4. Use the house price data set for this question. Remember that in Chapter 4 we carried out Bayesian inference using this data and a Normal linear regression model, in the present chapter we added heteroskedasticity of a known form

(see Section 6.3.2) and independent Student-t errors (see Section 6.4.3). In all cases, we used an independent Normal-Gamma prior for þ and h with values for hyperparameters as in Chapter 4, Section 4.2.7. Use this prior throughout this question. For the other parameters (i.e. Þ for the heteroskedasticity extension and ¹½ for the Student-t errors extension) use an informative prior of your choosing.

(a) Write programs for carrying out Bayesian inference in these three models

(or obtain programs from the website associated with this book and modify as appropriate). Suppose interest centers on the marginal effect of lot size on house price (i.e. þ2). Calculate the posterior mean and standard deviation of this marginal effect using Bayesian model averaging.

Background: Bayesianmodel averaging is briefly described in Chapter 2, equation 2.42 for the case of prediction. Chapter 11 will discuss it in more detail. Briefly, if g.þ2/ is a function of þ2, the rules of conditional probability imply that E[g.þ2/jy] D XR rD1 E[g.þ2/jy; Mr ]p.Mr jy/

and, thus, the Bayesian should calculate E[g.þ2/jy] by averaging results obtained from all models where the weights in the average are given by p.Mr jy/.

(b) Write a program which incorporates all of the extensions of part

(a) into one model. That is, develop a posterior simulator for the linear regression model with Student-t errors which exhibit heteroskedasticity of a known form. Use this program to calculate the posterior mean and standard deviation of þ2. Compare your result with that of part (a).