Multivariate Gram-Charlier Densities

Del Brio, Esther B. and Ñíguez, Trino-Manuel and Perote, Javier (2008): Multivariate Gram-Charlier Densities. Published in: Documentos de Trabajo FUNCAS, No. 381 (2008)

Abstract

This paper introduces a new family of multivariate distributions based on Gram-Charlier and Edgeworth expansions. This family encompasses many of the univariate seminonparametric densities proposed in the financial econometrics as marginal distributions of the different formulations. Within this family, we focus on the specifications that guarantee positivity so obtaining a well-defined multivariate density. We compare different "positive" multivariate distributions of the family with the multivariate Edgeworth-Sargan, Normal and Student’s t in an in- and out-sample framework for financial returns data. Our results show that the proposed specifications provide a quite reasonably good performance being so of interest for applications involving the modelling and forecasting of heavy-tailed distributions.

• Item Type: MPRA Paper
• Language: English
• Keywords: Multivariate distributions; Gram-Charlier and Edgeworth-Sargan densities; MGARCH models; financial data

References

Azzalini, A. and A. Dalla Valle, 1996. The Multivariate Skew Normal Distribution. Biometrika, 83, 715-726. Barton D.E. and K.E.R. Dennis, 1952. The Conditions under which Gram-Charlier and Edgeworth Curves are Positive Definite and Unimodal. Biometrika, 39, 425-427. Bauwens, L., S. Laurent and J. V. K. Rombouts, 2005. Multivariate GARCH Models: A Survey. Journal of Applied Econometrics, 21, 79-109. Bollerslev, T., 1986. Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31, 307-327. 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. Bollerslev, T. and J. Wooldridge, 1992. Quasi Maximum Likelihood Estimation and Inference in Dynamic Models with Time-Varying Covariances. Econometric Reviews, 11, 143-172. Charlier C. V., 1905. Uber Die Darstellung Willkurlicher Funktionen. Arvik fur Mathematik Astronomi och Fysik, 9, 1-13. Corrado, C. J. and T. Su, 1996. Skewness and Kurtosis in S&P 500 Index Returns Implied by Option Prices. Journal of Financial Research, 19, 175-192. Davidson, R., and J. G. MacKinnon, 1998. Graphical Methods for Investigating the Size and Power of Hypothesis Tests. The Manchester School of Economic & Social Studies, 66, 1- 26. Diebold, F. X., T. A. Gunther, and S. A. Tay, 1998. Evaluating Density Forecasts with Applications to Financial Risk Management. International Economic Review, 39, 863-883. Diebold, F. X., J. Hahn, and S. A. Tay, 1999. Multivariate Density Forecasts Evaluation and Calibration in Financial Risk Management: High-Frequency Returns of Foreign Exchange. Review of Economics and Statistics, 81, 661-673. Draper, N. and D. Tierny, 1972. Regions of Positive and Unimodal Series Expansion of the Edgeworth and Gram-Charlier Approximations. Biometrika, 59, 463-465. 16 Edgeworth, F. Y., 1896. The Asymmetrical Probability Curve. Philosophical Magazine, 5th Series, 41. Edgeworth F. Y., 1907. On the representation of statistical frequency by series. Journal of the Royal Statistical Society, Series A, 80. Engle, R. F., 1982. Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflations. Econometrica, 50, 987-1007. Fenton, V. and A. R. Gallant, 1996. Qualitative and Asymptotic Performance of SNP Density Estimators. Journal of Econometrics, 74, 77-118. Fiorentini, G., E. Sentana and G. Calzolari, 2003. Maximum Likelihood Estimation and Inference in Multivariate Conditional Heteroskedastic Dynamic Regression Models with Student t Innovations. Journal of Business and Economic Statistics, 21, 532-546. Gallant, R. and D. Nychka, 1987. Seminonparametric Maximum Likelihood Estimation. Econometrica, 55, 363-390. Gallant R., G. Tauchen, 1989. Seminonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications. Econometrica, 57, 1091- 1120. Harvey C. R. and A. Siddique, 1999. Autoregressive Conditional Skewness. Journal of Financial and Quantitative Analysis, 34, 465-487. Henery, R.J., 1981. An Approximation to Certain Multivariate Normal Probabilities. Journal of the Royal Statistical Society, Series B, 43, 81-85. Jondeau, E. and M. Rockinger, 2001. Gram-Charlier Densities. Journal of Economic Dynamics and Control, 25, 1457-1483. Kendall, M. and A. Stuart, 1977. The Advanced Theory of Statistics, Vol. I, 4th edn., Griffin & Co, London. Kotz, S. and S. Nadarajah, 2004. Multivariate T-Distributions and Their Applications, Cambridge University Press. León A., G. Rubio, and G. Serna, 2005. Autoregressive Conditional Volatility, Skewness and Kurtosis. Quarterly Review of Economics and Finance, 45, 599-618. León A, J. Mencia and E. Sentana, 2006. Parametric properties of semi-nonparametric distributions, with applications to option valuation. Discussion Paper 5435, C.E.P.R.. Malevergne, Y. and D. Sornette, 2004. Multivariate Weibull Distributions for Asset Returns: I. Finance Letters, 2, 16-32. Mauleón I. and J. Perote, 2000. Testing Densities with Financial Data: An empirical Comparison of the Edgeworth-Sargan Density to the Student's t. European Journal of Finance, 6, 225-239. Nabeya, S., 2001. Approximation to the Limiting Distribution of t- and F- Statistics in Testing for Seasonal Unit Roots. Econometric Theory, 17, 711-748. Nishiyama, Y. and P. M. Robinson, 2000. Edgeworth Expansions for Semiparametric Averaged Derivatives. Econometrica, 68, 931-980. Ñíguez, T.M. and J. Perote, 2004. Forecasting the Density of Asset Returns. STICERD Econometrics Discussion Paper 479. London School of Economics. Olcay, A., 2005.A New Class of Multivariate Distributions: Scale Mixture of Kotz-Type Distributions. Statistics & Probability Letters, 75, 18-28. Perote, J., 2004. The Multivariate Edgeworth-Sargan Density. Spanish Economic Review, 6, 77-96. Perote, J. and E.B. Del Brio, 2003. Measuring Value-at-Risk under the Conditional Edgeworth-Sargan Distribution. Finance Letters, 1, 23-40. Prucha, I. and H. Kelejian, 1984. The Structure of Simultaneous Equation Estimation: A Generalization towards Nonnormal Disturbances. Econometrica, 52, 721-731. 17 Sargan, J. D., 1975. Gram-Charlier Approximations Applied to t Ratios of k-Class Estimators. Econometrica, 43, 327-347. Sargan J. D., 1976. Econometric Estimators and the Edgeworth Approximation. Econometrica, 44, 421-448. Velasco, C. and P. M. Robinson, 2001. Edgeworth Expansions for Spectral Density Estimates and Studentized Simple Mean. Econometric Theory, 17, 497-539. Verhoeven, P. and M. McAleer, 2004. Fat Tails in Financial Volatility Models. Mathematics and Computers Simulation, 64, 351-362. Xiaohong, C., F. Yanqin and V. Tsyrennikov, 2006. Efficient Estimation of Semiparametric Multivariate