The three tests are asymptotically equivalent. If the restricted model is adequate, then the restriction function evaluated at the unrestricted MLE should not significantly differ from zero (which is the value of the restriction function at the restricted MLE). If the restricted model is adequate, then the slope of the tangent of the loglikelihood function at the restricted MLE (indicated by T 0 in the figure) should not significantly differ from zero (which is the slope of the tangent of the loglikelihood function at the unrestricted MLE, indicated by T). ![]() If the restricted model is adequate, then the difference between the maximized objective functions, l ( θ A M L E ) − l ( θ 0 M L E ), should not significantly differ from zero.
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