Testing for the Transformation of the Dependent Variable

We propose a family of transformations which, unlike the Box-Cox transformation, can sensibly be applied to variables of either sign which may be near or far from zero. We derive two forms of Lagrange multiplier test for the null hypothesis that the dependent variable has not been transformed against the alternative that a transformation of this family has been applied to it. One form is based on the double length artificial regression, the other on the outer product of the gradient artificial regression. These tests turn out to be closely related to one version of the well-known RESET test; the latter can be thought of as an approximation to the former. We provide Monte Carlo evidence on the performance of the two forms of our test under the null. We also provide evidence, based on asymptotics and sampling experiments, on the power of our test and of competing tests against certain alternatives.