The Relation Among the Likelihood Ratio-, Wald-, and Lagrange Multiplier Tests and Their Applicability to Small Samples

by Daniel F. Kohler


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This paper shows that in a regression model with linear constraints the Likelihood Ratio (LR) test, the Lagrange Multiplier (LM) test, and the Wald (W) test are all monotonic transformations of the same test. The tests are thus equivalent if properly applied. Furthermore, since the LM test and the LR test can be transformed into a W test, for which there is a small sample equivalent, all three tests can, after proper modification, be applied to any sample.

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