19.14 Consider the regression model Y = XB + U. Partition X as [X1 X2] and B as [B1 B2], where X1 has k1 columns and X2 has k2 columns. Suppose that X2Y = 0k2 * 1. Let R = 3Ik1 0k1 * k24.

a. Show that B n (XX)B n

= (RB n)3R(XX)-1R4-1(RB n).

b. Consider the regression described in Equation (12.17).

Let W = [1 W1 W2 c Wr], where 1 is an n * 1 vector of 1’s, W1 is the n * 1 vector with ith element W1i, and so forth. Let Un TSLS denote the vector of two stage least squares residuals.

i. Show that WUn TSLS = 0.

ii. Show that the method for computing the J-statistic described in Key Concept 12.6 (using a homoskedasticity-only F-statistic) and that using the formula in Equation (19.63) produce the same value for the J-statistic.

[Hint: Use the results in (a), (b.i), and Exercise 19.13.]