Specification Tests Based on Artificial Regressions

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Abstract
Many specification tests can be computed with artificial linear regressions designed to be used as calculating devices to obtain test statistics and other quantities of interest. This article discusses the general principles that underlie all artificial regressions, and the use of such regressions to compute Lagrange multiplier and other specification tests based on estimates under the null hypothesis. The generality and power of artificial regressions as a means of computing test statistics is demonstrated; how Durbin–Wu–Hausman, conditional moment, and other tests that are not explicitly Lagrange multiplier tests may be computed is shown; and several special cases that illustrate the general results and can be useful in practice are discussed. These include tests of parameter restrictions in nonlinear regression models and tests of binary-choice models such as the logit and probit models.

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