18.14 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 = [Ik1 0k1 * k2].

a. Show that B n (XX)B n

= (RB n)[R(XX)-1R]-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 ones, 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 the formula in Equation (18.63) produce the same value for the J-statistic.

[Hint: Use the results in (a),

(b, i), and Exercise 18.13.]