Many specification tests can be computed by means of artificial linear regressions. These are linear regressions designed to be used as calculating devices to obtain test statistics and other quantities of interest. In this paper, we discuss the general principles which 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. We demonstrate the generality and power of artificial regressions as a means of computing test statistics, show how Durbin-Wu-Hausman, conditional moment, and other tests which are not explicitly Lagrange Multiplier tests may be computed, and discuss a number of special cases which serve to illustrate the general results and can also be very useful in practice. These include tests of parameter restrictions in nonlinear regression models and tests of binary choice models such as the logit and probit models.
Durbin-Wu-Hausman test conditional moment test Lagrange Multiplier test LM test artificial regression