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WORKI NGPAPERS ERI ESNO957 /NOVEMBER2008MODELINGAUTOREGRESSIVECONDITIONAL SKEWNESS AND KURTOSIS WITH MULTI-QUANTILE CAViaRby Halbert White,Tae-Hwan Kim and Simone ManganelliWORKI NGPAPERS ERI ESNO957/ NOVEMBER2008In 2008 all ECB publications feature a motif taken from the 10 banknote.MODELING AUTOREGRESSIVE CONDITIONAL SKEWNESS AND KURTOSIS WITH MULTI-QUANTILE CAViaR 1by Halbert White, 2 Tae-Hwan Kim 3 and Simone Manganelli 4This paper can be downloaded without charge fromhttp://www.ecb.europa.eu or from the Social Science Research Networkelectronic library at http://ssrn.com/abstract_id=1291165.1 The views expressed in this paper are those of the authors and do not necessarily reflect those of the European Central Bank.2 Department of Economics, 0508 University of California, San Diego 9500 Gilman Drive La Jolla, California 92093-0508, USA;e-mail: [email protected] School of Economics, University of Nottingham, University Park Nottingham NG7 2RD, U.K. andYonsei University, Seoul 120-749, Korea; e-mail: [email protected] European Central Bank, DG-Research, Kaiserstrasse 29, D-60311 Frankfurt am Main, Germany;e-mail: [email protected] European Central Bank, 2008Address Kaiserstrasse 29 60311 Frankfurt am Main, Germany Postal address Postfach 16 03 19 60066 Frankfurt am Main, Germany Telephone +49 69 1344 0 Website http://www.ecb.europa.euFax +49 69 1344 6000 All rights reserved. Anyreproduction,publicationand reprintintheformofadifferent publication,whetherprintedor producedelectronically,inwholeorin part,ispermittedonlywiththeexplicit writtenauthorisationoftheECBorthe author(s). The views expressed in this paper do not necessarily reect those of the European Central Bank.ThestatementofpurposefortheECB WorkingPaperSeriesisavailablefrom the ECB website, http://www.ecb.europa.eu/pub/scientific/wps/date/html/index.en.htmlISSN 1561-0810 (print) ISSN 1725-2806 (online)3ECBWorking Paper Series No 957November 2008Abstract4Non-technical summary51Introduction62The MQ-CAViaR process and model73MQ-CAViaR estimation: consistency and asymptotic normality94Consistent covariance matrix estimation135Quantile-based measures of conditional skewness and kurtosis146Application and simulation156.1Time-varying skewness and kurtosisfor the S&P500156.2Simulation187Conclusion19References19Mathematical appendix21Tables and gures32European Central Bank Working Paper Series37CONTENTS4ECBWorking Paper Series No 957November 2008Abstract Engle and Manganelli (2004) propose CAViaR, a class of models suitable for estimating conditional quantiles in dynamic settings. Engle and Manganelli apply their approach to the estimation of Value at Risk, but this is only one of many possible applications. Here we extend CAViaR models to permit joint modeling of multiple quantiles, Multi-Quantile (MQ) CAViaR. We apply our new methods to estimate measures of conditional skewness andkurtosisdefinedintermsofconditionalquantiles,analogoustotheunconditional quantile-based measures of skewness and kurtosis studied by Kim and White (2004). We investigate the performance of our methods by simulation, and we apply MQ-CAViaR to study conditional skewness and kurtosis of S&P 500 daily returns. Keywords: Asset returns; CAViaR; Conditional quantiles; Dynamic quantiles; Kurtosis; Skewness. JEL Classifications: C13, C32. 5ECBWorking Paper Series No 957November 2008Non-technical Summary Highermomentsofdistributionsoffinancialvariables,suchasskewnessandkurtosis, canbeimportanttoassesstheriskofaportfolio,complementingtraditionalvariance measures,aswellasforgenerallyimprovingtheperformanceofvariousfinancial models.Respondingtothisrecognition,researchersandpractitionershavestartedto incorporatethesehighermomentsintotheirmodels,mostlyusingtheconventional measures,e.g.thesampleskewnessand/orthesamplekurtosis.Modelsofconditional counterparts of the sample skewness and the sample kurtosis, based on extensions of the GARCH model, have also been developed and used; see, for example, Leon, Rubio, and Serna(2004).KimandWhite(2004)pointoutthatbecausestandardmeasuresofskewness and kurtosis are essentially based on averages, they can be sensitive to one or a few outliers - a regular feature of financial returns data - making their reliability doubtful. Todealwiththis,KimandWhite(2004)proposetheuseofmorestableandrobust measuresofskewnessandkurtosis,basedonquantilesratherthanaverages. Nevertheless,KimandWhite(2004)onlydiscussunconditionalskewnessandkurtosis measures. In this paper, we extend the approach of Kim and White (2004) by proposing conditionalquantile-basedskewnessandkurtosismeasures.Forthis,weextendEngle and Manganellis (2004) univariate Conditional Autoregressive Value at Risk (CAViaR)modeltoamulti-quantileversion.Thisallowsforageneralvectorautoregressive structure in the conditional quantiles, as well as the presence of exogenous variables. We thenusethismodeltospecifyconditionalversionsofthemorerobustskewnessand kurtosismeasuresdiscussedinKimandWhite(2004).Weapplyourmethodologytoa sampleofS&P500dailyreturns.Wefindthatconventionalestimatesofbothskewness and kurtosis tend to be dwarfed by a few outliers, which typically plague financial data. Ourmorerobustmeasuresshowmoreplausiblevariability,raisingdoubtsaboutthe reliability of unrobust measures. A Monte Carlo simulation is carried out to illustrate the finite sample behavior of our method. 6ECBWorking Paper Series No 957November 20081 IntroductionIt is widely recognized that the use of higher moments, such as skewness and kur-tosis, can be important for improving the performance of various nancial models.Responding to this recognition, researchers and practitioners have started to in-corporate these higher moments into their models, mostly using the conventionalmeasures, e.g.the sample skewness and/or the sample kurtosis.Models of con-ditional counterparts of the sample skewness and the sample kurtosis, based onextensions of the GARCH model, have also been developed and used; see, for ex-ample, Leon, Rubio, and Serna (2004). Nevertheless, Kim and White (2004) pointout that because standard measures of skewness and kurtosis are essentially basedon averages, they can be sensitive to one or a few outliers a regular feature ofnancial returns data making their reliability doubtful.To deal with this, Kim and White (2004) propose the use of more stable androbust measures of skewness and kurtosis, based on quantiles rather than averages.Nevertheless, Kim and White (2004) only discuss unconditional skewness andkurtosis measures. In this paper, we extend the approach of Kim and White (2004)by proposing conditional quantile-based skewness and kurtosis measures. For this,we extend Engle and Manganellis (2004) univariate CAViaR model to a multi-quantile version, MQ-CAViaR. This allows for both a general vector autoregressivestructure in the conditional quantiles and the presence of exogenous variables. Wethen use the MQ-CAViaR model to specify conditional versions of the more robustskewness and kurtosis measures discussed in Kim and White (2004).The paper is organized as follows. In Section 2, we develop the MQ-CAViaRdata generating process (DGP). In Section 3, we propose a quasi-maximum likeli-hood estimator for the MQ-CAViaR process and prove its consistency and asymp-totic normality. In Section 4, we show how to consistently estimate the asymptoticvariance-covariance matrix of the MQ-CAViaR estimator. Section 5 species con-ditional quantile-based measures of skewness and kurtosis based on MQ-CAViaRestimates. Section 6 contains an empirical application of our methods to the S&P500 index. We also report results of a simulation experiment designed to examinethe nite sample behavior of our estimator. Section 7 contains a summary andconcluding remarks. Mathematical proofs are gathered into the MathematicalAppendix.7ECBWorking Paper Series No 957November 20082 The MQ-CAViaR Process and ModelWe consider data generated as a realization of the following stochastic process.Assumption 1 The sequence {(1t. A0t) : t = 0. 1. 2. .... } is a stationary andergodic stochastic process on the complete probability space (. F. 10), where 1tis a scalar and At is a countably dimensioned vector whose rst element is one.Let Ft1 be the o-algebra generated by 2t1= {At. (1t1. At1). ...}. i.e.Ft1 = o(2t1). We let 1t() = 10[1t < |Ft1] dene the cumulative distri-bution function (CDF) of 1t conditional on Ft1.Let 0 < o1 < ... < oj < 1. For , = 1. .... j. the o)th quantile of 1t conditionalon Ft1. denoted
),t, is
),t = inf{ : 1t() = o)}. (1)and if 1t is strictly increasing,
),t = 11t(o)).Alternatively,
),t can be represented asZq
d1t() = 1[1[Yq
]|Ft1] = o). (2)where d1t() is the Lebesgue-Stieltjes probability density function (PDF) of 1tconditional on Ft1, corresponding to 1t().Our objective is to jointly estimate the conditional quantile functions
),t. , =1. 2. .... j. For this we write
t = (
1,t. ....
j,t)0 and impose additional appropriatestructure.First, we ensure that the conditional distribution of 1t is everywhere contin-uous, with positive density at each conditional quantile of interest,
),t. We let,t denote the conditional probability density function (PDF) corresponding to 1t.In stating our next condition (and where helpful elsewhere), we make explicit thedependence of the conditional CDF 1t on . by writing 1t(.. ) in place of 1t().Realized values of the conditional quantiles are correspondingly denoted
),t(.).Similarly, we write ,t(.. ) in place of ,t().After ensuring this continuity, we impose specic structure on the quantiles ofinterest.Assumption 2 (i) 1t is continuously distributed such that for each t and each. . 1t(.. ) and ,t(.. ) are continuous on R; (ii) For given 0 < o1 < ...