7. Multicollinearity. Consider the Normal linear regression model with likelihood, prior and posterior as described Sections 3.3, 3.4 and 3.5, respectively.

Assume in addition that Xc D 0 for some non-zero vector of constants c.

Note that this is referred to as a case of perfect multicollinearity. It implies the matrix X is not of full rank and .X0X/31 does not exist (see Appendix A on matrix algebra for relevant definitions):

(a) Show that, despite this pathology, the posterior exists if V is positive definite. Define

Þ D c0V31þ

(b) Show that, given h, the prior and posterior distributions of Þ are both identical and equal to:

N(CVB, hcVc) Hence, although prior information can be used to surmount the problems caused by perfect multicollinearity, there are some combinations of the regression coefficients about which learning does not occur.