The Asymptotic Covariance Matrix of the Least Squares Estimator in the Stochastic Linear Regression Model: The Case of Elliptically Symmetric Distribution

Abstract

This article considers the unconditional asymptotic covariance matrix of the least squares estimator in the linear regression model with stochastic explanatory variables. The asymptotic covariance matrix of the least squares estimator of regression parameters is evaluated relative to the standard asymptotic covariance matrix when the joint distribution of the dependent and explanatory variables is in the class of elliptically symmetric distributions. An empirical example using financial data is presented. Numerical examples and simulation experiments are given to illustrate the difference of the two asymptotic covariance matrices.

Keywords:

• Asymptotic robustness
• Non normality
• Unconditional Inference

Mathematics Subject Classification:

• Primary 62J05
• Secondary 62F12

Acknowledgments

The author would like to thank the referees of this journal, Takeaki Kariya, John Knight, and Frank Samaniego, for helpful comments on previous versions of the article. This research was supported by Nomura Foundation for Social Science and Grant-in-Aid for Scientific Research (KAKENHI(14530036, 19530184)).

Notes

¹In the family of multivariate generalized t distributions introduced by Arslan (2004), the excess kurtosis for any element of an elliptically symmetric vector may not be the same as that for the original random vector. Such distributions are ruled out from the class of elliptically symmetric distributions we consider.

²We are grateful to Ken French for making the database available.

The table gives univariate and multivariate skewness and kurtosis measures for Fama and French benchmark portfolios and factors from July 1963 to December 2006. It also gives p-values under univariate and multivariate normal and t distribution with six degrees of freedom. The word MKT denotes the factor of the excess market return.

Note: Numbers given in the table denote the ratio of the mean of the standard (true) asymptotic variance estimators, obtained based on 20,000 iterations, and the mean of the true variance estimators, obtained based on 100,000 iterations. The word std stands for the ratio of the mean of the standard asymptotic variance estimators and true variance estimators and the word true stands for the ratio of the mean of the true asymptotic variance estimators and true variance estimators.

Note: Numbers given in the table denote the ratio of the mean of the standard (true) asymptotic variance estimators, obtained based on 20,000 iterations, and the mean of the true variance estimators, obtained based on 100,000 iterations. The word std stands for the ratio of the mean of the standard asymptotic variance estimators and true variance estimators and the word true stands for the ratio of the mean of the true asymptotic variance estimators and true variance estimators.

Note: Numbers given in the table denote the ratio of the mean of the standard (true) asymptotic variance estimators, obtained based on 20,000 iterations, and the mean of the true variance estimators, obtained based on 100,000 iterations. The word std stands for the ratio of the mean of the standard asymptotic variance estimators and true variance estimators and the word true stands for the ratio of the mean of the true asymptotic variance estimators and true variance estimators.