Invariance, Nonlinear Models and Asymptotic Tests
Several Asymptotic Tests Proposed in the Literature Are Shown Not to Be Invariant to Changes in Measurement Units Or, More Generally to Various Transformations Which Leave Both the Model and the Null Hypothesis Invariant. the Test Involved Include the Wald Test, a Variant of the Lagrange Multiplier Test, Neyman's 'C' Test, Durbin's Procedure (1970), Hausman-Type Tests and a Number of Tests Suggested by White (1982). for All These Procedures, Simply Changing Measurement Units in a Way That Leaves Both the Form of the Model and the Null Hypothesis Invariant Can Lead to Vastly Different Answers. This Problem Is Illustrated by Considering Regression Models with Box-Cox Transformations on the Variables. We Observe, in Particular, That Various Consistent Estimators of the Information Matrix Lead to Test Procedures with Different Invariance Properties. We Then Establish General Sufficient Conditions Which Ensure That Neyman's Test Is Invariant to Transformations Which Leave Invariant the Form of the Model Further, We Give Conditions Under Which a Generalized 'C' Test Applicable to General Restrictions, Is Invariant to the Algebraic Formulation of the Restrictions. in Many Practical Cases Where Wald-Type Tests Lack Invariance, We Find That a Modification of the 'C' Test Is Invariant and Hardly More Costly to Compute Than Wald Tests. This Computational Simplicity Stands in Contrast with Other Invariant Tests Such As the Likelihood Ratio Test. We Conclude That Non-Invariant Asymptotic Tests Should Be Avoided Or Used with Great Care. Further in Many Situations, the Suggested Modification of the 'C' Test Yields an Attractive Substitute to the Wald Test and to Other Invariant Tests.
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