The Exact General Formulas for the Moments of a Ridge Regression Estimator when the Regression Error Terms Follow a Multivariate t Distribution

Abstract

Huang (1999) proposed a feasible ridge regression (FRR) estimator to estimate a specific regression coefficient. Assuming that the error terms follow a normal distribution, Huang (1999) examined the small sample properties of the FRR estimator. In this article, assuming that the error terms follow a multivariate t distribution, we derive an exact general formula for the moments of the FRR estimator to estimate a specific regression coefficient. Using the exact general formula, we obtain exact formulas for the bias, mean squared error (MSE), skewness, and kurtosis of the FRR estimator. Since these formulas are very complex, we compare the bias, MSE, skewness, and kurtosis of the FRR estimator with those of ordinary least square (OLS) estimator by numerical evaluations. Our numerical results show that the range of MSE dominance of the FRR estimator over the OLS estimator is widen under a fat tail distributional assumption.

Keywords:

• Mean squared error
• Multivariate Student's t error terms
• Ordinary least square estimator
• Ridge regression

2000 Mathematics Subject Classification:

• 62J05
• 62J07

Acknowledgments

My heartfelt appreciation goes to Prof. K. Ohtani, Prof. H. Tanizaki, and Prof. S. Hamori whose comments and suggestions were of inestimable value for my study. I would also like to thank the anonymous referees for valuable comments which greatly improved the present version of the article.