The Lagrange multiplier test of the hypothesis R q = 0 is equivalent to aWald test

The Lagrange multiplier test of the hypothesis Rβ − q = 0 is equivalent to aWald test of the hypothesis that λ = 0, where λ is defined in (6-14). Prove that χ2 = λ'{Est.Var[λ]} −1 λ = (n − K) [e'*e*/e'/e – 1]. Note that the fraction in brackets is the ratio of two estimators of σ2. By virtue of (6-19) and the preceding discussion, we know that this ratio is greater than 1. Finally, prove that the Lagrange multiplier statistic is equivalent to JF, where J is the number of restrictions being tested and F is the conventional F statistic given in (6-6).