The score of conditionally heteroskedastic dynamic regression models with student t innovations, an lm test for multivariate normality

We provide numerically reliable analytical expressions for the score of conditionally heteroskedastic dynamic regression models when the conditional distribution is multivariate $t$. We also derive one-sided and 2-sided LM tests for multivariate normality versus multivariate $t$ based on the first two moments of the (squared) norm of the standardised innovations evaluated at the Gaussian quasi-ML estimators of the conditional mean and variance parameters. We reinterpret them as specification tests for multivariate excess kurtosis, and show that they have power against leptokurtic alternatives. Finally, we analyse UK stock returns, and confirm that their conditional distribution has fat tails.

[1]  D. Andrews.  Testing When a Parameter Is on the Boundary of the Maintained Hypothesis , 2001 .

[2]  Enrique Sentana,et al.  Factor Representing Portfolios in Large Asset Markets , 2000 .

[3]  Lutz Kilian,et al.  Residual-Based Tests for Normality in Autoregressions: Asymptotic Theory and Simulation Evidence , 2000 .

[4]  Bruce D. McCullough,et al.  The Numerical Reliability of Econometric Software , 1999 .

[5]  Enrique Sentana,et al.  Testing for GARCH effects: a one-sided approach , 1998 .

[6]  Douglas G. Steigerwald,et al.  Asymptotic Bias for Quasi-Maximum-Likelihood Estimators in Conditional Heteroskedasticity Models , 1997 .

[7]  M. Hashem Pesaran,et al.  Working with Microfit 4.0 : interactive econometric analysis , 1997 .

[8]  Gabriele Fiorentini,et al.  Analytic Derivatives and the Computation of Garch Estimates , 1996 .

[9]  Enrique Sentana.  Quadratic Arch Models , 1995 .

[10]  J. MacKinnon,et al.  Estimation and inference in econometrics , 1994 .

[11]  Ian T. Jolliffe,et al.  Introduction to Multiple Time Series Analysis , 1993 .

[12]  Maxwell L. King,et al.  A Locally Most Mean Powerful Based Score Test for ARCH and GARCH Regression Disturbances , 1993 .

[13]  G. C. Tiao,et al.  An introduction to multiple time series analysis. , 1993, Medical care.

[14]  A. Harvey,et al.  Unobserved component time series models with Arch disturbances , 1992 .

[15]  R. Chou,et al.  ARCH modeling in finance: A review of the theory and empirical evidence , 1992 .

[16]  J. Wooldridge,et al.  Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances , 1992 .

[17]  Alberto Holly,et al.  A General Kronecker Formula for the Moments of the Multivariate Normal Distribution , 1990 .

[18]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[19]  R. Baillie,et al.  The Message in Daily Exchange Rates , 1989 .

[20]  T. Bollerslev,et al.  A CONDITIONALLY HETEROSKEDASTIC TIME SERIES MODEL FOR SPECULATIVE PRICES AND RATES OF RETURN , 1987 .

[21]  Anil K. Bera,et al.  A test for normality of observations and regression residuals , 1987 .

[22]  A. Bera,et al.  Additivity and separability of the Lagrange multiplier, likelihood ratio and Wald tests , 1985 .

[23]  R. Engle.  Wald, likelihood ratio, and Lagrange multiplier tests in econometrics , 1984 .

[24]  James G. MacKinnon,et al.  Small Sample Properties of Alternative Forms of the Lagrange Multiplier Test , 1983 .

[25]  N. Kiefer,et al.  Testing Normality in Econometric Models , 1982 .

[26]  Philip E. Gill,et al.  Practical optimization , 1981 .

[27]  C. Gouriéroux,et al.  Kuhn-Tucker, likelihood ratio and Wald tests for nonlinear models with inequality constraints on the parameters , 1981 .

[28]  Anil K. Bera,et al.  Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo Evidence , 1981 .

[29]  Glenn M. MacDonald,et al.  Some Large-Sample Tests for Nonnormality in the Linear Regression Model , 1980 .

[30]  Anil K. Bera,et al.  Efficient tests for normality, homoscedasticity and serial independence of regression residuals , 1980 .

[31]  M. Crowder.  Maximum Likelihood Estimation for Dependent Observations , 1976 .

[32]  Franklin A. Graybill,et al.  Introduction to the Theory of Statistics, 3rd ed. , 1974 .

[33]  A. Zellner.  An Introduction to Bayesian Inference in Econometrics , 1971 .

[34]  K. Mardia.  Measures of multivariate skewness and kurtosis with applications , 1970 .

[35]  David M. Miller,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[36]  J. Wolfowitz,et al.  An Introduction to the Theory of Statistics , 1951, Nature.