Predictive Distribution of Regression Vector and Residual Sum of Squares for Normal Multiple Regression Model

Abstract

This article proposes predictive inference for the multiple regression model with independent normal errors. The distributions of the sample regression vector (SRV) and the residual sum of squares (RSS) for the model are derived by using invariant differentials. Also, the predictive distributions of the future regression vector (FRV) and the future residual sum of squares (FRSS) for the future regression model are obtained. Conditional on the realized responses, the FRV is found to follow a multivariate Student t distribution, and that of the residual sum of squares follows a scaled beta distribution. The new results have been applied to the market return and accounting rate data to illustrate its application.

Key Words:

• Multiple regression model
• Regression vector and residual sum of squares
• Noninformative prior
• Future regression model
• Predictive inference
• Future regression vector
• Multivariate normal
• Student t
• Beta and gamma distributions

AMS 1991 Subject Classification:

• Primary 62H10
• Secondary 62J12

Acknowledgments

The author thankfully acknowledges some valuable suggestions from Professor Lehana Thabane, McMaster University, Canada that improved the quality and content of the article significantly. This work was initiated while the author was visiting the Institute of Statistical Research and Training, University of Dhaka, Bangladesh.