-sample period ranges

from 3rd January 2011 to 31st December 2013, a total of 754 trading days.

The daily option data for each selected stock is downloaded from the Option-

Metrics database during the same data window in which the stock returns are col-

lected. For each stock or for the S&P 500 Index, I select all the put and call options

which meet the following restrictions as in Bakshi et al. (2003) and Driessen et al.

(2009): 1) when calculating the model-free implied variance for individual options,

only the OTM (out-of-money) options with maturities of 8 to 120 days are selected

to minimize the effect of early exercise premia on American-style option prices; 2)

for the trading strategy design, only options with Black-Scholes delta higher than

0.15 for calls and lower than -0.05 for puts are included, and options must allow

for straddle strategies to be formed; 3) options with zero bids, zero open interest,


and missing implied volatility or delta recordings are excluded. The mid-quotes

are used as the market prices. The risk-free rate is taken as the 1-month maturity

commercial paper rate from the Federal Reserve Bank.

4.4.2 Empirical results under the physical measure

The parameter estimates for each sector-portfolio are presented in Table 4.1. The

loadings on the MV factor are statistically significant and positive for all the stocks

in each sector-portfolio. The average individual stocks variance exposure to the

market variance is 1.85, meaning that the average size of the spillover effect is 1.85

times the market variance. The loadings on the VR factor show strong significance

for most stocks. Table 4.2 shows similar estimates for the Dow Jones portfolio,

where all 25 stocks have statistically significant and positive loadings on the MV

factor, and 15 out of the 25 stocks have statistically significant and positive loadings

on the VR factor. The results show strong evidence that the two-factor variance

model is necessary to capture the variance dynamics, especially due to the fact that

the loadings on the VR factor are in most cases statistically significant (92 out of

105 cases).

Figure 4.6 shows scatter plots of estimated factor loadings and the uncondi-

tional variance levels for the sector-portfolios and the Dow Jones portfolio. The

plot on the left shows that the MV factor loadings and the unconditional variance

levels are strongly positively correlated at a level of 0.9375. The correlation between

the VR factor loadings and the unconditional variance levels is at a moderate level

of 0.6289, as depicted in the middle plot. The plot on the right shows that the factor

loadings are also positively correlated at a moderate level of 0.4952. The plots lead

to the conclusion that even though the unconditional variance level of an individ-

ual stock is largely due to its exposure to the market variance, its relation with

the exposure to the VR factor is not as strong. Figure 4.7 presents the estimated

factors for the sector-portfolios and the Dow Jones portfolio. The estimated MV

factors for each group of stocks are very similar, which is a good indication that

the market variance dynamics are successfully identified. Except for the energy


Tabl

e4.

1:In

-sam

ple

stoc

k-sp

ecifi

ces

tim

ates

unde

rth

eph

ysic

alm

easu

rem

odel

(sec

tor-

port

folio

s)

Con

sum

erD

iscr

etio

nary

Con

sum

erSt

aple

sEn

ergy

Hea

lth

Car

e

Ass

etN

umbe

rα

iβ

mi

βxi

αi

βm

iβ

xiα

iβ

mi

βxi

αi

βm

iβ

xi

Ass

et1

2.59

810.

6203

***

0.93

55**

*1.

4569

0.37

83**

*0.

0351

2.90

150.

8246

***

0.25

25**

*1.

5946

0.48

05**

*0.

4573

***

Ass

et2

3.83

531.

1732

***

1.18

25**

*1.

6456

0.57

52**

*0.

4107

***

5.26

571.

4211

***

1.33

94**

*2.

7192

0.81

31**

*0.

5730

***

Ass

et3

4.13

310.

8961

***

0.47

79**

1.75

810.

5209

***

0.29

30**

*5.

7565

1.57

65**

*1.

0731

***

3.07

360.

7000

***

0.20

64**

Ass

et4

4.88

511.

0541

***

1.23

06**

1.77

130.

5926

***

0.42

14**

*6.

3003

1.84

96**

*1.

5661

***

3.57

670.

9815

***

0.93

14**

*

Ass

et5

5.30

661.

3061

***

1.30

71**

1.81

310.

6600

***

0.27

83**

*6.

7884

1.43

90**

*0.

9387

***

3.77

471.

1214

***

0.28

93**

*

Ass

et6

5.94

251.

7046

***

1.42

32**

1.99

370.

5281

***

0.21

96**

*7.

0516

2.14

84**

*1.

0667

***

4.03

271.

0240

***

0.62

61**

*

Ass

et7

6.39

151.

7030

***

1.01

01**

2.23

130.

7025

***

0.48

12**

*7.

7846

1.96

70**

*0.

4481

***

4.36

571.

2234

***

1.01

59**

*

Ass

et8

6.79

421.

9091

***

2.37

95**

*2.

5171

0.69

97**

*0.

3653

***

8.13

502.

4622

***

1.47

66**

*5.

1953

1.56

81**

*-0

.636

3***

Ass

et9

7.64

962.

2859

***

1.28

47**

2.97

610.

9112

***

0.50

72**

*8.

5522

2.37

98**

*1.

0723

***

7.35

322.

0454

***

0.61

34**

*

Ass

et10

9.33

922.

8827

***

0.01

723.

3205

0.86

16**

*0.

4755

***

10.7

384

3.24

75**

*1.

8009

***

7.68

852.

1377

***

2.10

96**

*

S&P

500

1.89

900.

6002

***

1.89

900.

6885

***

1.89

900.

7390

***

1.89

900.

7155

***

Indu

stri

als

Info

rmat

ion

Tech

nolo

gyM

ater

ials

Uti

liti

es

Ass

etN

umbe

rα

iβ

mi

βxi

αi

βm

iβ

xiα

iβ

mi

βxi

αi

βm

iβ

xi

Ass

et1

2.82

830.

7484

***

0.39

79**

*2.

8784

0.62

77**

*0.

7537

***

2.73

060.

6000

***

0.00

671.

4257

0.42

64**

*0.

0688

**

Ass

et2

2.92

350.

8474

***

0.36

66**

*4.

3313

0.95

30**

*1.

2424

***

3.25

500.

8997

***

0.02

181.

6184

0.50

82**

*0.

0961

***

Ass

et3

2.94

880.

7837

***

0.28

38**

*5.

8676

1.20

64**

*2.

0781

***

3.35

580.

7054

***

0.30

02**

*2.

0402

0.73

79**

*-0

.181

2***

Ass

et4

3.30

771.

0795

***

0.68

94**

*7.

4587

1.54

44**

*3.

4050

***

3.57

660.

9143

***

-0.2

335*

**2.

1851

0.70

35**

*-0

.029

2

Ass

et5

3.91

991.

0935

***

1.02

54**

*7.

5671

1.77

87**

*3.

5039

***

3.73

730.

9876

***

0.31

63**

*2.

5255

0.83

47**

*-0

.135

4***

Ass

et6

3.96

001.

1415

***

-0.1

186

8.12

881.

2552

***

2.39

09**

*3.

7550

0.80

95**

*0.

2821

***

2.68

390.

8824

***

0.05

84

Ass

et7

4.17

001.

1041

***

-0.1

731*

*8.

5958

1.80

64**

*3.

0182

***

4.57

571.

0884

***

-0.4

463*

**3.

0600

0.93

80**

*0.

2929

***

Ass

et8

4.28

831.

1848

***

0.10

999.

3761

2.22

63**

*2.

2718

***

5.78

431.

3940

***

-0.9

303*

**3.

2154

1.04

99**

*-0

.336

2***

Ass

et9

4.65

041.

5186

***

0.66

33**

*9.

4857

1.81

35**

*4.

4482

***

5.82

691.

5050

***

0.98

27**

*3.

3090

0.90

91**

*0.

9260

***

Ass

et10

5.63

551.

5807

***

0.10

9810

.361

92.

0976

***

4.73

48**

*7.

0318

1.18

74**

*0.

6696

***

4.15

601.

3653

***

0.57

75**

*

S&P

500

1.89

900.

6796

***

1.89

900.

5935

***

1.89

900.

5860

***

1.89

900.

6861

***

Not

e:Th

ista

ble

pres

ents

the

in-s

ampl

est

ock-

spec

ific

esti

mat

esun

der

the

phys

ical

mea

sure

mod

els

for

the

sect

or-p

ortf

olio

s,w

here

the

cond

itio

nalv

aria

nces

ofin

divi

dual

stoc

kre

turn

s

are

assu

med

tofo

llow

Λit=

αi+

βm

ifm

t+

βxi

f xt.

Each

sect

or-p

ortf

olio

cont

ains

10st

ocks

whi

char

ela

bele

d‘A

sset

1’to

‘Ass

et10

’acc

ordi

ngto

the

unco

ndit

iona

lvar

ianc

ele

velα

i,su

ch

that

αi

of‘A

sset

1’is

the

smal

lest

and

that

of‘A

sset

10’i

sth

ela

rges

t.A

llα

i’sar

eca

lcul

ated

byth

esa

mpl

em

ean

ofsq

uare

dre

turn

s.Th

eco

lum

nsun

der

βm

ian

dβ

xire

pres

ent

the

MV

and

the

VR

fact

orlo

adin

gsof

each

stoc

k.T

hein

-sam

ple

data

win

dow

rang

esfr

omJa

nuar

y20

01to

Dec

embe

r20

10,a

tota

lof

2515

trad

ing

days

.Th

ete

stst

atis

tics

that

are

sign

ifica

ntat

1%,5

%an

d10

%le

vela

rede

note

dw

ith

3,2,

1as

teri

sks

resp

ecti

vely

.


Table 4.2: In-sample stock-specific estimates under the physical measure model (the Dow

Jones portfolio)

αi βmi βxi

Stickers Estimates Estimates Std.Err Estimates Std.Err

JNJ 1.5940 0.2738*** (0.0369) 0.3584*** (0.0553)

PG 1.5954 0.2424*** (0.0201) 0.1069*** (0.0285)

KO 1.8109 0.3509*** (0.0310) 0.2249*** (0.0422)

WMT 2.2286 0.3205*** (0.0360) 0.3010*** (0.0539)

MMM 2.3847 0.3714*** (0.0281) 0.0933** (0.0414)

MCD 2.5980 0.2931*** (0.0461) 0.4454*** (0.0726)

XOM 2.9014 0.3856*** (0.0287) -0.0748* (0.0439)

IBM 2.9091 0.5607*** (0.0542) 0.4417*** (0.0790)

PFE 2.9668 0.4236*** (0.0392) 0.2409*** (0.0564)

CVX 3.0587 0.3709*** (0.0313) -0.1938*** (0.0464)

UTX 3.4387 0.6801*** (0.0587) 0.4201*** (0.0825)

DD 3.5743 0.6666*** (0.0407) -0.0567 (0.0519)

NKE 3.6623 0.6489*** (0.0483) 0.2246*** (0.0648)

MRK 3.8010 0.5445*** (0.0420) 0.0068 (0.0570)

MSFT 4.0725 0.8458*** (0.0681) 0.4360*** (0.0921)

BA 4.2883 0.6623*** (0.0516) 0.1916*** (0.0680)

DIS 4.5536 0.9018*** (0.0889) 0.7389*** (0.1265)

HD 4.6293 0.8704*** (0.0687) 0.4134*** (0.0985)

GE 4.8156 1.1403*** (0.0665) 0.0601 (0.0731)

CAT 4.8180 0.6905*** (0.0500) -0.1898*** (0.0671)

UNH 5.1954 0.7695*** (0.0584) -0.4026*** (0.0846)

INTC 6.7354 1.2300*** (0.1556) 1.5079*** (0.2287)

GS 6.9506 1.4265*** (0.0905) -0.5478*** (0.1160)

AXP 7.2527 1.7181*** (0.0971) -0.5281*** (0.1142)

CSCO 7.8029 1.5191*** (0.2205) 2.3160*** (0.3285)

S&P 500 1.8990 0.4290*** (0.0253)

Note: This table presents the in-sample stock-specific estimates under the physical measure models

for the Dow Jones portfolio, where the conditional variances of individual stock returns are assumed

to follow Λit = αi + βmi fmt + βxi fxt. The portfolio contains 25 Dow Jones components. All αi’s are

calculated by the sample mean of squared returns. The columns under βmi and βxi represent the

MV and the VR factor loadings of each stock. The in-sample data window ranges from January

2001 to December 2010, a total of 2515 trading days. The test statistics that are significant at 1%, 5%

and 10% level are denoted with 3, 2, 1 asterisks respectively.


Alpha0 5 10 15

MV

-Loa

ding

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Alpha's vs MV-Loadings

Alpha0 5 10 15

VR

-Loa

ding

s

-1

0

1

2

3

4

5Alpha's vs VR-Loadings

MV-Loading0 1 2 3 4 5

VR

-Loa

ding

s

-1

0

1

2

3

4

5MV-Loadings vs VR-Loadings

Correlation: 0.6289 Correlation: 0.4952Correlation: 0.9375

Figure 4.6: Factor loadings and variance levels.

This figure shows the scatter plots of the parameter estimates for the sector-portfolios and the

Dow Jones portfolio. The three graphs show the plots of the unconditional variances against the

MV factor loadings, the unconditional variances against the VR factor loadings, and the MV factor

loadings against the VR factor loadings. All MV factor loadings are standardized by the loadings on

the market portfolio, such that the values displayed are βimv/βmmv. The number of stocks included

is 80 in the sector-portfolios (8 groups of 10 stocks), and 25 in the Dow Jones portfolio.

group, the VR factors also share a great level of similarity. Moreover, there is no

obvious dependence between the two factors, which corresponds to the statistical

independence setting imposed on the factors. Figure 4.8 shows the portfolio-wise

correlation series which share a degree of upward trend, with an average level

around 0.5. Figure 4.9 shows the estimated conditional variances for the 105 stocks

that cover a wide range of variance levels. The average individual stock variance

level (the white line) is higher than the index variance (the black line) at all times,

so that the low average variance risk premia of individual stocks are not caused by

the low variance levels.

4.4.3 Empirical results under the risk-neutral measure

By applying the Kalman filter procedure introduced in Section 4.3.2 on the MFIVs

of the individual stocks, the factor premium parameters μm and μx can be esti-

mated. Table 4.3 shows that the estimated MV factor premium parameters for all

portfolios (except for the materials group) are statistically significant and positive,


01/2001 01/2003 01/2005 01/2007 01/2009 12/2010

-5

0

5

10

Consumer Discretionary

MV-factor VR-factor

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010

-5

0

5

10

Consumer Staples

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010

-5

0

5

10

Energy

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010

-5

0

5

10

Health Care

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010

-5

0

5

10

Industrials

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010

-5

0

5

10

Information Technology

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010

-5

0

5

10

Materials

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010

-5

0

5

10

Utilities

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010

-5

0

5

10

Dow Jones Components

Figure 4.7: Estimated MV and VR factors.

This figure shows the estimated factors for 8 sector-portfolios and the Dow Jones portfolio. The

black lines represent the estimated MV factors; the grey lines represent the estimated VR factors.

The average persistence parameter bm of the MV factor is 0.9882, with an averaged half-life of 62

days; the average persistence parameter bx of the VR factor is 0.9957, with an average half-life of 194

days. The data window ranges from January 2001 to December 2010, a total of 2515 trading days.

which corresponds to the empirical findings in the literature that the index variance

risk premium is statistically significant and positive. The VR factor premium pa-

rameters are, in line with the pricing implication conjecture, statistically significant

and negative for 8 out of 9 portfolios, indicating that the VR factor premium serves

as an offsetting force compensating the excess risk premium spillover from the

market variance to individual stock variances. The average MV factor premium is

1.1857, which can be translated to an annual factor premium of 2.99%. The average

VR factor premium is -1.1533, which implies an annual factor premium of -2.91%.

The daily conditional factor premia can be calculated by taking the difference be-

tween the factor values under the risk-neutral and the physical measures, such that

the conditional MV factor premium series is{

f Qmt − f P

mt

}T

t=1, and the conditional

VR factor premium series is{

f Qxt − f P

xt

}T

t=1.

Figure 4.10 shows the filtered factor premia of the 9 portfolios. Pronounced

4.4. Empirical Results 111Ta

ble

4.3:

In-s

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oups

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ctor

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and

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pre

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Jone

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roup

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ete

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med

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lio.

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mns

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the

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edpe

rsis

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epa

ram

eter

sfo

rth

etw

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ctor

s,

whi

char

efil

tere

dac

cord

ing

tof m

,t+1=

(1−

b2 m)1

/2 s m

t+

b mf m

tan

df x

,t+1=

(1−

b2 x)1/

2 s xt+

b xf x

t.Th

eha

lf-l

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ofth

etw

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ctor

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calc

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edby

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)’an

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oadi

ng(V

R)’.

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colu

mns

unde

r‘μ

m’

and

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pres

ent

the

fact

orpr

emiu

mpa

ram

eter

s,s.

t.,μ

m=

EQ[f

Q mt]−

EP[f

P mt]

and

μx=

EQ[f

Q xt]−

EP[f

P xt].

The

colu

mn

unde

r‘V

RP’

pres

ents

the

aver

age

vari

ance

risk

prem

ium

ofal

lthe

stoc

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calc

ulat

edby

the

diff

eren

ce

betw

een

the

MFI

Vs

and

the

real

ized

vari

ance

s.Th

eco

lum

nun

der

‘VR

P%’

pres

ents

the

vari

ance

risk

prem

ium

scal

edby

the

vari

ance

leve

l,s.

t.,

VR

P%

=(M

FIV−

RV)/

MF

IV.

The

data

win

dow

rang

esfr

omJa

nuar

y20

01to

Dec

embe

r20

10,a

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days

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ete

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tics

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resp

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vely

.


01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.25

0.5

0.75

Consumer Discretionary

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.25

0.5

0.75

Consumer Staples

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.25

0.5

0.75

Energy

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.25

0.5

0.75

Health Care

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.25

0.5

0.75

Industrials

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.25

0.5

0.75

Information Technology

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.25

0.5

0.75

Materials

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.25

0.5

0.75

Utilities

01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

0.25

0.5

0.75

Dow Jones Components

Sector Correlation

Figure 4.8: Estimated portfolio-wise correlations.

This figure shows the estimated portfolio-wise correlations for 8 sector-portfolios and the Dow

Jones portfolio. The most correlated sector is the energy sector, with an unconditional correlation

of 0.6901. The consumer discretionary sector represents the lowest correlation group with an un-

conditional correlation of 0.2350. The Dow Jones portfolio has an average correlation of 0.1133. The

data window ranges from January 2001 to December 2010, a total of 2515 trading days.

offsetting patterns can be highlighted by the fact that the VR factor premia in most

cases move in opposite direction to the MV factor premia. Figure 4.11 shows the

average premium plot. The correlation between the MV and the VR factor premia

is -0.2197, and this negative correlation can be clearly seen from the graph. The MV

factor premium is positive throughout the in-sample period, while the VR factor

premium is positive before year 2003 and negative afterwards. The observation is

further supported by the OLS regressions that regress the average variance risk pre-

mia of each of the 105 stocks on their own factor loadings in two periods, namely,

2001-2003 and 2003-2010.

Period 2001-2003 : VRPi = 0.0171∗∗∗ · βmi + 0.0245∗∗ · βxi,

(0.0036) (0.0111)

Period 2004-2010 : VRPi = 0.0212∗∗∗ · βmi − 0.0071∗∗ · βxi.

(0.0019) (0.0031)


01/2001 01/2003 01/2005 01/2007 01/2009 12/20100

5

10

15

20

25

1%-20% & 80%-100%20%-40% & 60%-80%40%-60%Average varianceMarket variance

Figure 4.9: Estimated conditional variances.

This figure shows the quintile plots of the estimated conditional variances for 105 stocks. The aver-

age conditional variance of the 105 stocks is displayed in the white line. The conditional variance

for the index return is presented in the black line. The data window ranges from January 2001 to

December 2010, a total of 2515 trading days.

The above two regression results confirm the conjecture on individual stocks that

the MV factor premium serves as a positive driving force for the variance risk

premium, while the VR factor after year 2003 provides a negative force.

The factor premia can be used to calculate the variance risk premia of individual

stocks. Specifically, for stock i, the daily conditional variance risk premium series

is

VRPit = βmi( f Qmt − f P

mt) + βxi( f Qxt − f P

xt).

Following the recurrence (4.12), the s-day ahead predicted variance risk premium

given the current daily predicted factor values is

VRPi,t,t+s =1s

s

∑n=1

(βmibn−1

m ( f Qmt − f P

mt) + βxibn−1x ( f Q

xt − f Pxt) + μn

),

where

μn = βmiμm(1− bn−1m ) + βxiμx(1− bn−1

x ).

As the prediction period s increases, VRPi,t,t+s approaches its unconditional value

βmiμm + βxiμx.


01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-10

0

10

20Consumer Discretionary

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-5

0

5

10

15Consumer Staples

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-10

0

10

20

30Energy

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-10

0

10

20Health Care

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-10

0

10

20

30Industrials

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-10

0

10

20Information Technology

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-10

0

10

20

30Materials

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-10

0

10

20Utilities

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-20

0

20

40

60Dow Jones Components

MV-factor VR-factor

Figure 4.10: Estimated factor premia.

This figure shows the factor premia of 8 sector-portfolios and the Dow Jones portfolio. The premia

are calculated by taking the differences between the factors filtered under the risk-neutral and the

physical measure. The black and the grey areas represent separately the MV factor premia and the

VR factor premia. The prediction horizon is one trading day. The data window ranges from January

2001 to December 2010, a total of 2515 trading days.

4.4.4 Implications on the term structure of variance

Table 4.3 provides the estimated persistence parameters bm and bx of the MV and

the VR factors across 9 different portfolios. All portfolios suggest that bx is closer

to 1 than bm. This corresponds to the average half-life of the MV and the VR factors

being 62 days and 193 days. The difference in the half-life implies that the VR

factor dominates the long-run variance dynamics. Furthermore, since bm and bx

are assumed equal under different probability measures, the VR factor premium

also dominates the long-run variance risk premia of individual stocks. Figure 4.12

shows the comparison of the variance risk premia over prediction periods of 30

and 200 trading days for 4 selected stocks. The 4 stocks represent the stocks with

high, medium, low, and negative VR factor loadings. In the short run, both factors

are important inputs. From Figure 4.10, the MV premia across different groups

are in general larger than the VR factor premia in terms of magnitudes, resulting


01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-5

0

5

10

15

20

25Average Factor Premium Comparison

Average MV Factor PremiumAverage VR Factor Premium

V RPi = 0.0214× βmi − 0.0066× βxi

(0.0018) (0.0031)

V RPi = 0.0157× βmi + 0.0258× βxi

(0.0039) (0.0116)

Figure 4.11: Average factor premia.

This figure shows the average factor premia of the MV and the VR factors across 9 portfolios. The

sub-sample OLS regression results are shown. The first period is from January 2001 to Decem-

ber 2002, the second period is from January 2003 to December 2010. The regression equation is

VRPi = λmβmi + λxβxi + εi, where λm and λx are the factor premia to be estimated. All estimates

are significant at 5% level in both sub-samples. The data window ranges from January 2001 to

December 2010, a total of 2515 trading days.

in positive variance risk premia. As the predictive period gets longer, the MV

factor premium dwindles at a faster speed than the VR factor premium, thus the

variance risk premia of individual stocks with positive VR factor loadings start

to decrease and finally become negative. The stock with a negative VR factor

loading however, does not have a negative variance risk premium. The predicted

variance risk premium of this stock only approaches zero. The difference in the

factor memories indicates that the individual variance prediction does not converge

to its unconditional mean at a speed suggested by the one-factor model, since the

VR factor is still governing the variance dynamics given the lessening impact of the

MV factor.


01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-1.5

-1

-0.5

0

0.5

1

1.5

2

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-0.5

0

0.5

1

1.5

2

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-0.5

0

0.5

1

1.5

2

01/2001 01/2003 01/2005 01/2007 01/2009 12/2010-1

-0.5

0

0.5

1

1.5

2

2.5

Ticker: SREMV Loading: 0.9380VR Loading: 0.2929

Ticker: PEGMV Loading: 1.0499VR Loading: -0.3362

Ticker: TEMV Loading: 1.3653VR Loading: 0.5775

Ticker: DUKMV Loading: 0.9091VR Loading: 0.9260

Figure 4.12: Variance risk premium term structure.

This figure shows the term structures of the variance risk premia for 4 selected stocks which have

respectively high, medium, low and negative VR factor loadings. The black areas represent the

variance risk premia predicted for the next 30 trading days, and the grey areas represent the same

measures for the next 200 trading days. The data window ranges from January 2001 to December

2010, a total of 2515 trading days.

4.5 Option Portfolio Design

This section shows how option portfolios can be constructed using the previously

estimated variance risk premia. The straddle option strategy is used for the portfo-

lio. A straddle is an investment strategy that consists of buying one call option and

one put option with the same strike and maturity. The return on a delta-hedged

short straddle is empirically reconciled as a measure of variance risk premium,

such as in Bakshi and Kapadia (2003), Driessen et al. (2009) and Vasquez (2014), as

it exploits the difference between the implied and the realized volatilities.

4.5.1 Straddle returns for different maturities

Buying a delta-hedged straddle implies taking negative exposure to the variance

risk premium of the underlying stock or the index. Under the two-factor model as-

4.5. Option Portfolio Design 117

sumption, a long position on a delta-hedged individual stock straddle has negative

exposure to both the MV and the VR factor premia, assuming the stock has positive

VR factor loading. The negative exposure to the MV factor incurs potential losses,

since the MV factor premium is positive. On the other hand, the negative exposure

to the VR factor generates profits given that the VR factor premium is negative.

Therefore, the two-factor model suggests that the return of buying a delta-hedged

straddle has two components. The first component contributes a negative return

and is delivered by the exposure to the MV factor. The second component con-

tributes a positive return and is caused by the exposure to the VR factor. In the

short-term delta-hedged straddle returns for individual stocks, both components

are significant. Since the MV factor premium is larger than that of the VR factor

in terms of absolute magnitude, the MV factor premium dominates the sum of

the two components. The return of buying a delta-hedged straddle is thus mainly

determined by the negative exposure to the MV factor which generates negative

returns.

As indicated in the previous section, the MV factor has a shorter memory com-

pared to the VR factor. Therefore, the exposure to the MV factor in the long-term

delta-hedged straddles of individual stocks has a much lesser effect than the VR

factor does. The delta-hedged straddle returns of individual stocks are then dom-

inated by the positive return contributions from the negative exposure to the VR

factor. This conclusion states that buying long-term delta-hedged straddles are on

average more profitable than buying short-term delta-hedged straddles. Note that

this conclusion can also be applied on the delta-hedged index straddles, for which

the only component in the delta-hedged straddle returns is the negative exposure

to the MV factor premium. Buying a short-term (long-term) delta-hedged index

straddles thus incur higher (lower) losses. Since this component always provides

negative return contributions, one can only generate positive returns by shorting

the index straddles, especially the short-term ones. Figure 4.13 shows averaged

delta-hedged straddle returns for both the index and individual straddles against

their corresponding maturities. The delta-hedging procedure is implemented daily.


Maturity

50 100 150 200 250 300

Del

ta-h

edge

d S

trad

dle

Ret

urn

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Average Return Individual StraddlesAverage Return Index Straddles

Figure 4.13: Average straddle returns against maturities.

This figure shows the average delta-hedged straddle returns against different maturities. The delta

hedging is implemented on a daily basis. The straddles are held until maturities. The average

returns of individual stock straddles are presented in black line, and the index straddle returns are

presented in the grey line. The data window ranges from January 2001 to December 2010, a total of

2515 trading days.

The conclusion based on the model is well in line with the empirical results: 1) buy-

ing long-term individual straddles generates positive returns which are higher than

the returns of buying short-term individual straddles, 2) shorting short-term index

straddles generates positive returns which are higher than the returns of shorting

long-term index straddles.

4.5.2 VRP sorting strategy

The above analysis makes it straightforward to conclude that buying long-term

delta-hedged individual straddles and shorting short-term delta-hedged index strad-

dles render an appealing strategy to acquire both factor premia in a profitable way.

I first adopt two model-free strategies that are designed based on this conclusion.

The first strategy invests evenly in all the long-term individual stock straddles, and

shorts evenly all the short-term index straddles each trading day. The strategy is

called the average strategy and is labeled ‘AVE’ in the following analysis. The sec-


ond strategy buys the individual straddles with the longest maturity and shorts the

index straddle with the shortest maturity. The strategy is called the extreme strategy

and is labeled ‘EXT’ in the following analysis. Both strategies are implemented on

a daily basis starting from a daily initial wealth of 1. All the straddles are held

to their maturities. The delta-hedged straddle return is calculated by dividing the

sum of the straddle payoff and the delta-hedged gain by the initial straddle price.

The 60-day maturity is used as a threshold to separate long- and short-term strad-

dle maturities. Straddles with less than 6 days to maturity are excluded.

A model-based strategy is also implemented using the sorted variance risk pre-

mia. Since the focus of the portfolio strategy is on straddle returns, instead of

payoffs, the variance risk premium measure is scaled by the variance level, s.t.

VRPscalei,t,t+s =

1s

s

∑n=1

(βmibn−1

m ( f Qmt − f P

mt) + βxibn−1x ( f Q

xt − f Pxt) + μn

)/σ2

i,t+n−1,t+n,

(4.18)

where σ2i,t+n−1,t+n is the predicted variance level between time (t + n− 1, t + n).5

The strategy is called the variance risk premium strategy and is labeled ‘VRP’ in the

following analysis. Note that the two-factor model suggests that the negative pre-

mium of the VR factor dominates the long-term variance risk premia of individual

stocks, which is the reason for buying long-term individual straddles. The buy-side

of the VRP strategy requires buying the individual straddle with the lowest pre-

dicted variance risk premium according to the predictions obtained using (4.18) for

each of the long-term individual straddles. Correspondingly, the short-side of the

VRP strategy requires shorting the the index straddle with the highest predicted

variance risk premium.

4.5.3 Empirical performance

The in-sample portfolio performance is analyzed over the same period as the esti-

mation procedure in Section 4.4, with an out-of-sample period ranging from Jan-

5The variance level σ2i,t+n−1,t+n is the variance predicted under the physical measure, such that

σ2i,t+n−1,t+n = αi + βimbn

m fmt + βixbnx fxt.


uary 2011 to December 2013, a total of 754 trading days. Table 4.4 shows results for

three portfolio strategies (VRP, AVE, EXT) based on the idea to long the long-term

individual straddles and to short the short-term index straddles. The three strate-

gies are applied in each of the 9 portfolios separately. The row ‘Market Average’

presents the results for the market average portfolio which invests equally in each

of the portfolios following the specific strategies. In all three strategies, the long-

term individual straddles always outperform the short-term individual straddles.

The VRP strategy delivers the best performance for the market average portfolio.

The return per contract for the market average portfolio is 6.49%, which is higher

than 4.39% of the AVE strategy and 4.11% of the EXT strategy. In the out-of-sample

performance, the long-short strategy does not indicate clear differences but one can

still notice the superiority of the VRP strategy in selecting straddles.

To further test whether sorting according to model-predicted variance risk pre-

mia is indeed necessary, CAPM results for the portfolios containing only the se-

lected individual straddles are reported in Table 4.6. In both in- and out-of-sample

analyses, the highest CAPM alphas of the market average portfolio returns are ob-

tained by the VRP strategy. Table 4.5 shows a return comparison among the three

strategies. The returns are scaled to the monthly frequency for comparison. The

best performers are highlighted in bold type. The results show that both in- and

out-of-sample, the VRP strategy outperforms the other two model-free strategies.

For the market average portfolio, the monthly return of the VRP strategy is 4.30%,

followed by the EXT strategy with 3.33% and the AVE strategy with 2.97%. The

best out-of-sample performance is also obtained by the VRP strategy, under which

the market average portfolio has an average monthly return of 7.07%, followed by

the AVE strategy with 6.46% and the EXT strategy with 6.23%.

4.5.4 Robustness checks

To confirm the validity of the results, several robustness checks are conducted.

The analyses above have already shown the superiority of the VRP strategy across

portfolios of different sectors, different sizes, both in- and out-of-sample. In a


Table 4.4: Straddle returns under different sorting schemes

VRP AVE EXT

In-sample Group Low VRP High VRP Long Mty Short Mty Max Mty Min Mty

Consumer Discretionary 7.50% 2.91% 6.86% 1.86% 8.58% 0.68%

Consumer Staples -1.03% -4.30% -0.63% -3.70% -0.97% -5.99%

Energy 3.69% 3.16% 9.20% 6.35% 8.89% 4.32%

Health Care 2.80% -0.86% 0.63% -3.73% 1.07% -7.72%

Industrials 2.55% 0.83% 4.61% 0.86% -2.38% -7.46%

Information Technology 12.18% 2.87% 5.51% 2.74% 6.15% -1.73%

Materials 8.72% 3.29% 7.15% 1.35% 7.97% -2.83%

Utilities 8.67% 0.38% 3.22% 2.00% 1.61% 0.80%

Dow Jones 25 Stocks 13.34% 6.85% 2.92% 1.13% 6.08% -4.23%

S&P 500 -1.08% -2.46% -2.84% -2.84% -1.40% -2.94%

Market Average 6.49% 1.68% 4.39% 0.98% 4.11% -2.69%

Out-of-sample Group Low VRP High VRP Long Mty Short Mty Max Mty Min Mty

Consumer Discretionary 0.31% -3.30% -1.94% -1.76% -0.68% -1.22%

Consumer Staples 11.78% -6.41% 9.01% 1.67% 6.14% -10.02%

Energy -0.11% 3.74% -0.59% 3.72% -8.51% -2.12%

Health Care 11.95% -2.82% 11.08% 2.25% 16.81% 2.28%

Industrials 2.19% 8.72% 3.84% 3.69% 17.96% -2.89%

Information Technology -6.02% -3.51% 7.22% 9.91% 16.95% 10.76%

Materials 7.14% 8.64% 6.05% 5.42% 12.69% 16.56%

Utilities 3.10% 11.17% 3.05% 6.14% -2.41% 12.91%

Dow Jones 25 Stocks 8.88% 2.47% 1.11% 2.62% -2.95% -3.92%

S&P 500 -4.17% -5.53% -5.73% -8.43% -4.17% -5.53%

Market Average 4.36% 2.08% 4.32% 3.74% 6.22% 2.48%

Note: The table presents the straddle returns when sorted by the VRP strategy, the AVE strategy

and the EXT strategy. The returns represent the average life-time return of buying a straddle and

hold to maturity. The columns under ‘VRP’ presents the straddle returns when sorted by the

predicted variance risk premium. The columns under ‘AVE’ present the average straddle returns

when sorted by the AVE strategy. The columns under ‘EXT’ present the straddle returns sorted

by their maturities. The row ‘Market Average’ presents the sorted straddle returns when investing

equally in each group. The in-sample data window ranges from January 2001 to December 2010, a

total of 2515 trading days; The out-of-sample data window ranges from January 2011 to December

2013, a total of 754 trading days.


Table 4.5: Monthly portfolio returns of different strategies

Strategy Return Monthly

In-sample Group VRP AVE EXT Diff.AVE Diff.EXT

Consumer Discretionary 5.41% 3.54% 3.86% 0.08% 0.52%

Consumer Staples 2.63% 1.86% 2.69% 3.04% 5.49%

Energy 4.34% 3.96% 3.99% -3.53% -2.90%

Health Care 3.56% 2.16% 2.94% 3.39% 7.40%

Industrials 3.83% 3.24% 2.51% 0.61% 6.80%

Information Technology 4.39% 3.22% 3.60% -1.02% 2.58%

Materials 5.39% 3.49% 3.85% 0.33% 3.78%

Utilities 4.22% 2.65% 2.97% -1.04% -0.54%

Dow Jones 25 Stocks 4.95% 2.61% 3.54% -0.16% 4.92%

S&P 500 2.54% 2.14% 2.80%

Market Average 4.30% 2.97% 3.33% 0.19% 3.12%

Out-of-sample Group VRP AVE EXT Diff.AVE Diff.EXT

Consumer Discretionary 5.57% 4.94% 5.35% 1.23% 1.46%

Consumer Staples 9.04% 7.70% 6.34% 1.57% 11.04%

Energy 5.52% 5.53% 4.36% -2.90% 0.99%

Health Care 8.56% 7.66% 7.62% 0.23% -0.07%

Industrials 7.70% 6.39% 7.69% -1.78% 4.90%

Information Technology 4.74% 7.12% 7.55% -6.29% -8.25%

Materials 8.63% 6.85% 7.03% -2.80% -14.37%

Utilities 5.62% 6.09% 5.02% -4.32% -12.47%

Dow Jones 25 Stocks 8.24% 5.83% 5.15% -1.53% 3.55%

S&P 500 5.45% 5.43% 5.45%

Market Average 7.07% 6.46% 6.23% -1.84% -1.47%

Note: This table presents the portfolio returns of three strategies: VRP, AVE and EXT. The returns

are obtained first by calculating the realized return over the life-time of the chosen straddle, and

then scaling it to monthly. At each trading day, the VRP strategy buys the individual straddle

with the lowest predicted variance risk premium, and short the index straddle with the highest

predicted variance risk premium. The AVE strategy buys all the long-term individual straddles and

shorts all the short-term index straddles. The Diff.AVE strategy buys all the long-term individual

straddles and shorts all the short-term individual straddles. The EXT strategy buys the individual

straddle with the longest maturity, and shorts the index straddle with the shortest maturity. The

Diff.EXT strategy buys the individual straddle with the longest maturity and shorts the individual

straddle with the shortest maturity. The row ‘Market Average’ represents the average monthly

return when investing equally in each group. The in-sample data window ranges from January

2001 to December 2010, a total of 2515 trading days; The out-of-sample data window ranges from

January 2011 to December 2013, a total of 754 trading days.


Table 4.6: CAPM analysis on individual straddles portfolios

CAPM Alpha CAPM Beta

In-sample Group VRP AVE EXT VRP AVE EXT

Consumer Discretionary 0.0287*** 0.0140*** 0.0105*** -0.1582 -0.1664** -0.0955

Consumer Staples 0.0009 -0.0027 -0.0012 -0.0380 -0.0838 -0.0708

Energy 0.0180*** 0.0183*** 0.0119*** -0.1427 -0.1151 -0.0774

Health Care 0.0102** 0.0002 0.0014 0.0655 -0.1242** -0.1104

Industrials 0.0129** 0.0111*** -0.0029 -0.2223 -0.1103** -0.0495

Information Technology 0.0185*** 0.0109*** 0.0079*** -0.0489 -0.0565 -0.0909

Materials 0.0285*** 0.0136*** 0.0105*** -0.1161 -0.2164*** -0.0675

Utilities 0.0169*** 0.0052** 0.0016 -0.2986* -0.2114** -0.1266*

Dow Jones 25 Stocks 0.0241*** 0.0048*** 0.0074*** -0.0075 -0.0933 0.0706

Market Average 0.0176*** 0.0084*** 0.0052*** -0.1074 -0.1308** -0.0687*

Out-of-sample Group VRP AVE EXT VRP AVE EXT

Consumer Discretionary 0.0015 -0.0049* -0.0010 -0.5898 -0.1357 0.1098

Consumer Staples 0.0360*** 0.0228*** 0.0086* -0.1301 -0.0356 0.4182*

Energy 0.0006 0.0010 -0.0108*** 0.1208 0.0190 -0.2277

Health Care 0.0308*** 0.0224*** 0.0216*** 0.4742 -0.0490 0.1231

Industrials 0.0225*** 0.0096*** 0.0224*** -0.0843 -0.0245 0.0263

Information Technology -0.0071* 0.0169*** 0.0210*** 0.0798 -0.0255 -0.1355

Materials 0.0316*** 0.0144*** 0.0156*** 0.3308 -0.1748** 0.2876

Utilities 0.0023 0.0067** -0.0043 -0.9151 -0.1455 0.0426

Dow Jones 25 Stocks 0.0277** 0.0041* -0.0030 0.2831 -0.0323 -0.0735

Market Average 0.0162*** 0.0103*** 0.0078*** -0.0478 -0.0671 0.0634

Note: This table presents the CAPM regression results of the monthly returns of the individual

straddle portfolios constructed by the VRP strategy, the AVE strategy, and the EXT strategy. The

CAPM alphas and betas are estimated from the regression equation rjt = αj + βjrmt + η

jt , where

rjt is the monthly return of strategy j, j =VRP, AVE, EXT; rmt is the monthly return of the S&P

500 Index. The significance levels of 10%, 5%, and 1% are labeled as 1, 2, and 3 asterisks. The

in-sample data window ranges from January 2001 to December 2010, a total of 2515 trading days;

The out-of-sample data window ranges from January 2011 to December 2013, a total of 754 trading

days.

month-by-month examination, the market average portfolio under the VRP strategy

achieves the highest return in 81 out of 156 months, while the market average

portfolios under the AVE and EXT strategies achieve the highest return in only

33 and 42 months. These results indicate that the good performance of the VRP


strategy is not driven by extreme monthly returns. Different variance risk premium

calculation methods are also checked. First, similar results are found when using

the variances filtered under the risk-neutral measure as the scaling parameter σ2i,t,t+s

instead of using the variances filtered under the physical measure. Second, similar

performance can be found when the straddles are sorted according to the variance

risk premia calculated by

VRPscalei,t,t+s =

(βmibs−1

m ( f Qmt − f P

mt) + βxibs−1x ( f Q

xt − f Pxt) + μs

)/σ2

i,t+s−1,t+s.

In this way the variance risk premium over the time (t, t + s) is only determined

by the end-of-period factor premia. The measure provides a more volatile term-

structure of the variance risk premia. Another calculation method that uses rolling-

window premia estimates as replacements for μm and μx is also checked, still,

similar results are found.

4.5.5 Implications on dispersion trade

As mentioned in Section 4.2, the existence of the VR factor premium sheds light on

a particular kind of dispersion trading strategy that only requires a small number

of individual straddles. A dispersion trading strategy usually involves shorting

the index straddles and buying the individual straddles. When the underlying

stocks of the individual straddles represent all the components in the index, the

dispersion trading strategy is then designed to capture the positive correlation risk

premium embedded in the index. However, when the number of individual strad-

dles included is small, the correlation risk premium is hard to identify. I propose

here a new approach in quantifying the dispersion trading strategy with a limited

number of stocks, which generates positive returns in a theoretical framework.

Assume that the unexpected shock on the individual stock return follows

dSi

Si−E

[dSi

Si

]= v1/2

i dWi,

and the unexpected shock on the index return follows

dSm

Sm−E

[dSm

Sm

]= v1/2

m dWm,


where Wi and Wm are the standard Wiener processes. The symbols vi and vm denote

the instantaneous variance processes of the individual stocks and the index. Define

fi and fm to be the instantaneous factors such that fmt = Et−s

[∫ tt−s fm(u)du

]and

fxt = Et−s

[∫ tt−s fx(u)du

], and assume each factor follows an Ito process:

d fm −E[ fm] = σf mdWf m,

d fx −E[ fx] = σf xdWf x,

where Wf m and Wf x are standard Wiener processes. Under the variance-factor

model, the instantaneous variance processes can then be written as

dvi −E[vi] = βmiσf mdWf m + βxiσf xdWf x,

dvm −E[vm] = βmmσf mdWf m.

Denote by Oi and Om the prices of the at-the-money straddles for each stock i and

the index, then

dOi

Oi−E

[dOi

Oi

]=

Si

OiΔiv1/2

i dWi +1

2Oiv−1/2

i κiβmiσf mdWf m +1

2Oiv−1/2

i κiβxiσf xdWf x,

dOm

Om−E

[dOm

Om

]=

Sm

OmΔmv1/2

m dWm +1

2Omv−1/2

m κmβmmσf mdWf m,

where Δi and κi are the corresponding Black-Scholes delta and vega for stock i and

the index (when i = m). Assume the portfolio weight on each of the individual

straddles as

zi =1N

2Oiv1/2i

κiβxi,

where N is the number of stocks in the portfolio. Assume by yi the portfolio weight

on individual stock i, ym the weight on the index, and zm the weight on the index

straddle. The delta-hedging conditions for each stock and the index require

ziSi

OiΔiv1/2

i + yiv1/2i = 0,

zmSm

OmΔmv1/2

m + ymv1/2m = 0.

Then the hedging of the diffusion term of the MV factor fm provides enough re-

strictions to identify all portfolio weights:

zm

2Omv−1/2

m κmβmm +N

∑i=1

zi

2Oiv−1/2

i κiβmi = 0.


Denote the value of the portfolio as DVR, then

dDVR

DVR−E

[dDVR

DVR

]= σf xdWf x.

This portfolio requires taking long positions in individual straddles and index, and

short positions in individual stocks and index straddles. This portfolio only has

negative exposure to the VR factor. Since the VR factor premium is shown to be

negative, this strategy generates positive returns.

4.6 Conclusion

I show that the idiosyncratic variance comovement can be alleviated when using

a variance-factor model with two factors, namely, the market variance factor and

the variance residual factor. The market variance factor serves to mimic the risk

premium spillover from the market variance to individual stock variances and

the variance residual factor aims to capture the remaining dynamics in individ-

ual stock variances. To filter the factors, I implement a multivariate score-driven

model based on daily return series of individual stocks and the index. The model is

able to identify the market variance factor as a standardized measure of the market

variance, and to filter the independent variance residual factor series. Moreover,

the estimation results strongly support the existence of the variance residual factor

in the sense of statistically significant factor loadings. Upon existence, the variance

residual factor has its own pricing implication on variance risk. A Kalman filter

approach on the model-free implied variance shows that the variance residual fac-

tor carries a negative premium that compensates the excess risk premium spillover

from the market variance to individual stock variances.

A further analysis reads that the variance residual factor possesses a longer

memory than the market variance factor, so that it dominates the long-run variance

risk premia of individual stocks. An option portfolio strategy is implemented based

on the predicted variance risk premia. The strategy buys long-term individual

straddles with the lowest predicted variance risk premium, and shorts short-term

4.6. Conclusion 127

index straddles with the highest predictions. The results of portfolio performance

show that this strategy outperforms two competing model-free strategies.

5 | Summary

This thesis, entitled “Time-Varying Correlation and Common Structures in Volatil-

ity”, investigates the dynamics of correlations and volatilities for multivariate series

of financial returns. Specifically, the thesis develops answers to the following three

questions: 1) Given that the correlation between two return series is time-varying

at a daily frequency, is it possible to characterize the dynamic properties of corre-

lations at intraday frequencies, and if so, how can we use this intraday information

to improve the accuracy of daily correlation estimates? 2) When the volatilities of

multiple return series follow a strong common structure, how can we specify an

observation-driven model to estimate the common factors in volatilities, and what

are the statistical properties of the parameter estimators of such a model? 3) If the

estimated common factors in volatilities are statistically significant, what are the

pricing implications of the common factors and what is the relation between the

risk premia they carry and the variance risk premium?

To answer the first question, Chapter 2 investigates the intraday correlation dy-

namics by developing a new class of correlation models which aggregate intraday

information extracted from high frequency returns. We find sufficient evidence

that the correlation between two return series can be time-varying at intraday fre-

quencies, and that incorporating intraday dynamics brings significant improve-

ments for model fit and for correlation timing.

In Chapter 3, we shift our focus from correlation to volatility, answering the sec-

ond question by proposing a class of variance-factor models with common factors

imposed on conditional variances. The model specification gives a straightforward

way to explain the comovement feature in the conditional variances of multiple

129

130 Chapter 5. Summary

return series. Given that the model is observation-driven, its parameters can be

easily estimated through the maximum likelihood method.

Both the statistical significance and the economic implications of a two-factor

formulation of the variance-factor model are discussed in Chapter 4, in order to

provide an answer to the third question. We find that the common dynamics of

the conditional variances of multiple return series can generally be explained by

two factors: the market variance factor and the variance residual factor. An em-

pirical analysis using option data shows that both factors embed risk premia, a

combination of which constitutes the variance risk premia of individual stocks.

This finding is further supported by evidence stemming from an option portfolio

trading strategy that aims at collecting both factor premia in a profitable way.

For future work, one possible extension for the correlation model in Chapter

2 is to change the Gaussian copula to a Student’s t copula to capture the tail de-

pendence structure of two return series, in which case an aggregation procedure

for the degrees of freedom parameter of the Student’s t copula should be derived.

Regarding the variance-factor model in Chapter 3, one promising extension is to

include intraday information such as realized covariance matrices in the recurrence

equation for variance factors. It would also be interesting to explore the relation

between the strategy in Chapter 4 and the volatility based strategies widely used

by practitioners such as VIX futures basis strategies and smart beta strategies.

6 | Samenvatting

Dit proefschrift, getiteld “Time-Varying Correlation and Common Structures in

Volatility”, bestudeert de dynamiek van correlaties en volatiliteiten in multivari-

ate tijdreeksmodellen voor financiële rendementen. In het bijzonder wordt een

antwoord gegeven op de volgende drie vragen: 1) Gegeven dat de correlatie op

dagbasis tussen twee rendementsreeksen tijdsvariërend is, is het dan mogelijk

om de intra-dag dynamiek van correlaties te karakteriseren, en zo ja, hoe kun-

nen we deze intra-dag informatie gebruiken om de dagelijkse correlatiebewegin-

gen nauwkeuriger te voorspellen? 2) Als volatiliteiten van verschillende rende-

mentsreeksen een sterke gezamenlijke structuur vertonen, hoe kunnen we dit ver-

schijnsel dan reproduceren en schatten in een observation-driven tijdreeksmodel, en

wat zijn de statistische eigenschappen van de parameterschattingen in een dergelijk

model? 3) Als we statistische significante gezamenlijke factoren in volatiliteiten vin-

den, wat zijn dan de gevolgen hiervan voor de waardering van aandelen, en wat

is de relatie tussen de risicopremies op de gezamenlijke factoren en de variantie-

risicopremie?

Ter beantwoording van de eerste vraag wordt in Hoofdstuk 2 de intra-dag

dynamiek van correlaties onderzocht binnen een nieuwe klasse van correlatie-

modellen, waarin de intra-dag informatie verkregen uit hoogfrequente rendements-

waarnemingen wordt geaggregeerd tot dagelijke correlatievoorspellingen. Toepas-

sing van het model biedt ondersteuning voor de hypothese dat correlaties binnen

een dag variëren, en laat zien dat gebruik van deze intra-dag correlatie dynamiek

leidt tot significante verbeteringen in model fit en in het dateren van veranderingen

in correlaties.

131

132 Chapter 6. Samenvatting

In Hoofdstuk 3 wordt de aandacht verlegd van correlaties naar volatiliteiten, en

wordt antwoord gegeven op de tweede vraag door een klasse van variantie-factor

modellen te ontwikkelen waarin volatiliteiten worden beïnvloed door gezamenlijke

factoren. De specificatie van het model biedt een eenvoudige beschrijving van het

feit dat conditionele varianties in multivariate rendementsreeksen een gezamenlijk

patroon vertonen. Aangezien het een zogeheten observation-driven model betreft,

kunnen de parameters op relatief eenvoudige wijze geschat worden met de maxi-

mum likelihood methode.

De statistische significantie en de economische implicaties van een twee-factor

versie van het variantie-factor model worden onderzocht in Hoofdstuk 4, waarmee

een antwoord wordt gegeven op de derde vraag. We zien hier dat de gezamenlijke

dynamiek van conditionele varianties van multivariate rendementsreeksen over het

algemeen goed beschreven kan worden door twee factoren: de marktvariantie fac-

tor een de zogenaamde variantie-residuele factor. Een empirische analyse van op-

tiegegevens laat zien dat beide factoren een risicopremie met zich meebrengen, en

dat een combinatie van beide premia doorwerkt in de variantierisicopremie van

individuele aandelen. Verdere ondersteuning hiervoor wordt verkregen uit de re-

sultaten van een optieportefeuillestrategie, geconstrueerd om beide factorpremies

te exploiteren.

Tenslotte bespreken we een aantal mogelijke uitbreidingen van de analyse in

dit proefschrift. Een mogelijke uitbreiding van het correlatiemodel uit Hoofdstuk

2 is om de Gaussische copula te vervangen door een Student’s t copula, om hier-

mee de staartafhankelijkheid tussen twee reeksen beter te beschrijven; dit vereist

een nieuw aggregatieresultaat voor de vrijheidsgradenparameter van de Student’s

t copula. Een veelbelovende uitbreiding van het variantie-factor model uit Hoofd-

stuk 3 is om intra-dag informatie zoals de realized covariance matrix op te nemen in

de vergelijkingen voor de variantiefactoren. Tenslotte is het de moeite waard om de

relatie te onderzoeken tussen enerzijds de strategie uit Hoofdstuk 4 en anderzijds

in de praktijk gebruikte volatiliteitsstrategieën zoals VIX futures basis en smart beta

strategieën.

Bibliography

Alexander, C. (2001), “Orthogonal GARCH, ” Market Risk, Financial Times - Prentice

Hall, 2, 21–38.

Andersen, T. G. and Bollerslev, T. (1998), “Deutsche Mark-Dollar Volatility: Intra-

day Activity Patterns, Macroeconomic Announcement, and Longer Run Depen-

dencies,” Journal of Finance, 53(1), 219–265.

Ang, A., Hodrick, R., Xing, Y. and Zhang, X. (2012), “High Idiosyncratic Volatility

and Low Returns: International and Further U.S. Evidence,” Journal of Financial

Economics, 91, 1–23.

Asness, C. S., Moskowitz, T. J., and Pedersen, L. H. (2013), “Value and Momentum

Everywhere,” Journal of Finance, 68(3), 929–985.

Baba, Y., Engle, R., Kraft, D. F., and Kroner, F. (1991), “Multivariate Simultaneous

Generalized ARCH,” manuscript, Department of Economics, UCSD.

Bakshi, G., Kapadia, N. and Madan, D. (2003), “Stock Return Characteristics, Skew

Laws, and the Differential Pricing of Individual Equity Options,” Review of Fi-

nancial Studies, 16(1), 101–143.

Bakshi, G. and Kapadia, N. (2003), “Delta-Hedged Gains and the Negative Market

Volatility Risk Premium,” Review of Financial Studies, 16, 527–566.

Bakshi, G. and Madan, D. (2006), “A Theory of Volatility Spreads,” Management

Science, 52, 1945–1956.

Barndorff-Nielsen, O. E. and Shephard, N. (2004), “Econometrics Analysis of Real-

ized Covariance: High Frequency Based Covariance, Regression, and Correlation

in Financial Economics,” Econometrica, 73(3), 885–925.

Blasques, F., Koopman, S. J., and Lucas, A. (2014a), “Maximum Likelihood Estima-

133

134 Bibliography

tion for Generalized Autoregressive Score Models,” Working Paper

Blasques, F., Koopman, S. J., and Lucas, A. (2014b), “Stationarity and Ergodicity

of Univariate Generalized Autoregressive Score Processes,” Electronic Journal of

Statistics, 8, 1088–1112.

Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedasticity,”

Journal of Econometrics, 31, 307–327.

Bollerslev, T., Engle, R., and Wooldridge, M. (1988), “A Capital Asset Pricing Model

with Time-Varying Covariances,” The Journal of Political Economy, 96, 116–131.

Bollerslev, T. (1990), “Modelling the coherence in short-run nominal exchange rates:

A multivariate generalized ARCH model,” Review of Economics and Statistics, 72,

498–505.

Boswijk, P. and Liu, Y. (2014), “Correlation Aggregation in Intraday Financial

Data,” Working Paper.

Boswijk, P. and Liu, Y. (2015), “Score-Driven Variance-Factor Models,” Working

Paper.

Britten-Jones, M. and Neuberger, A. (2000), “Option Prices, Implied Price Processes,

and Stochastic Volatility,” Journal of Finance, 55, 839–866.

Buss, A. and Vilkov, G. (2012), “Measuring Equity Risk with Option-implied Cor-

relations,” Review of Financial Studies, 25(10), 3113–3140.

Cao, J. and Han, B. (2013), “Cross Section of Option Returns and Idiosyncratic

Stock Volatility,” Journal of Financial Economics, 108, 231–249.

Carr, P. and Wu, L. (2012), “Variance Risk Premiums,” Review of Financial Studies,

22(3), 1311–1341.

Chen, Z. and Petkova, R. (2012), “Does Idiosyncratic Volatility Proxy for Risk Ex-

posure?,” Review of Financial Studies, 29(5), 2745–2787.

Christodoulakis, G. A. and Satchell, S. E. (2002), “Modeling the Time-Varying Con-

ditional Correlation Between Financial Asset Returns,” European Journal of Oper-

ational Research, 139(2), 351–370.

Christoffersen, P., Mathieu, F., and Jacobs, K. (2013), “The Factor Structure in Equity

Options,” Working Paper.

Bibliography 135

Cox. D. R., Gudmundsson, G., Lindgren, G., Bondesson, L., Harsaae, E., Laake, P.,

Juselius, K., and Lauritzen, S. L. (1981), “Statistical Analysis of Time Series: Some

Recent Developments,” Scandinavian Journal of Statistics, 8, 93–115.

Creal, D., Koopman, S. J., and Lucas, A. (2011), “A dynamic multivariate heavy-

tailed model for time-varying volatilities and correlations,” Journal of Business and

Economic Statistics, 29, 552–563.

Creal, D., Koopman, S. J., and Lucas, A. (2013), “Generalized Autoregressive Score

Models with Applications,” Journal of Applied Econometrics, 28(5), 777–795.

Della Corte, P., Sarno, L., and Tsiakas, I. (2008), “Correlation Timing in Asset Allo-

cation with Bayesian Learning,” Working Paper.

Dorst, F. and Nijman, T. (1993), “Temporal Aggregation of GARCH Process,” Econo-

metrica, 61(4), 909–927.

Driessen, J., Maenhout, P. J., and Vilkov, G. (2009), “The Price of Correlations Risk:

Evidence from Equity Options,” Journal of Finance, 64(3), 1377–1406.

Duan, J-C. and Wei, J. (2009), “Systematic Risk and the Price Structure of Individual

Equity Options,” Review of Financial Studies, 22, 1981–2006.

Dunn, J. (1973), “A Note on Sufficient Condition for Uniqueness of a Restricted

Factor Matrix,” Psychometrika, 38, 141–143.

Engle, R. (2002), “Dynamic Conditional Correlation-A Simple Class of Multivariate

GARCH Models,” Journal of Business and Economics Statistics, 20, 339-350.

Engle, R., Victor, NG., and Rothschild, M. (1990), “Asset pricing with a factor-

ARCH covariance structure,” Journal of Econometrics, 45, 213–237.

Engle, R. and Kelly, B. (2012), “Dynamic Equicorrelation,” Journal of Business and

Economics Statistics, 30(2), 212-228.

Engle, R. and Kroner, F. (1995), “Multivariate Simultaneous Generalized ARCH,”

Econometric Theory, 11, 122–150.

Fama, E. and French, K. (1993), “Common Risk Factors in the Returns on Stocks

and Bonds,” Journal of Financial Economics, 33, 3–56.

Fama, E. and MacBeth, J. D. (1973), “Risk, Return, and Equilibrium: Empirical

Tests,” Journal of Political Economy, 81, 607–636.

136 Bibliography

Fu, F. (2009), “Idiosyncratic Risk and the Cross-Section of Expected Stock Returns,”

Journal of Financial Economics, 91, 24–37..

Glosten, L., Jagannathan, R., and Runkle, E. (1993), “On the Relation Between the

Expected Value and the Volatility of the Nominal Excess Return on Stocks, ”

Journal of Finance, 48(5), 1779–1801.

Hamilton, J. (1994), “Time Series Analysis,” Princeton University Press

Hansen, B. (1994), “Autoregressive Conditional Density Estimation,” International

Economic Review, 35(3), 705–730.

Herskovic, B., Kelly, B., Lustig, H., and van Nieuwerburgh, S. (2014), “The Com-

mon Factors in Idiosyncratic Volatility: Quantitative Asset Pricing Implications,”

NBER Working Paper, 20076.

Jacquier, E., Polson, N. G., and Rossi, P. (1999), “Stochastic Volatility: Univari-

ate and Multivariate Extensions,” Technical Report, Society for Computational Eco-

nomics.

Jaeckel, P. and Rebonato, R. (2000), “The Most General Methodology for Creating

a Valid Correlation Matrix for Risk Management and Option Pricing Purposes, ”

Journal of Risk, 2 (2), 17–28.

Jagannathan, R. and Wang, Z. (1996), “The Conditional CAPM and Cross-section

of Expected Returns,” The Journal of Finance, 51(1), 3–53.

Jiang, J. and Tian, S. (2005), “The Model-Free Implied Volatility and Its Information

Content,” Review of Financial Studies, 18, 1305–1342.

Kupiec, P. (1995), “Techiniques for Verfying the Accuracy of Risk Measurement

Models,” Journal of Derivatives, 2, 173–184.

Lee, T. H. and Long, X. (2009), “Copula-based Multivariate GARCH Model with

Uncorrelated Dependent Errors,” Journal of Econometrics, 150(2), 207–218.

Lanne, M. and Saikkonen, P. (2007), “A Multivariate Generalized Orthogonal Factor

GARCH Model,” Journal of Business & Economic Statistics, 25, 61–75.

Ling, S. and Li, W. K. (1997), “Diagnostic Checking of Nonlinear Multivariate Time

Series with Multivariate ARCH Errors,” Journal of Time Series Analysis, 18, 447–

464.

Bibliography 137

Li, W. K. and Mak, T. K. (1994), “On the Squared Residual Autocorrelations in

Nonlinear Time Series with Conditional Heteroskedasticity,” Journal of Time Series

Analysis, 15, 627–636.

Liu, Y. and Xiao. X. (2016), “Factor Premia in Variance Risk,” Working Paper.

Markowitz, H. M. (1952), “Portfolio Selection,” Journal of Finance, 7, 77-91.

Marquering, W. and Verbeek, M. (2004), “The Economic Value of Predicting Stock

Index Returns and Volatility, ” Journal of Financial and Quantitative Analysis, 39(2),

407–429.

Mclnish, T. H. and Wood, R. A. (1992), “An Analysis of Intraday Pattern in Bid/Ask

Spreads for NYSE Stocks, ” Journal of Finance, 47(2), 753–764.

Merton, R.C. (1973), “An Intertemporal Capital Asset Pricing Model,” Econometrica,

41, 867–887.

Moskowitz, T. J., Ooi, Y. H., and Pedersen, L. H. (2012), “Time Series Momentum,”

Journal of Financial Economics, 104(2), 228–250.

Newey, W. K. and West, K. (1987), “A Simple, Positive Semi-definite, Heteroskedas-

ticity and Autocorrelation Consistent Covariance Matrix,” Econometrica, 59(3),

817–858.

Noureldin, D., Shephard, N., and Sheppard, K. (2012), “Multivariate High-

Frequency-Based Volatility (HEAVY) Model,” Journal of Applied Econometrics,

27(6), 907–933.

Patton, A. J. (2006), “Modeling Asymmetric Exchange Rate Dependence,” Interna-

tional Economic Review, 47(2), 527–556.

Rebonato, R. and Jackel, P. (2011), “The Most General Methodology to Create a Valid

Correlation Matrix for Risk Management and Option Pricing Purposes, Working Paper.

Ross, S. A. (1976), “The Arbitrage Theory of Capital Asset Pricing,” Journal of Eco-

nomic Theory, 13, 341–360.

Salvatierra, I. D. L. and Patton, A. J. (2014), “Dynamic Copula Models and High

Frequency Data,” Journal of Empirical Finance, 30, 120–135, 2014.

Schepsmeier, U. and Stober, J. (2014), “Derivatives and Fisher Information of Bi-

variate Copulas,” Statistical Papers, 55(2), 525–542, 2014.

138 Bibliography

Serban, M., Lehoczky, J., and Seppi, D. (2008), “Cross-sectional Stock Option Pric-

ing and Factor Models of Returns,” Working Paper, CMU.

Tse, Y. K. and Tsui, A. K. (2002), “A Multivariate Generalized Autoregressive Condi-

tional Heteroscedasticity Model with Time-varying Correlations,” Journal of Busi-

ness & Economic Statistics, 20, 351–362.

Vasquez, A. (2014), “Equity Volatility Term Structures and the Cross-Section of

Option Returns,” Working Paper.

van der Weide, R. (2002), “GOGARCH: A Multivariate Generalized Orthogonal

GARCH Model,” Journal of Applied Econometrics, 17, 549–564.

White, H. (1982), “Maximum Likelihood Estimation of Misspecified Models, ”

Econometrica, 50(1), 1–25.

Wood, R. A., McInish, T. H., and Ord, J. K. (1985), “An Investigation of Transaction

Data for NYSE Stocks, ” Journal of Finance, 40(3), 723–739.

The Tinbergen Institute is the Institute for Economic Research, which was founded

in 1987 by the Faculties of Economics and Econometrics of the Erasmus University

Rotterdam, University of Amsterdam and VU University Amsterdam. The Institute

is named after the late Professor Jan Tinbergen, Dutch Nobel Prize laureate in

economics in 1969. The Tinbergen Institute is located in Amsterdam and Rotterdam.

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