In the model y = X???? + u (T × 1) where X is T × k, consider the test statistic W = (R????̂ − c)′(R(X′X)−1R′)−1(R????̂ − c)

rs2 where R (r × k) and c (r × 1) are fixed and s2 = ̂u′ ̂u∕(T − k).

(a) For W to have the F(r, T − k) distribution requires that the numerator and denominator are distributed independently. State definitions of the quantities

????̂ and û and appropriate assumptions under which this condition holds, and explain the underlying reasoning.

(b) Also justify, under the same definitions/assumptions, the claim that the numerator and denominator have rescaled chi-squared distributions.