This and the next two exercises are based on the test statistic usually used to test a set of J linear restrictions in the generalized regression model: where β is the GLS estimator. Show that if ?? is known, if the disturbances are normally distributed and if the null hypothesis, Rβ = q, is true, then this statistic is exactly distributed as F with J and n ?? K degrees of freedom. What assumptions about the regressors are needed to reach this conclusion? Need they be nonstochastic?

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(R$- q)'[R(X'2-¹X)-¹R']-¹(R-q)/J F[J,n-K] = (N – XÂY9-(y – X)/(n – K)