(Solution) - The Lagrange multiplier test of the hypothesis R q -(2025 Original AI-Free Solution)

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The Lagrange multiplier test of the hypothesis R? ? q = 0 is equivalent to aWald test of the hypothesis that ? = 0, where ? is defined in (6-14). Prove that ?2 = ?'{Est.Var[?]} ?1 ? = (n ? K) [e'*e*/e'/e ? 1]. Note that the fraction in brackets is the ratio of two estimators of ?2. 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).