Francq, Christian and Jiménez Gamero, Maria Dolores and Meintanis, Simos (2015): Tests for sphericity in multivariate garch models.

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Abstract
Tests for spherical symmetry of the innovation distribution are proposed in multivariate GARCH models. The new tests are of KolmogorovSmirnov and Cram\'ervon Misestype and make use of the common geometry underlying the characteristic function of any spherically symmetric distribution. The asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions. It is shown that both the finite sample and the asymptotic null distribution depend on the unknown distribution of the Euclidean norm of the innovations. Therefore a conditional Monte Carlo procedure is used to actually carry out the tests. The validity of this resampling scheme is formally justified. Results on the behavior of the test in finitesamples are included, as well as an application on financial data.
Item Type:  MPRA Paper 

Original Title:  Tests for sphericity in multivariate garch models 
Language:  English 
Keywords:  Extended CCCGARCH; Spherical symmetry; Empirical characteristic function; Conditional Monte Carlo test 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C12  Hypothesis Testing: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C15  Statistical Simulation Methods: General C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables > C32  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C58  Financial Econometrics 
Item ID:  67411 
Depositing User:  Christian Francq 
Date Deposited:  23. Oct 2015 12:29 
Last Modified:  23. Oct 2015 12:52 
References:  Amengual, D. and Sentana, E. (2011) Inference in multivariate dynamic models with elliptical innovations. Econometrics Seminar, TSE, Toulouse. ArellanoValle, R.B., Azzalini, A. (2008) The centred parametrization for the multivariate skewnormal distribution. J. Multivariate Anal., 99, 13621382. Bai, J. and Chen, Z. (2008) Testing multivariate distributions in GARCH models. J. Econometr., 143, 1936. Bai, J. and Ng, S. (2001) A test for conditional symmetry in time series models. J. Econometr., 103, 225258. Baringhaus, L. (1991) Testing for spherical symmetry of a multivariate distribution. Ann. Statist., 19, 899917. Bauwens, L. and Laurent, S. (2005) A new class of multivariate skew densities, with application to GARCH models. J. Bus. Econom. Statist., 23, 346354. Berk, J. (1997) Necessary conditions for the CAPM. J. Econom. Theor., 73, 245257. Billingsley, P. (1968) em Convergence of probability measures. John Wiley & Sons. Bollerslev, T. (1990) Modelling the coherence in shortrun nominal exchange rates: a multivariate generalized ARCH model. Rev. Econ. Stat., 72, 498505. Cacoullos, T. (2014) Polar angle tangent vectors follow Cauchy distributions under spherical symmetry. J. Multivar. Anal., 128, 147153. Chen, Q., Gerlach, R. and Lu, Z. (2012) Bayesian valueatrisk and expected shortfall forecasting via the asymmetric Laplace distribution. Comput. Statist. Dat. Anal., 56, 34983516. Csorgo, S. (1981) Limit behavior of the empirical characteristic function. Ann. Probab., 9, 130144. Delgado, M., and Escanciano, J.C. (2007) Nonparametric tests for conditional symmetry in dynamic models. J. Econometr., 141, 652682. Delgado, M., and Stute, W. (2008) Distributionfree specification tests of conditional models. J. Econometr., 143, 3755. De Luca, G., Genton, M.G., and Loperfido, N. (2006) A multivariate skewGARCH model. In D. Terrell (Ed.) Advances in Econometrics: Econometric Analysis of Economic and Financial Timee Series, Part A, Elsevier, pp. 3357. Diks, C. and Tong, H. (1999) A test for symmetries of multivariate probability distributions. Biometrika, 86, 605614. Embrechts, P., McNeil, A. and Straumann, D. (2002) Correlation and dependence in risk management: properties and pitfalls. In M.A.H. Dempster (Ed)Risk Management: Value at Risk and Beyond, Cambridge University Press, London, pp. 176223. Fang, K.T., Kotz, S. and Ng, K.W. (1990) em Symmetric Multivariate and Related Distributions. Chapman and Hall, London. Francq, C., Horváth, L., and Zako{"ian, J.M. (2015) Variance targeting estimation of multivariate GARCH models. Journal of Financial Econometrics, published online, DOI: 10.1093/jjfinec/nbu030. Francq, C., and Zako{"ian, J.M. (2010) em GARCH Models: Structure, Statistical Inference and Applications. Wiley, London. Francq, C., and Zako{"ian, J.M. (2012) QML estimation of a class of multivariate asymmetric GARCH models. Econom. Theory, 28, 179206. Ghosh, S. and Ruymgaart, F.H. (1992) Applications of empirical characteristic functions in some multivariate problems. Canad. J. Statist., 20, 429440. Giacomini, R., Politis, D.N., White, H. (2013) A warpspeed method for conducting Monte Carlo experiments involving bootstrap estimators. Econometr. Theory, 29, 567589. Haas, M., Mittnik, S, and Paolella, M.S. (2009) Asymmetric multivariate normal mixture GARCH. Comput. Statist. Dat. Anal., 53, 21292154. Hafner, C.M. and Rombouts, J.V.K. (2007) Semiparametric multivariate volatility models. Econometr. Theor., 23, 251280. Hallin, M. and Paindaveine, D. (2002) Optimal tests for multivariate location based on interdirections and pseudoMahalanobis ranks. Ann. Statist., 30, 11031133. Hallin, M. and Werker, B.J.M. (2003) Semiparametric efficiency, distributionfreeness and invariance. Bernoulli, 9, 137165. Henze, N., Hl'avka, Z., and Meintanis, S.G. (2014) Testing for spherical symmetry via the empirical characteristic function. Statisics, 48, 12821296. Jeantheau, T. (1998) Strong consistency of estimators for multivariate ARCH models. em Econometr. Theor., 14, 7086. Jones, M. C. (2008) The distribution of the ratio $X/Y$ for all centered elliptically symmetric distributions. J. Multivar. Anal., 99, 572573. Kariya, T. and Eaton, M.L. (1977) Robust tests for spherical symmetry. Ann. Statist., 5, 206215. Koltchinskii, V.I. and Li, L. (1998) Testing for spherical symmetry of a multivariate distribution. J. Multivar. Anal., 65, 228244. Kreiss, J.P., Paparoditis, E. and Politis, D.N. (2011) On the range of validity of the autoregressive sieve bootstrap. Ann. Statist., 39, 21032130. Lee, J., Lee, S. and Park, S. (2014) Maximum entropy test for GARCH models. Statist. Methodol., 22, 816. Lee, S., Park, S. and Lee, T. (2010) A note on the JarqueBera normality test for GARCH innovations. J. Kor. Statist. Soc., 39, 93102. Liu, S., Heyde, C.C. and Wong, W.K. (2011) Moment matrices in conditional heteroskedastic models under elliptical distributions with applications in ARARCH models. Statist. Pap., 52, 621632. Meintanis, S.G. and NgatchouWandji, J. (2012) Recent tests for symmetry with multivariate and structured data: A review. In Nonparametric Statistical Methods and Related Topics 3573, World Scientific Publishing Company, London. Mittnik, S. and Paolella, M.S. (2000) Conditional density and valueatrisk prediction of Asian currency exchange rates. J. Forecasting, 19, 313333. Reinsel, G.C. (1997) Elements of Multivariate Time Series Analysis. SpringerVerlag, New York. Shimizu, K. (2013) The bootstrap does not always work for heteroscedastic models. Statistics & Risk Modeling, 30, 189204. Silvennoinen, A. and Ter{"asvirta, T. (2009) Multivariate GARCH models. In T.G. Andersen et al. (eds.): Handbook of Financial Time Series 201229, Springer, Berlin. Trindade, A.A. and Zhu, Y. (2007) Approximating the distributions of estimators of financial risk under an asymmetric Laplace law. Comput. Statist. Dat. Anal., 51, 34333447. Tsay, R.S. (2002) Analysis of Financial Time Series. Wiley, New York. Tsay, R.S. (2014) Multivariate Time Series Analysis: with R and Financial Applications. Wiley. Zhu, L.X. (2005) Asymptotics of goodnessoffit tests for symmetry. In: Lecture Notes in Statistics 20,NonParametric Monte Carlo Tests and Their Applications, 2743, Springer, New York. Zhu, L.X. and Neuhaus, G. (2000) Nonparametric Monte Carlo tests for multivariate distributions. Biometrika, 87, 919928. Zhu, D., ZindeWalsh, V.(2009) Properties and estimation of asymmetric exponential power distribution. J. Econometr., 148, 8699. Zuo, Y. and Serfling, R. (2000) General notions of statistical depth function. Ann. Statist., 28, 461482. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/67411 