Abstract

This paper shows that if the errors in a multiple regression model are heavy-tailed, the ordinary least squares (OLS) estimators for the regression coefficients are tail-dependent. The tail dependence arises, because the OLS estimators are stochastic linear combinations of heavy-tailed random variables. Moreover, tail dependence also exists between the fitted sum of squares (FSS) and the residual sum of squares (RSS), because they are stochastic quadratic combinations of heavy-tailed random variables.

Highlights

  • Mikosch and de Vries (2013) show that in an ordinary least squares (OLS) regression— an econometric method often applied to financial data—the estimator for the regression coefficient is heavy-tailed if the errors are heavy-tailed

  • We study the tail dependence between OLS estimators for the regression coefficients when the error terms in the regression are heavy-tailed

  • We show the presence of tail dependence and provide an explicit formula relating the level of tail dependence to the distribution of the regressors

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Summary

INTRODUCTION

Mikosch and de Vries (2013) is the only study to provide small sample analytical results on the distribution of the OLS estimator when the errors follow heavy-tailed distributions. They find that the OLS estimator is heavy-tailed in the case of a simple linear regression model with additive or multiplicative errors. The FSS and the RSS are stochastically weighted sums of squares and cross products of the errors Their tail dependence can be established in a similar way.

Model Setup
Tail Dependence Between the OLS Estimators
Tail Dependence Between the FSS and the RSS
SIMULATION STUDY
Bivariate Tail Dependence Between the OLS Estimators
Tail dependence estimates
PROOFS
Extension of the Feller Theorem
Proofs of Theorems 1 and 2
Proof of Theorem 3
Proof of Proposition 1
CONCLUSION
Full Text
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