Suppose we would like to test Ho; 2 = 0 in the following unrestricted model given also in (7.8) y = X + u =

Suppose we would like to test Ho; β2 = 0 in the following unrestricted model given also in (7.8)

y = Xβ + u = X1β1 + X2β2 + u

(a) Using the FWL Theorem, show that the URSS is identical to the residual sum of squares obtained from ¯ PX1y = ¯ PX1X2β2 + ¯ PX1u. Conclude that URSS = y

 ¯ PXy = y

 ¯ PX1y − y

 ¯ PX1X2(X



2

¯ PX1X2)−1X



2

¯ PX1 y.

(b) Show that the numerator of the F-statistic for testing Ho; β2 = 0 which is given in (7.45), is y ¯ PX1X2(X

2

¯ PX1X2)−1X

2

¯ PX1y/k2.

Substituting y = X1β1+u under the null hypothesis, show that the above expression reduces to u ¯ PX1X2(X

2

¯ PX1X2)−1X

2

¯ PX1u/k2.

(c) Let v = X
2 ¯ PX1u, show that if u ∼ IIN(0, σ2) then v ∼ N(0, σ2X
2 ¯ PX1X2). Conclude that the numerator of the F-statistic given in part

(b) when divided by σ2 can be written as v[var(v)]−1v/k2 where v[var(v)]−1v is distributed as χ2 k2under Ho. Hint: See the discussion below lemma 1.

(d) Using the result that (n − k)s2/σ2 ∼ χ2 n−k where s2 is the URSS/(n − k), show that the F-statistic given by (7.45) is distributed as F(k2, n − k) under Ho. Hint: You need to show that u ¯ PXu is independent of the quadratic term given in part (b), see problem 11.

(e) Show that the Wald Test for Ho; β2 = 0, given in (7.41), reduces in this case to W = β

2[R(XX)−1R]−1β2/s2 were R = [0, Ik2 ], β2 denotes the OLS or equivalently the MLE of β2 from the unrestricted model and s2 is the corresponding estimate of σ2 given by URSS/(n − k). Using partitioned inversion or the FWL Theorem, show that the numerator of W is k2 times the expression in part (b).

(f) Show that the score form of the LM statistic, given in (7.42) and (7.44), can be obtained as the explained sum of squares from the artificial regression of the restricted residuals (y−X1β1,RLS) deflated by s on the matrix of regressors X. In this case, s2 = RRSS/(n−k1)
is the Mean Square Error of the restricted regression. In other words, obtain the explained sum of squares from regressing ¯ PX1y/s on X1 and X2.