An alternative way to test the hypothesis R ?? q = 0 is to use a Wald

An alternative way to test the hypothesis Rβ ?? q = 0 is to use a Wald test of the hypothesis that λ?? = 0, where λ?? is defined in (5-23). Prove that

the fraction in brackets is the ratio of two estimators of σ². By virtue of (5-28) and the preceding discussion, we know that this ratio is greater than 1. Finally, prove that this test statistic is equivalent to JF, where J is the number of restrictions being tested and F is the conventional F statistic given in (5-16). Formally, the Lagrange multiplier test requires that the variance estimator be based on the restricted sum of squares, not the unrestricted. Then, the test statistic would be LM = nJ/[(n ?? K)/F + J ]. See Godfrey (1988).

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e'e, x² = X,{Est. Var[2.]}¯2, = (n – K) e'e