Dissertation
zur Erlangung des Grades
Doktor der Wirtschaftswissenschaften (Dr. rer. pol.) am Fachbereich Wirtschaftswissenschaften
der Universit¨at Konstanz
vorgelegt von:
Roxana-Mihaela Chiriac
Tag der m¨undlichen Pr¨ufung: 26. April 2010
1. Referent: Prof. Dr. Winfried Pohlmeier 2. Referent: Prof. Dr. Ralf Br¨uggemann
Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-117681
URL: http://kops.ub.uni-konstanz.de/volltexte/2010/11768/
The completion of this thesis would not have been possible without the contribution and support of many people, to whom I would like to express my deepest gratitude.
Many thanks go to Prof. Dr. Winfried Pohlmeier for his constant support, profes- sional advice and guidance from the very early stage of my studies in Germany. I am profoundly grateful to him for believing in me and giving me the chance of starting a professional career in the fields of econometrics. Moreover, I am very thankful to him for giving me the opportunity to work in the outstanding and stimulating academic environment of the Chair of Econometrics at the University of Konstanz.
His professionalism, enthusiasm and understanding have continuously inspired and enriched my growth as student, researcher and in general as person.
I also want to thank Prof. Dr. Ralf Br¨uggemann for his kindness in providing pro- fessional advice on many of my research questions. Many thanks go to Valeri Voev for the motivating and pleasant collaboration. To my colleagues Fabian Kr¨uger, Hao Liu, Sandra and Ingmar Nolte, R´emi Piatek, Derya Uysal and Laura Wichert I am very thankful for the friendly atmosphere and the support they offered to me when I needed it most. Special thanks go to Li Lidan for her effort on improving the English version of my thesis.
This work would not have been possible without the support of my family. I want to thank my parents for the continuous encouragement and for giving me ”roots and wings - roots to know where home is and wings to fly off and practice what has been taught” (Jonas Salk). Without the determination and ambition inspired by my father and the patience learnt from my mother I would have never succeeded.
Last, but not least I want to express my deepest gratitude to my fianc´e Andreas for accepting me as I am. His intelligence, kindness, calmness, patience, unconditional support and love have given to me much confidence and helped me to get back after very difficult times.
Introduction 8
Zusammenfassung 12
1 Properties of the Realized Volatility Wishart Autoregressive Model 16
1.1 Introduction . . . 16
1.2 The Model . . . 18
1.2.1 The Model under Stationarity Assumptions . . . 20
1.2.2 The Model under Nonstationarity Assumptions . . . 22
1.2.3 The Model under Specific Distributional Settings . . . 25
1.3 Data . . . 27
1.4 Empirical Results . . . 29
1.5 Conclusion . . . 32
Bibliography . . . 34
Appendix . . . 35
2 Modelling and Forecasting Multivariate Realized Volatility 42 2.1 Introduction . . . 42
2.2 Dynamic Conditional Covariance Models . . . 45
2.2.1 VARFIMA(p, d, q) Model . . . 46
2.2.2 Heterogeneous Autoregressive (HAR) Model . . . 49
2.2.3 Wishart Autoregressive (WAR) Model . . . 49
2.2.4 (Fractionally Integrated) Dynamic Conditional Correlation Model 50 2.3 Forecasting . . . 51
2.3.1 Forecasting with the VARFIMA Model . . . 53
2.3.2 Forecasting with the HAR Model . . . 54
2.3.3 Forecasting with the WAR Model . . . 55
2.4 Empirical Application . . . 56
2.4.1 Data . . . 57
2.4.2 Estimation Results . . . 58
2.4.3 Forecasting Results and Evaluation . . . 59
2.5 Conclusion . . . 64
Bibliography . . . 66
Appendix . . . 70
3 Forecasting Multivariate Volatility using the VARFIMA Model on Realized Covariance Cholesky Factors 77 3.1 Introduction . . . 77
3.2 The Model . . . 79
3.3 Empirical Applications . . . 80
3.3.1 Data . . . 80
3.3.2 Forecasting using the VARFIMA-Cholesky Model with Het- eroscedastic Error Terms . . . 82
3.3.3 Economic Evaluation of VARFIMA-Cholesky Forecasts: A Comparison Approach . . . 87
3.4 Conclusion . . . 93
Bibliography . . . 95
Appendix . . . 98
4 How Risky is the Value at Risk? 102 4.1 Introduction . . . 102
4.2 Set-Up of the Meta Study . . . 106
4.3 Meta Study Results . . . 112
4.4 Combining VaR Forecasts . . . 119
4.4.1 Conditional Coverage Optimization Method (CCOM) . . . 121
4.4.2 Conditional Quantile Optimization Method (CQOM) . . . 123
4.4.3 Empirical Results . . . 123
4.5 Conclusions . . . 129
Bibliography . . . 131
Appendix . . . 133
Complete Bibliography 137
1.1 Summary statistics of realized variances and realized covariances . . . 28 1.2 Estimation results of WAR(1) model, n= 3 . . . 31 1.3 Estimation results of WAR(1) model on different time samples, n= 3 31 2.1 Number of parameters for the general VARFIMA(p, d, q) model and
its restricted specifications . . . 49 2.2 RMSE based on the Frobenius norm of the forecasting error . . . 61 2.3 Annualized realized conditional standard deviations of ex-post global
minimum variance portfolios (in %) . . . 64 3.2 Annualized Sharpe ratios and standard deviations of out-of-sample
realized portfolio returns . . . 90 3.3 P-values of the LMW and KRS tests for 2nd order stochastic dominance 93 4.1 Backtest results for the calm period . . . 113 4.2 Backtest results for the crisis period . . . 114 4.3 Backtest results for the crash period . . . 115 4.4 Results of OLS Meta Regressions: Percentage number of violations . 117 4.5 Results of OLS Meta Regressions: P-value of conditional coverage test118 4.6 Percentage rate of violations based on CCOM, Assessment Stage 1 . . 126 4.7 Percentage rate of violations based on CQOM, Assessment Stage 1 . . 127 4.8 Percentage rate of violations, Assessment Stage 2 . . . 128
1.1 Kernel density of the ˆK estimator under stationarity assumptions, K = 3 . . . 21 1.2 Kernel density of the ˆK estimator under nonstationary assumptions,
K = 3 . . . 24 1.3 Kernel densities of realized volatilities . . . 29 2.1 Mean-variance plots . . . 63
Introduction
This dissertation investigates the topic of measuring financial risks and it comprises four stand-alone research papers on assessing, modelling and forecasting risks during different financial times. The existing risk measures largely ignore the systematic risk induced by correlations among financial assets, financial markets or financial agents and are increasingly exposed to model risks stemming from model misspecifi- cation, estimation risk or measurement error. In view of these facts, this thesis aims at extensively analyzing the sources of risks in measuring financial risks. Moreover, it aims at developing new approaches which appropriately measure the systematic dependencies among different financial variables and minimize the impact of model risk on the accuracy of resulting risk models.
The thesis is organized as follows: the first three chapters focus on analyzing and measuring financial systematic risks by means of parsimoniously modelling and ac- curately forecasting financial correlations within flexible multivariate frameworks.
Of particular interest is modelling and forecasting the joint dynamics of covariance matrix components, which are estimated based on high-frequency financial data and which are known in the literature as the “realized covariance” estimators. More pre- cisely, the first chapter investigates the performance of the Wishart Autoregressive (WAR) model, which is one of the first approaches in this direction, applied to a series of realized covariance matrices of stock returns. As an alternative to the WAR model, chapter two introduces a new approach of modelling and forecasting multi- variate realized volatilities that can be easily applied to covariance matrices of any dimension without imposing heavy parametrization structures. A thorough analysis of the empirical performance of this model subject to different evaluation criteria and market conditions is carried out in the third chapter of the dissertation. The last chapter deals with assessing the impact of model risk on the performance of popular univariate risk models and suggests new risk measures which remain robust
The first chapter thoroughly analyzes the properties of the WAR model proposed by Gourieroux, Jasiak & Sufana (2009) to capture the dynamics of realized covariance matrices on the basis of the sample variance-covariance matrix distribution, known in literature as the Wishart distribution, computed from i.i.d. multivariate autore- gressive Gaussian processes. Although easy to implement, this approach loses its interpretation as sum of squared autoregressive Gaussian processes when applied to a series of multivariate realized volatilities with large variance and volatility clus- tering. The main contribution of the chapter is to show by means of theoretical reasoning and empirical facts that there are irreconcilable discrepancies between the model assumptions and the statistical properties of the underlying data. This fact leads to estimates of the model’s degrees of freedom that describe degenerate Wishart distributions for the underlying covariance processes. Sound estimates of Wishart degrees of freedom are obtained from series with smooth dynamics, such as realized volatilities of interest and exchange rates or short samples of data, such as one month. However, when applied on large samples of realized covariances of stock returns, which usually exhibit large variation and volatility clustering, the standard estimators of Wishart degrees of freedom are inconsistent and converge to values close to zero. Moreover, the assumption of Wishart distributed realized covariance matrices leads to the assumption of Gamma distributed realized portfolio volatili- ties. The fact that this assumption is incompatible with the empirically observed log-normal distribution has a negative impact on the parameter estimates and their interpretation: the Wishart degrees of freedom, which can be computed directly from coefficients of variation, are consequently time dependent and are smaller than the dimension of the process during periods of large variation, thus describing de- generate Wishart distributions for the underlying covariance process.
The second chapter is a reprint of the article “Modelling and Forecasting Multivari- ate Realized Volatility”, which is jointly written with Valeri Voev and is forthcom- ing in the Journal of Applied Econometrics. This paper proposes a new approach of jointly modelling the dynamics of realized covariance matrices, which alleviates most of the drawbacks of existent multivariate volatility models: lack of parsimony, large parameter restrictions to guarantee positive definiteness of matrix forecasts, inflexible dependence patterns, etc. This new approach models the Cholesky fac- tors of realized covariance matrices by means of the Vector Fractionally Integrated Autoregressive Moving Average (VARFIMA) specification, which automatically as-
sures the positive definiteness of resulting forecasts, explicitly accounts for the long memory property of financial volatility and allows for the inclusion of exogenous predictive variables, such as: traded volume, interest rates, historical returns, etc.
Within a comprehensive empirical application, we show that this approach generates more precise multivariate volatility forecasts than other available specifications at various horizons and according to different evaluation criteria. Besides the smallest mean squared forecast errors, the new approach also provides portfolios with the best mean-variance trade-offs.
The empirical implementation of the model described in chapter two is based on a set of choices and assumptions, which are empirically tested in chapter three: first, regardless of the dimension of the covariance matrix, it involves only three param- eters to estimate. Second, it ignores the bias correction of the covariance matrix forecasts stemming from the non-linear transformation implied by the reverse op- eration of Cholesky decomposition. Third, the economic evaluation of covariance forecasts relies on heavy distributional assumptions of portfolio returns or certain functional forms of investors’ utility functions. In this paper, which is another joint work with Valeri Voev, we show that restricted versions of the model provide the best covariance forecasts in the absence of bias correction. Moreover, we show by means of stochastic dominance tests that any risk averse investor could achieve larger expected utilities from investing in portfolios optimally chosen on basis of volatility forecasts stemming from this model compared to the ones originating in alternative multivariate volatility models. This result holds for any return distribution or form of the utility function.
The fourth chapter of the thesis is a joint work with Winfried Pohlmeier and aims at assessing the robustness of one of the most popular risk measures, namely the Value at Risk (VaR), with respect to different sources of model risk before and during the recent financial crisis. We show that using sampling windows of at least two years of data is sufficient to accurately estimate the downward risks before the recent financial crisis. During turbulent times, accounting for the information on previous crises plays a crucial role in precisely estimating the future extreme losses.
Moreover, more parametrical model specifications and non-standard distributional assumptions, such as General Extreme Value distribution, significantly improve the
Furthermore, we bring valuable empirical evidence that the performance of VaR dif- fers across various degrees of market capitalization of the underlying stocks. As a result of the instability of VaR measures with respect to the model choice, distri- butional assumptions, estimation windows and asset choices, we propose two new approaches of measuring financial risks, which remain robust with respect to these dimensions. The new approaches consist of optimally combining two different VaR estimators by maximizing the coverage rate implied by the resulting VaR estimator or by minimizing the distance between population and empirical quantiles of under- lying returns. Regardless of the method, the optimal combination of VaR estimators radically improves the performance of the stand-alone estimators, especially of the ones often applied by practitioners, such as RiskMetrics or Historical Simulation.
Moreover, the new risk measures exhibit overall stability across the above-mentioned sources of risks and across stocks with different degrees of market capitalization.
Diese Dissertation untersucht die Messung finanzieller Risiken und besteht aus vier eigenst¨andigen Forschungspapieren ¨uber die Analyse, Modellierung und Vorhersage solcher Risiken in verschiedenen wirtschaftlichen Szenarien. Die gegenw¨artigen Risi- komaße ignorieren gr¨oßtenteils das systematische Risiko, das durch Korrelationen von Finanzanlagen, Finanzm¨arkten oder Finanzinvestoren induziert wird und sind in zunehmendem Maße Modellrisiken ausgesetzt, die infolge von falschen Modellspe- zifikationen, Sch¨atzrisiken oder Messfehlern entstehen. Vor diesem Hintergrund ver- sucht diese Arbeit, die Risiken in der Messung finanzieller Risiken ausf¨uhrlich zu analysieren. Dar¨uber hinaus wird versucht, neue Ans¨atze zu entwickeln, welche hinreichend genau die systematischen Zusammenh¨ange zwischen verschiedenen fi- nanziellen Variablen messen und den Einfluss von Modellrisiken auf die Pr¨azision von daraus resultierenden Risikomodellen minimieren.
Diese Arbeit ist wie folgt aufgebaut: Die ersten drei Kapitel konzentrieren sich auf die Analyse und Messung systematischer finanzieller Risiken durch sparsame Model- lierung und genaue Prognose finanzieller Korrelationen innerhalb flexibler multivari- ater Rahmenbedingungen. Von besonderem Interesse ist dabei die Modellierung und Vorhersage der gemeinsamen Dynamiken von Kovarianzmatrizen, welche auf Basis von hochfrequenten Finanzdaten gesch¨atzt werden und in der Literatur unter dem Begriff “realized covariance” bekannt sind. Das erste Kapitel untersucht anhand einer Zeitreihe realisierter Kovarianzmatrizen von Aktienrenditen die Performanz des Wishart Autoregressive (WAR) Modells, das einen der ersten Ans¨atze in dieser Richtung darstellt. Als eine Alternative zum WAR - Modell stellt Kapitel 2 einen neuen Ansatz zur Modellierung und Prognose multivariater realisierter Volatilit¨aten vor, welcher problemlos auf Kovarianzmatrizen jeglicher Dimension angewandt wer- den kann, ohne dabei eine zu starke Parametrisierung einzuf¨uhren. Eine gr¨undliche Analyse der empirischen Performanz des Modells nach Maßgabe verschiedener Be-
sertation durchgef¨uhrt. Das letzte Kapitel besch¨aftigt sich mit der Bewertung des Einflusses von Modellrisiken auf die Performanz g¨angiger univariater Risikomodelle und schl¨agt neue Risikomaße vor, die robust gegen verschiedene Risikoquellen sind.
Das erste Kapitel analysiert die Eigenschaften des von Gourieroux, Jasiak & Sufana (2009) entworfenen WAR - Modells zur Erfassung der Dynamik realisierter Kovari- anzmatrizen auf Basis der Verteilung der Stichproben - Varianz - Kovarianzmatrix, die in der Literatur als Wishart - Verteilung bekannt ist. Diese setzt sich aus un- abh¨angig und identisch verteilten, multivariaten autoregressiven Gauß - Prozessen zusammen. Obwohl leicht umzusetzen, verliert dieser Ansatz seine Interpretation als Summe quadrierter autoregressiver Gauß - Prozesse, wenn er auf eine Zeitreihe multi- variater realisierter Volatilit¨aten mit großer Varianz und Volatilit¨atsclustering ange- wandt wird. Der Hauptzweck dieses Kapitels ist es, anhand theoretischer Diskussio- nen und empirischer Gegebenheiten zu zeigen, dass unl¨osbare Diskrepanzen zwischen den Modellannahmen und den statistischen Eigenschaften der zugrunde liegenden Daten bestehen. Dieser Umstand f¨uhrt zu Sch¨atzungen von Modellfreiheitsgraden, die degenerierte Wishart - Verteilungen der zugrunde liegenden Kovarianzprozesse implizieren. Zuverl¨assige Sch¨atzungen der Wishart - Freiheitsgrade erh¨alt man von Zeitreihen mit gleichm¨aßiger Dynamik, wie zum Beispiel realisierten Volatilit¨aten von Zinsraten oder Wechselkursen, sowie durch Datenstichproben von kurzer L¨ange, wie etwa einem Monat. Allerdings sind die Standardsch¨atzer von Wishart - Frei- heitsgraden inkonsistent und konvergieren zu Werten nahe Null, wenn sie auf große Stichproben realisierter Kovarianzen von Aktienrenditen angewandt werden, welche
¨ublicherweise große Varianz und Volatilit¨atsclustering aufweisen. Dar¨uber hinaus f¨uhrt die Annahme Wishart - verteilter realisierter Kovarianzmatrizen zu der An- nahme dass, die realisierten Portfoliovolatilit¨aten Gamma - verteilt sind. Dass diese Annahme inkompatibel mit der empirisch beobachtbaren Log - Normal Verteilung ist, hat negative Auswirkungen auf die Parametersch¨atzungen und deren Interpre- tation: Die Wishart - Freiheitsgrade, die anhand der Variationskoeffizienten direkt berechnet werden k¨onnen, sind folglich zeitabh¨angig und innerhalb von Perioden großer Variation kleiner als die Dimension des Prozesses. Sie beschreiben daher degene-rierte Wishart - Verteilungen der zugrunde liegenden Kovarianzprozesse.
Das zweite Kapitel ist ein Nachdruck des Artikels “Modelling and Forecasting Multi- variate Realized Volatility”, welcher zusammen mit Valeri Voev verfasst wurde und
in K¨urze im “Journal of Applied Econometrics” erscheinen wird. Dieses Papier stellt einen neuen Ansatz f¨ur die gemeinsame Modellierung der Dynamik realisierter Kovarianzmatrizen vor, welcher die meisten Nachteile gegenw¨artiger multivariater Volatilit¨atsmodelle behebt: Starke Parametrisierung, un¨ubersichtliche Parameter- restriktionen zur Sicherung positiver Definitheit von Matrixvorhersagen, unflexible Abh¨angigkeitsstrukturen, etc. Der vorgeschlagene Ansatz modelliert die Cholesky - Faktoren realisierter Kovarianzmatrizen anhand der Vector Fractionally Intergrated Autoregressive Moving Average (VARFIMA) Spezifikation, welche automatisch die positive Definitheit resultierender Vorhersagen sichert, explizit der Langzeitged¨acht- nis - Eigenschaft von Finanzvolatilit¨at Rechnung tr¨agt und die Einbeziehung exo- gener Variablen, wie zum Beispiel Handelsvolumina, Zinss¨atzen, historischen Ren- diten, etc. erlaubt. Im Rahmen einer umfassenden empirischen Anwendung zeigen wir, dass dieser Ansatz pr¨azisere multivariate Volatilit¨atsvorhersagen als andere ver- f¨ugbare Spezifikationen zu verschiedenen Zeithorizonten und unterschiedlichen Be- wertungskriterien generiert. Neben den kleinsten quadratischen Prognosefehlern f¨uhrt der neue Ansatz zu Portfolios mit dem besten Verh¨altnis aus Risiko und Ren- diten.
Die in Kapitel 2 beschriebene empirische Anwendung des Modells basiert auf einer Reihe von Modellierungsentscheidungen, welche in Kapitel 3 empirisch erarbeitet werden: Das Modell beinh¨alt – unabh¨angig von der Dimension der Kovarianzma- trix – lediglich nur drei Parametern; es ignoriert zudem eine Bias - Korrektur, die aus theoretischer Sicht notwendig ist, da nicht die Kovarianzmatrix selbst, sondern dessen Cholesky - Zerlegung modelliert wird. Dar¨uber hinaus basiert die ¨okonomi- sche Auswertung der Kovarianz - Vorhersagen auf starken Verteilungsannahmen von Portfoliorenditen oder bestimmten Funktionsformen der Nutzenfunktionen von In- vestoren. In dieser Arbeit, die ebenfalls in Zusammenarbeit mit Valeri Voev ent- stand, zeigen wir, dass restringierte Versionen des Modells ohne Bias - Korrek- tur die besten Kovarianz - Vorhersagen bieten. Dar¨uber hinaus zeigen wir durch Tests auf stochastische Dominanz, dass jeder risikoaverse Investor einen h¨oheren Erwartungsnutzen erreichen k¨onnte, wenn er in optimale Portfolios investiert, die auf Basis dieses Modells ausgew¨ahlt wurden. Die Aussagekraft dieses Ergebnisses beruht darauf, dass sie weder eine Wahrscheinlichkeitsverteilung der Renditen, noch eine spezifische Nutzenfunktion des Investors annehmen.
Das vierte Kapitel dieser Dissertation ist eine gemeinsame Arbeit mit Winfried Pohlmeier und unternimmt eine Bewertung der Robustheit eines in der Praxis po- pul¨aren Risikomaßes, des Value at Risk (VaR), hinsichtlich verschiedener Quellen von Modellrisiko vor und w¨ahrend der aktuellen Finanzkrise. Wir zeigen, dass ein Stich- probenfenster von mindestens zwei Jahren gen¨ugt, um genaue Sch¨atzungen des Ab- w¨artsrisikos vor der aktuellen Finanzkrise zu erlangen. In turbulenten Zeiten spielt die Ber¨ucksichtigung von Informationen aus fr¨uheren Krisen einer wichtigen Rolle bei der Sch¨atzung zuk¨unftiger extremer Verluste. Zudem erh¨ohen parametrische Modellspezifikationen und nicht - standard Verteilungsannahmen, wie zum Beispiel die Extremwertverteilung, die Genauigkeit von VaR - Sch¨atzungen in Zeiten in de- nen diese am meisten gebraucht werden. Weiterhin pr¨asentieren wir empirische Belege daf¨ur, dass sich die Performanz des VaR zwischen verschiedenen Graden der Marktkapitalisierung der zugrundeliegenden Aktien unterscheidet. Aufgrund der Instabilit¨at des VaR - Maßes im Hinblick auf Modellwahl, Verteilungsannahmen, Sch¨atzungszeitraum und Wahl des Anlagegegenstandes pr¨asentieren wir zwei neue Ans¨atze zur Messung finanzieller Risiken, welche robust bez¨uglich dieser Dimen- sionen sind. Die neuen Ans¨atze bestehen aus der optimalen Kombination zweier verschiedener VaR - Sch¨atzer, die durch Maximierung der durch den resultierenden VaR - Sch¨atzer implizierten Deckungsrate oder durch Minimierung des Abstands zwischen den Quantilen der Grundgesamtheit und den empirischen Quantilen der zugrundeliegenden Renditen erreicht wird. Unabh¨angig von der Optimierungsme- thode verbessert die Kombination der VaR - Sch¨atzer die Performanz der individu- ellen Sch¨atzer. Dies gilt insbesondere f¨ur Sch¨atzer die in der Praxis oft verwendet werden, wie etwa RiskMetrics oder Historical Simulation. Dar¨uber hinaus weisen die Ergebnisse der neuen Risikomaße eine umfassende Stabilit¨at in Hinblick auf die oben erw¨ahnten Risikoquellen und Aktien mit verschiedenen Graden der Marktka- pitalisierung auf.
Properties of the Realized
Volatility Wishart Autoregressive Model
1.1 Introduction
Modelling multivariate volatility is of particular relevance in financial applications such as risk management, portfolio management and asset pricing. In the existing literature, the most applied multivariate volatility models are multivariate general- ized autoregressive conditional heteroscedasticity (MGARCH) models (see Bauwens, Laurent & Rombouts (2006)) and multivariate stochastic volatility models, MSVM (reviewed in Asai, McAleer & Yu (2006)). MGARCH and MSVM models tradi- tionally use daily data to model the unobserved daily volatility. One drawback of these models is the estimation, which is often not easy due to the large number of parameters involved, or to the complexity of constraints imposed to assure pos- itive definitiveness (positivity) of the covariance matrix. As an alternative to the MGARCH and MSVM models, the multivariate realized (co)variance (RCoV) mod- els (reviewed in McAleer & Medeiros (2008)) have been developed in recent years;
the RCoV models exploit the information contained in high-frequency data in or- der to obtain highly precise estimates of the covariance of the underlying assets.
There are only a few approaches (Bauer & Vorkink (2007), Chiriac & Voev (2009)) which model the time series of the observed RCoV matrices, preserve their positivity and allow for a flexible dynamic specification. Gourieroux et al. (2009) proposed the
an observed multivariate RCoV series on the basis of the sample variance-covariance matrix distribution, known in the literature as the Wishart distribution, computed from i.i.d. multivariate Gaussian processes. One important property of the model, is that the number of Gaussian processes, which gives the number of the degrees of freedom (d.f.) of the Wishart distribution, has to be larger than the dimension of the matrix process in order to describe a non-degenerate WAR process for the underlying data. However, empirical results from applying the model to series of re- alized covariance matrices of stock returns reveal degenerate Wishart distributions of the underlying covariance matrix series.
Except for a brief empirical illustration of the model in Gourieroux et al. (2009), there are just a few studies which analyze the WAR model beyond its theoretical specification since it was introduced in the literature: Bonato (2009) brings em- pirical evidence that extreme observations in the covariance matrix process induce Wishart d.f. estimates, which are much smaller than the dimension of the process;
Chiriac & Voev (2009) and Bonato, Caporin & Ranaldo (2009) empirically apply the model in forecasting applications. Chiriac & Voev (2009) use this framework within a horse race study, where the forecasting qualities of different multivariate models are compared, by means of economic criteria. In their study, Chiriac & Voev (2009) refrain from reporting and interpreting the estimation results on the d.f. and simply apply the flexible autoregressive framework of the WAR(1) model to forecast daily covariance matrices. Bonato et al. (2009) obtain estimates of the d.f. larger than the dimension of the process when applying the WAR framework on series with smooth dynamics, such as realized volatilities of U.S. treasury bond future and exchange rate returns.
In this paper, we aim at solving the puzzle of estimating degenerate Wishart pro- cesses from series of realized covariance matrices of stock returns, which are subject to increased volatility and volatility clustering. We show that inconsistencies be- tween the model assumptions and the statistical properties of the underlying data may lead to estimates of Wishart d.f., which are smaller than the dimension of the process. We derive the properties of the Method of Moment estimator of the Wishart d.f., highlighted by Gourieroux et al. (2009), under different stationarity settings and we show that nonstationary (cointegrated) autoregressive Gaussian processes offer the appropriate theoretical setting for describing the empirical results. More-
over, we show that violations of the distributional assumptions of the underlying data severely affect the estimation results and the interpretation of the model.
The paper is structured as follows: Section 1.2 describes the model under station- arity, nonstationarity (cointegration) and different distributional settings. Section 1.3 presents the data used in the empirical application and Section 1.4 reports the estimation results from applying the WAR model. Section 1.5 concludes.
1.2 The Model
The notation in this section follows partially that used by Gourieroux et al. (2009).
After briefly presenting the model as it was introduced by Gourieroux et al. (2009), we describe its properties under stationarity (Section1.2.1) and nonstationarity and cointegration conditions (Section 1.2.2), as well as under specific distributional set- tings of the underlying data (Section1.2.3).
Letxk,t withk = 1, . . . , K be independent Gaussian VAR(1) processes of dimension n×1 with the same autoregressive parameter matrix M and innovation variance Σ:
xk,t =Mxk,t−1+εk,t εk,tiid
∼N (0,Σ), k = 1, . . . , K. (1.1)
Gourieroux et al. (2009) define the processYt of dimension n×n given by:
Yt= XK
k=1
xk,tx0k,t, (1.2)
to be a Wishart Autoregressive process of order 1 and dimension n, denoted by:
Wn(K, M,Σ). The process Yt is Wishart distributed with K d.f., latent autoregres- sive matrix M and innovation covariance matrix Σ. An important property of the Wishart distribution is that the matricesYt are positive semi-definite, if and only if K ≥n. Whenever K < n, the process Yt is given by a sequence of singular covari- ance matrices with degenerate Wishart distribution.
From Equation (1.1), we can derive the conditional distribution ofx :
where=t−1represents the information set consisting of all relevant information up to and includingt−1. It is necessary to mention that there is no financial or economic interpretation of the latent process xk,t, while Yt is an observed process. Thus, the specification of Yt in Equation (1.2) has the aim of expressing Yt theoretically in a quadratic form and of simplifying the mathematical proofs of its properties. Given that the processxk,t has a conditional mean not equal to zero, the process Yt has a non-centered Wishart probability density function. The non-centered Wishart dis- tribution has a density function when K > n−1. Moreover, the WAR(1) process can be extended to non-integer K.
The following Proposition gives the first two unconditional moments ofYtas a func- tion of the unconditional moments of xk,t:
Proposition 1.2.1. Denote V [xk,t] ≡ Σ(∞) and let α be a vector of dimension n×1. Then xk,tiid∼N (0,Σ(∞)) and
E [Yt] =KΣ(∞)≡Σ∗(∞) (1.4)
V [α0Ytα] = 2[α0Σ∗(∞)α]2
K (1.5)
Proof. See Appendix A.1.1.
Based on the results of previous proposition and given a sequence of covariance matrices Y1, . . . , YT, we can estimate K from the unconditional moments of Yt as follows (see Appendix A.1.1):
KˆM M(α) = 2 ˆE[α0Ytα]2
Vˆ[α0Ytα] = 2[α0Σˆ∗(∞)α]2
Vˆ[α0Ytα] , (1.6) whereαis of dimensionn×1 and interpreted as a portfolio allocation (see Gourieroux et al. (2009)). This estimator, denoted the Method of Moments (MM) estimator (Gourieroux et al. (2009)), is feasible, but not efficient, given that it depends on the choice of α. However, this paper concentrates on determining why this estimator takes values smaller than n and therefore it mainly focus on its consistency and less on its efficiency. Consequently in what follows we derive the asymptotic properties of the MM estimator, subject to different stationarity assumptions of the latent autoregressive Gaussian processes xk,t and distributional settings of the process Yt.
1.2.1 The Model under Stationarity Assumptions
The process defined in Equation (1.1) is (strictly) stationary, i.e. the WAR(1) process defined in Equation (1.2) is (strictly) stationary, if and only if the matrix M has roots with modulus (strictly) less than 1. The process Yt from Equation (1.2) can be written as:
Yt = XK
k=1
xk,tx0k,t = XK
k=1
(Mxk,t−1+εk,t)(Mxk,t−1+εk,t)0
= XK
k=1
(Mxk,t−1x0k,t−1M0+Mxk,t−1ε0k,t+εk,tx0k,t−1M0+εk,tε0k,t)
= MYt−1M0+ XK
k=1
Mxk,t−1ε0k,t+ XK
k=1
εk,tx0k,t−1M0+ XK
k=1
εk,tε0k,t (1.7)
and its conditional mean with respect to =t−1 is:
E [Yt|=t−1] =MYt−1M0+KΣ. (1.8) Moreover, under stationarity conditions, the unconditional mean of Yt is given by:
E [Yt] = ME [Yt−1]M0+ XK
k=1
ME£
xk,t−1ε0k,t¤ +
XK
k=1
E£
εk,tx0k,t−1¤ M0+
XK
k=1
E£
εk,tε0k,t¤ .
Since E [xk,t] = 0n×1, then
E [Yt]≡Σ∗(∞) =KΣ(∞) =MΣ∗(∞)M0+KΣ = KMΣ(∞)M0+KΣ. (1.9) Consequently Σ(∞)≡V [xk,t] satisfies the following condition:
Σ(∞) =MΣ(∞)M0 + Σ. (1.10)
Proposition 1.2.2. If xkt are stationary (k= 1, . . . , K), then the MM estimator of the Wishart d.f. is consistent and asymptotically normal distributed with:
√T( ˆK−K)→d N(0, X∞
j=−∞
γj), (1.11)
P
Figure 1.1: Kernel density of ˆK estimator under stationarity assumptions, the case ofK= 3
Figure 1.1 plots the kernel density of the estimated Wishart d.f. stemming from a Monte Carlo study where we simulate a total of 100000 WAR(1) processes of di- mension 2×2 defined on stationary latent processes described by an autoregressive matrix M with the roots in modulus strictly smaller than one1. Figure 1.1 shows that the distribution of ˆK is approximately normal2 with the mean approximately equal to the true value, K = 3, which confirms the theoretical results described above.
From Proposition 1.2.2, we can conclude that the stationarity assumption is sufficient for obtaining consistent estimators of the Wishart d.f. Thus whenever K > n, then (asymptotically) ˆK > n and ˆK describes a non-degenerate Wishart process for the underlying covariance matrix process. But is the stationarity assumption also necessary? We answer this question by relaxing this assumption and by considering latent autoregressive processes with unit roots, as described below.
1For this Monte Carlo study we arbitrarily chooseM =
µ 0.8 −0.1 0.4 0.3
¶
, with the roots 0.7 and 0.4 and Σ =
µ 2.5 −1.3
−1.3 7
¶ .
1.2.2 The Model under Nonstationarity Assumptions
We assume that the process xk,t defined in Equation (1.1) is not stationary and follows a multivariate random walk of the form:
xk,t =xk,t−1+εk,t εk,tiid
∼N (0,Σ). (1.12)
In this case, the processYt can be written as:
Yt = XK
k=1
xk,tx0k,t = XK
k=1
(xk,t−1+εk,t)(xk,t−1 +εk,t)0
= XK
k=1
(xk,t−1x0k,t−1+xk,t−1ε0k,t+εk,tx0k,t−1+εk,tε0k,t)
= Yt−1+ XK
k=1
xk,t−1ε0k,t+ XK
k=1
εk,tx0k,t−1+ XK
k=1
εk,tε0k,t (1.13)
and the conditional expectation with respect to =t−1 is:
E [Yt|=t−1] =Yt−1+KΣ (1.14) Consequently the process Yt can be written as:
Yt=Yt−1+KΣ +ξt. (1.15)
whereξtis a heteroscedastic error term of dimensionn×1 with zero conditional mean.
Rewriting Equation (1.15) recursively, we derive the unconditional expectation ofYt
to be:
E [Yt] =tKΣ≡Σ∗(∞) = KΣ(∞), (1.16) from where we can deduce that under unit root conditions:
Σ(∞) =tΣ. (1.17)
Proposition 1.2.3. If xkt have unit root (k = 1, . . . , K), then the MM estimator of
the Wishart d.f. is inconsistent and:
Kˆ →d K Z1
0
[W(s)]2ds, (1.18)
where W(·) stands for the standard Brownian motion.
Proof. See Appendix A.1.3.
From Equation (1.18), we notice that the estimator of the Wishart d.f., ˆK, derived under unit root assumptions, converges in distribution to a random variable with expectation smaller than K:
E
K Z1
0
[W(s)]2ds
= K Z1
0
E£
[W(s)]2¤
ds=K Z1
0
s ds= K
2 < K (1.19) Even though one may assume that K is larger than n, the WAR(1) process is not guaranteed to be non-degenerate, given that ˆK might be (asymptotically) smaller than n. This is not the case whenever K2 ≥n, which is a very restrictive condition.
Figure1.2 plots the distribution of ˆK given in Equation (1.18) forK = 3 andn= 2, based on 100000 simulations. It shows that the asymptotic distribution of ˆK is extremely right-skewed.
The model specification described so far implies that each element of the Yt matrix follows a random walk (see Equation (1.15)). However, when applying the WAR(1) framework on series of daily realized covariance matrices, this assumption is no longer realistic: realized volatilities generally follow a stationary process with slowly decaying autocorrelations (long-memory). Such series usually exhibit excess prob- ability on extreme observations, which is reflected by large, sometimes exploding, values of their variances. Nevertheless, the empirical evidence presented in Section 1.4 indicates that estimated Wishart d.f. smaller than the dimension of the process are always accompanied by the fact that we cannot reject at 5% that the rank of the matrix In−M is smaller than n.
Denote H =In−M. In the vector autoregressive framework, this result indicates that the latent autoregressive structurexk,t might be cointegrated with the following
Figure 1.2: Asymptotic distribution of ˆK estimator under nonstationary assumptions, the case ofK= 3
vector error correction representation:
∆xk,t = −Hxk,t−1 +εk,t εk,tiid∼N (0,Σ), (1.20)
where rank(H) = r < n. The autoregressive matrix M from Equation (1.1) be- comes M =In−H. Consequently the process xk,t has the following autoregressive representation:
xk,t = (In−H)xk,t−1+εk,t. (1.21)
Given that the matrix H has rank r, it can be written as the product of two full rank matrices B and Γ, both of dimension n×r, H = BΓ0. Therefore there exists a non-zero matrix C of dimension n×r such thatC0B = 0(r,r), i.e. C is orthogonal to B.
Proposition 1.2.4. The process C0YtC of dimensionr×r is a random walk process with conditional mean given by
E [C0YtC|=t−1] =C0Yt−1C+KC0ΣC (1.22) and unconditional mean given by
Proof. See Appendix A.1.4.
Moreover, the asymptotic properties of the MM estimator under the cointegration assumption are given by the following proposition:
Proposition 1.2.5. Let xkt be cointegrated of rank r as given in Equation (1.20).
Then the MM estimator of the Wishart d.f. is inconsistent and:
Kˆ →d K Z1
0
[W(s)]2ds, (1.24)
where W(·) stands for the standard Brownian motion.
Proof. See Appendix A.1.5.
Similar to the general non-stationary case, under cointegration assumptions, the estimated Wishart d.f. converge to values smaller than the true value, and often smaller than the dimension of the process. In such cases, the estimated Wishart d.f.
describe degenerate Wishart distributions of non-positive definite matrix processes.
Particulary, if the cointegration rank is equal to 1 (r= 1), then H =BΓ0 has rank 1 and we can write Γ = (1, γ2, . . . , γn)0 and B = (1,0, . . . ,0)0. The matrix C of di- mension n×1 orthogonal to matrix B, can be written as C= (0, α2, . . . , αn)0, with α2, . . . , αn6= 0 and the process C0YtC, which is a linear combination of Yt elements, C0YtC = α22Y22,t+. . .+α2nYnn,t follows a random walk with drift whose variance increases in time. Although arguable, this theory is confirmed in our empirical ap- plication presented in Section1.4. In what follows we discuss the result of estimating Wishart d.f. smaller than the dimension of the covariance process from the empirical perspective of using data which features distributional properties different from the ones required by the model.
1.2.3 The Model under Specific Distributional Settings
In real terms, the WAR(1) model is estimated directly from a series of observed re- alized covariance matrices (Y1, . . . , YT), which generally exhibits stationarity. Given that E [Yt] = Σ∗(∞), the MM estimator from Equation (1.6) can be derived from
the empirical moments of Yt as follows:
KˆM M(α) = 2 ˆE[α0Ytα]2
Vˆ[α0Ytα] = 2 µ√
Vˆ[α0Ytα]
E[αˆ 0Ytα]
¶2 = 2 ˆ
c2v(α0Ytα), (1.25)
wherecv stands for the coefficient of variation. Defined as the ratio of the standard deviation to the mean, the coefficient of variation compares the degrees of variation among data with different means. Given that the mean of realized volatilities is always positive and in general different from zero, the coefficient of variation is a coherent measure in analyzing the magnitude of estimated Wishart d.f.
Bonato (2009) shows that if Yt of dimension n×n follows a Wishart process with parameters K, M and Σ, then for any α6= 0, Yt∗ = α0Ytα follows a Gamma distri- bution with Yt∗ ∼ Ga¡K
2,2α0Σ(∞)α¢
, where Σ(∞) is defined in Proposition 1.2.1.
In this case:
E[Yt∗] = K
22α0Σ(∞)α (1.26)
V[Yt∗] = K
2(2α0Σ(∞)α)2 (1.27)
The coefficient of variation of a Gamma distributed variable is constant and is given by:
cv(Yt∗) =
pV[Yt∗] E[Yt∗] =
√2
√K, (1.28)
which indicates that the estimator from Equation (1.25) is valid only under the assumption that the true data generating process is a WAR(1) or it follows a distri- bution with the coefficient of variation equal to that of the Gamma distribution.
However when applying the WAR(1) framework to series of daily realized volatil- ities, this condition is generally not fulfilled. Numerous studies (Taylor (1986), Andersen, Bollerslev, Diebold & Ebens (2001) and Andersen, Bollerslev, Diebold &
Labys (2001) among others) show by means of empirical evidence that daily log real- ized volatilities are approximately normal distributed, which implies that the series of daily realized volatilities are approximately log-normal distributed. The corre- sponding coefficient of variation of log-normal distributed variables Yt∗ is given by
√
in Equation (1.25), we get that the estimated d.f. are equal to 2
eVˆ[ln(Yt∗)]−1 and are also time varying.
Moreover, subject to log-normal distributed realized volatility processes, the esti- mated K will be larger than the dimension n of the process if only if ˆV[ln(Yt∗)] <
ln¡2
n+ 1¢
. It implies that for a realized covariance process of dimension 2×2, the variance of the log-realized volatilities should be smaller thanln(2) ≈0.693 in order to estimate a non-degenerate WAR(1) processes on the series under considerations.
This is a very restrictive condition, especially for portfolios which include highly volatile stocks, such as Citigroup, or especially during turbulent periods such as the recent financial crisis, when the variance of the log-realized volatility estimator has reached extreme values (e.g. the variance of log-realized volatility of Citigroup was around 1.34).
From the empirical perspective, the implied matrix autoregressive process can be interpreted as a WAR(1) process only when either the portfolio under consideration entails stocks with low return volatility or the estimation period is characterized by small variation in the logarithm transformation of realized volatilities. These types of conditions are in particulary fulfilled in the empirical applications carried out by Gourieroux et al. (2009) and Bonato et al. (2009), who find estimates of the d.f. larger than the dimension of the process: Gourieroux et al. (2009) use one month of daily realized covariance matrices from October 1998, which represents an example of typical calm period with no extreme large financial events; after removing some extreme observations, Bonato et al. (2009) apply the WAR model to a series of realized covariance matrices of a portfolio of bonds and exchange rates, whose returns usually exhibit less variation than the ones of financial stocks. In what follows, we analyze the adequacy of the WAR model when applied to capture the dynamics of realized covariance matrices of stock returns, subject to asset choice and estimation window in the light of the theoretical results derived so far.
1.3 Data
The data is taken from the NYSE Trade and Quotations (TAQ) database and con- sists of tick-by-tick bid and ask quotes sampled from 9:45 until 16:00 over the period
January 1, 2001 to June 30, 2006 (1381 trading days).3 Furthermore, we select the following 6 stocks: American Express Inc. (AXP), Citigroup (C), Home Depot Inc. (HD), Hewlett-Packard (HWP), International Business Machines (IBM) and JPMorgan Chase & Co (JPM). All stocks are traded on the NYSE and their high liquidity motivates the choice. Before proceeding to construct the equally-spaced returns, we clean the data of erroneous entries and construct the mid-quotes series, by averaging over the bid and ask quotes. In order to get a regularly spaced sequence of mid-quotes, we use the previous-tick interpolation method. The mid-quotes are then sampled at 5-minute and daily frequencies4, from which we construct the 5- minute and daily log returns. This leads to 75 intraday observations available for computing the realized variance-covariance matrices for each day by following the procedure described by Andersen, Bollerslev, Diebold & Ebens (2001).
Table 1.1: Summary statistics of realized variances and realized covariances. The table reports the average values of descriptive statistics of realized variances and covariances across the six stocks considered.
Realized variances Realized covariances
Mean 3.03×10−3 1.03×10−3
Maximum 0.01546 0.00404
Minimum 1.17×10−5 82×10−5
Std. dev 0.00063 0.00021
Skewness 13.1 8.6
Kurtosis 297.3 153.1
Table 1.1 reports the average values of the summary statistics of realized variances and covariances of the six stocks considered. Both realized variance and covariance distributions are extremely right skewed and leptokurtic, as already documented by Andersen, Bollerslev, Diebold & Ebens (2001).
In Figure 1.3 we plot the kernel density of the realized volatilities of the six stocks along with the fitted log-normal distribution and fitted Gamma distribution. One can see that the log-normal distribution better fits the distribution of realized volatil- ities than the Gamma distribution does. Bonato et al. (2009) obtain similar results from plotting the density of realized volatilities of portfolios of bonds and exchange rates against the log-normal and fitted Gamma distribution.
3Although the NYSE opens at 9:30, we filter out the quotes recorded in the first 15 minutes in order
Figure 1.3: Kernel densities of realized volatilities. Dashed lines correspond to the fitted Gamma distributions, dotten lines correspond to fitted log-normal distributions and solid lines are the kernel densities of the underlying series.
1.4 Empirical Results
We estimate the model forn= 3 assets5 within different combinations of the selected stocks, by means of the nonlinear least square procedure as suggested by Gourieroux et al. (2009), which provides the MM estimator from Equation (1.6) and which is described at length in the Appendix A.1.6. Table 1.2reports estimation results for the stock combinations HD-C-IBM, HWP-C-IBM and HWP-JPM-HD with equal portfolio allocation, α = (1,1,1). The results are similar across the pairs of stocks.
In particular the Wishart d.f. are captured by values of the K-parameter around 0.9. The fact that the estimated d.f. are smaller than the dimension of the process indicates that the covariance matrix process follows a degenerate Wishart distribu- tion. Following the theoretical findings described in Section 1.2.2, such an outcome could be the result of a cointegrated latent structure of the estimated matrix process, which on real data materializes in large variance of the underlying series. The fact that across stock combinations we cannot reject at 5% significance level that the determinant of matrix ˆH ≡ I3 −Mˆ is zero (det( ˆH) = 0.04 with p-value6 of 0.499), indicates that the rank of ˆH is smaller than 3. This leads to the presumption that
5In order to make the results comparable with the ones of Gourieroux et al. (2009), we choose the same number of stocks.
the latent process might be cointegrated, which is further mirrored in the increased variance of the underlying series. As a result we focus on analyzing how the variance magnitude of realized volatilities influence the estimation outcomes, in particular the estimated value of the Wishart d.f. As a result, we carry out the above analysis on RCoV samples of different length, and thus of different variances.
Table 1.3 reports estimation results for the same stock combinations on samples covering time periods of one month of data (May 2006) containing 22 observations, one year of data (year 2003) containing 252 observations and six and a half years of data (from 01.01.2001 until 30.06.2006) containing 1381 observations. Table 1.3 also reports the variance of the diagonal elements of the covariance matrices for each sample period and the coefficients of variation of portfolio volatilities. The entries of Table 1.3 reveal that the variance of the underlying series plays a considerable role in the estimation and, thus, confirms the theoretical results derived in Section 1.2.2: during short periods, which are generally characterized by a small variance, the estimated d.f. take values larger than the dimension of the process.
However, on longer horizons, characterized by exploding variances, the estimated d.f. characterize degenerate Wishart distributions. Large variances characteristic to longer horizons are generally triggered by extreme events, such as September, 11, 2001. These results are in line with the empirical findings of Bonato (2009), who shows that extreme observations in the covariance matrix process induce bias towards zero of the estimated d.f.
Table 1.3 also reports the estimated values of the corresponding coefficients of vari- ation, which vary with the sampling window. This fact confirms the theoretical results derived in Section 1.2.3. Between the estimates based on one month of data and the ones on one year of data, there is no considerable difference, which indicates that assuming a Gamma distribution for the two periods is a realistic choice. Con- sequently, the corresponding d.f. are estimated to be larger than the dimension of the process, which indicates that in this case the WAR(1) is a reasonable choice for modelling series of realized covariance matrices.
Table 1.2: Estimation results of WAR(1) model,n= 3. P-values are reported in the brackets.
The values of Sigmai,j, i, j= 1, . . . ,3 are scaled up by 104.
Portfolio HD-C-IBM HWP-C-IBM HWP-JPM-HD
M11 0.635 (0.000) 0.389 (0.000) 0.478 (0.000)
M12 0.017 (0.286) 0.223 (0.000) 0.052 (0.077)
M13 0.223 (0.000) 0.501 (0.000) 0.454 (0.000)
M21 0.437 (0.000) 0.324 (0.000) -0.149 (0.000)
M22 0.547 (0.000) 0.643 (0.000) 0.593 (0.000)
M23 -0.008 (0.187) -0.201 (0.000) 0.579 (0.000)
M31 -0.166 (0.000) -0.027 (0.000) 0.130 (0.000)
M32 0.166 (0.000) 0.145 (0.000) -0.122 (0.000)
M33 0.707 (0.000) 0.580 (0.000) 0.666 (0.000)
Σ∗11 1.5171 (0.000) 3.1287 (0.000) 3.1595 (0.000) Σ∗12 -0.2221 (0.000) 0.0160 (0.000) -0.3025 (0.019) Σ∗22 0.9402 (0.000) 1.0551 (0.000) 0.8605 (0.000) Σ∗13 0.2213 (0.000) 0.1353 (0.387) -0.1865 (0.000) Σ∗23 0.5304 (0.000) 0.7673 (0.000) -0.4235 (0.055) Σ∗33 1.1675 (0.000) 1.4872 (0.000) 1.8603 (0.000)
K 0.83 (0.000) 0.90 (0.000) 0.94 (0.000)
Based on the whole sample of data, all estimated d.f. are smaller than the dimension of the process indicating that the realized covariance matrices follow a degenerate WAR(1) process. As Figure B.1.1 in the Appendix B.1 shows, the log-normal dis- tribution fits better the distribution of portfolio realized volatility than the Gamma distribution does. In this case, the estimated parameters cannot be interpreted as describing a WAR(1) model based on latent autoregressive Gaussian vectors, but as describing a simple autoregressive framework, which turns out to provide quali- tatively good forecasts of multivariate volatilities (Bonato et al. (2009), Chiriac &
Voev (2009)).
Table 1.3: Estimation results of WAR(1) model on different time samples, n = 3. All ˆK estimators are significant at 1%. All variances are scaled up by 109.
Time length Stocks K V [Y11,t] V [Y22,t] V [Y33,t] cv(Yt∗)
HD-C-IBM 5.830 6.228 1.114 2.100 0.588
one month HWP-C-IBM 6.162 18.00 1.114 2.100 0.577
HWP-JPM-HD 6.139 18.00 3.272 6.228 0.583
HD-C-IBM 4.938 12.180 11.859 3.671 0.636
12 months HWP-C-IBM 4.053 70.434 11.859 3.671 0.698
HWP-JPM-HD 3.748 70.434 22.333 12.18 0.724
HD-C-IBM 0.831 140.911 325.227 81.363 1.543
66 months HWP-C-IBM 0.902 597.558 325.227 81.363 1.477
HWP-JPM-HD 0.946 597.558 441.598 140.91 1.446
1.5 Conclusion
In this paper, we study in detail the properties of Wishart Autoregressive model of order one, WAR(1), which was introduced by Gourieroux et al. (2009) to cap- ture the dynamics of quadratic and symmetric matrices, such as series of realized variance-covariance matrices on the basis of the sample variance-covariance matrix distribution, known in the literature as the Wishart distribution. In particular, we analyze the properties of the Wishart d.f. estimator subject to different stationarity conditions on the latent vector autoregressive processes and to different distribu- tional settings of the covariance matrix itself.
Non-degenerate Wishart distributions are characterized by d.f. larger than the di- mension of the process. The estimation results from fitting the WAR(1) model to different series of realized covariance matrices reveal that the estimated values of the d.f. are much smaller than the dimension of the process, indicating a degenerate Wishart process for the realized covariance matrices. These results diverge from the ones reported by Gourieroux et al. (2009), which estimated sound values (larger than the dimension of the process) based on one month sample of data.
One possible explanation for this discrepancy lies in the inconsistency between the stationarity assumption imposed by Gourieroux et al. (2009) to identify the model and the type of data used to estimate the model. As a result, we relax the station- arity assumption on the latent VAR processes and show that under nonstationary (cointegration) conditions, the estimated Wishart d.f. exhibit a downward bias and converge in probability to a value that might be smaller than the dimension of the process. Although mathematically sound, this theory becomes questionable when applied to realized (co)variance series: although they exhibit highly persistent dy- namics with large probability mass on extreme observations, there is no empirical or theoretical evidence that realized covariance processes are nonstationary.
A more plausible explanation for estimating Wishart d.f. smaller than the dimen- sion of the process when applied to daily realized covariance matrices, is given by the divergence between the distributional assumption on the underlying covariance process comprised in the WAR(1) specification and the statistical properties of re-
observations, typically detected in realized covariance series. Although in general minor, this fact has in this case a dramatic effect on the estimated parameters of the WAR(1) model: the d.f. estimates based on samples which include extreme events describe degenerate WAR(1) processes of covariance matrices. Our empirical results show that reasonable estimates might be obtained only from short samples or samples stemming from calm financial periods.
Moreover, the WAR(1) specification assumes that portfolio realized volatilities are Gamma distributed, which implies that their coefficient of variation is a constant.
But based on our empirical findings and numerous previous results, the distribution of realized volatilities is closer to the log-normal specification with a coefficient of variation dependent on the mean, but also on the variance of the process, which in this case is time varying (volatility of volatility effect described by Corsi et al.
(2008)). Thus during periods of high and clustered volatility, estimating a WAR(1) process on series, which are in fact approximately log-normally distributed, generates unsound results, which are difficult to interpret: the estimates of the d.f. are smaller than the dimension of the covariance process and consequently describe a degenerate Wishart distribution for the underlying realized covariance matrices. Although in these cases, it loses its interpretation as a WAR(1) model, the corresponding autore- gressive specification reveals high potential in forecasting multivariate volatilities (Chiriac & Voev (2009)).
