4.10 For the k-variable regression model, it can be shown that the variance of the kth (k = 2, 3, …, K) partial regression coeffi cient given in Eq. (4.10) can also be written as: 2 2 2 2 1 1 var( ) 1 y k k k R b n k R V § ·  ¨ ¸  ¨ ¸ V  © ¹ where 2 Vy = variance of Y, 2 Vk = variance of the kth regressor, 2 Rk = the coeffi - cient of determination from the regression of Xk on the remaining regressors, and R2 = coeffi cient of determination from the multiple regression of Y on all the regressors.

(a) Ceteris paribus, if 2 Vk increases, what happens to var(bk)? What are the implications for the multicollinearity problem?

(b) What happens to the preceding formula if collinearity is perfect?

(c) Evaluate the statement: Th e variance of bk decreases as R2 rises, so that the eff ect of a high 2 Rk can be off set by a higher R2.