An observation on regression-based specification tests

The use of regression-based specification tests, such as the nR² form of the Lagrange Multiplier test, has become quite widespread over the last 20 years. The popularization of the nR² form of the Lagrange Multiplier (LM) test, perhaps the most widely used class of regression-based tests, has come about in large part from the ease of its application to many tests of nonlinear restrictions and its asymptotic equivalence to Likelihood Ratio and Wald tests. Properly performed, these regression-based tests invariably include regressors which are orthogonal by construction to the dependent variable of the regression. The purpose of this paper is to motivate the inclusion of such variables by investigating implications for the test size and power if these regressors are erroneously omitted. It is straightforward to show that both the size and power of the test are adversely affected by omitting these regressors.