Chapter 13 Wald, likelihood ratio, and Lagrange multiplier tests in econometrics

The use of increasingly complex statistical models has led to heavy reliance on maximum likelihood methods for both estimation and testing. In such a setting, only asymptotic properties can be expected for estimators or tests. Often there are asymptotically equivalent procedures that differ substantially in computational difficulty and finite sample performance. In a maximum likelihood framework, the Wald, Likelihood Ratio, and Lagrange Multiplier (LM) tests are a natural trio. They all share the property of being asymptotically locally the most powerful invariant tests, and in fact all are asymptotically equivalent. However, in practice, there are substantial differences in the way the tests look at particular models. Frequently when one is very complex, another will be much simpler. Furthermore, this formulation guides the intuition as to what is testable and how best to formulate a model to test it. In terms of forming diagnostic tests, the LM test is frequently computationally convenient as many of the test statistics are already available from the estimation of the null. The application of these tests principles and particularly the LM principle to a wide range of econometric problems is a natural development of the field, and it is a development that is proceeding at a very rapid pace.