6. Ellipsoid Bound Theorems. Consider the Normal linear regression model with natural conjugate prior, ; h NG.; V ; s32; /: (a) Show that

6. Ellipsoid Bound Theorems. Consider the Normal linear regression model with natural conjugate prior, þ; h ¾ NG.þ; V ; s32; ¹/:

(a) Show that for any choice of V, the posterior mean of þ must lie in the ellipsoid:

.þ4þave/0X0X.þ4þave/ 

.bþ4þ/0X0X.bþ4þ/

4 where

þave D 1

2

.bþ C þ/

(b) For the case where k D 1, show that the result of

(a) implies the posterior mean must lie between the prior mean and OLS estimate.

(c) Suppose the prior covariance matrix is bounded between V 1  V  V 2, in the sense that V24V and V4V 1 are both positive definite matrices.

Derive an ellipsoid bound analogous to that of (a).

(d) Discuss how the results in

(a) and

(c) can be used to investigate the sensitivity of the posterior mean to prior assumptions.

Note: You may wish to read Leamer (1982) or Poirier (1995, pp.

526–537) for help with answering this question or more details about ellipsoid bound theorems.