2. Another popular approach which addresses issues similar to Bayesian model averaging is described in George and McCulloch (1993). This approach involves using the Normal linear regression model with independent Normal-

Gamma prior (see Chapter 4), with one minor alteration. This alteration is in the prior for the regression coefficients and is useful for the case where a large number of explanatory variables exist, but the researcher does not know which ones are likely to be important. To capture this, the prior for each regression coefficient is a mixture of two Normals, both with mean zero. One of the terms in the mixture has very small variance (i.e. it says the coefficient is virtually zero), and the other has a large variance (i.e. it allows the coefficient to be large). To be precise, for each coefficient þj for j D 1; : : : ; K, the prior is

þj j¼j ¾ .1½¼j /N.0; −2 j / C¼j N.0; c2 j − 2 j /

where cj and −j are known prior hyperparameters with −i being small and cj large. In addition,¼j D 0 or 1 with P¾¼j D 1/ D pj and 0¿pj¿1.

(a) Using a particular prior for each pj (e.g. a Uniform prior or the prior given in Chapter 10 (10.46)), derive a Gibbs sampler with data augmentation this model. Hint: This prior involves a two-component Normal mixture and the derivations of Chapter 10 (Section 10.3) are relevant here. If you are having trouble, you may wish to start with a simpler variant of this model where −jÀ−; cjÀc and¼jÀ¼(i.e. the same prior is used for every coefficient).

(b) Using the cross-country growth data set (see Exercise 1 for details) and your answer to part (a), carry out Bayesian inference using this approach.

With regards to the issue of prior elicitation, for the error precision you may wish to make a noninformative choice. The choice of the prior hyperparameters

−j and cj is a bit more tricky, since defining what is ‘small’

and ‘large’ depends upon the interpretation of the marginal effect (which depends upon the units the explanatory variables are measured in). To surmount this problem, you may wish to standardize each explanatory variable by subtracting off its mean and dividing by its standard deviation. This ensures that each coefficient measures the effect on the dependent variable of a one standard deviation change in the explanatory variable. With variables standardized in this way, in most applications it makes sense to set

−j  − and cj 

c. Start by setting − D 0:0001 and c D 1000, but experiment with various values for both. The ambitious reader may wish to look at George and McCulloch (1993, Section 2.2) for more sophisticated ways of choosing these prior hyperparameters.