Case Studies in Bayesian Statistical Modelling and Analysis

4

Bayesian Analysis of the Normal Linear Regression Model

Christopher M. Strickland and Clair L. Alston

Queensland University of Technology, Brisbane, Australia

4.1 Introduction

The normal linear regression model is a fundamental tool in statistical analysis. Bayesian analysis of this model is well developed; see for instance Zellner (1971), Judge et al. (1988), Koop (2003) and Marin and Robert (2007). The linear regression model relates k covariates, x_i,1, x_i,2, ..., x_i,k, to the ith observation y_i as follows:

(4.1)

for i = 1, 2, ..., n, where β_j, j = 1, 2, ..., k, are referred to as regression coefficients and ε_i is an error term, which throughout is assumed to be normally distributed.

This chapter aims to provide an introduction to the Bayesian analysis of the normal linear regression model, as given in Equation (4.1). We will use two case studies, presenting established data sets that have been widely analysed and are publicly available for researchers to use to develop their own models. These examples allow us to explore standard linear regression, variable selection, in this context, and to extend the standard model to account for serial correlation.

In this chapter, we analyse the basic linear regression model using natural conjugate priors, Jeffreys prior and Zellner's g-prior approach, illustrating the ease with which posterior estimation can be implemented. Sampling schemes ...