The distributions of the J and Cox non-nested tests in regression models with weakly correlated regressors

This paper examines the asymptotic null distributions of the J and Cox non-nested tests in the framework of two linear regression models with nearly orthogonal non-nested regressors. The analysis is based on the concept of near population orthogonality (NPO), according to which the non-nested regressors in the two models are nearly uncorrelated in the population distribution from which they are drawn. New distributional results emerge under NPO. The J and Cox tests tend to two different random variables asymptotically, each of which is expressible as a function of a nuisance parameter, c, a N(0, 1) variate and a χ2(q) variate, where q is the number of non-nested regressors in the alternative model. The Monte Carlo method is used to show the relevance of the new results in finite samples and to compute alternative critical values for the two tests under NPO by plugging consistent estimates of c into the relevant asymptotic expressions. An empirical example illustrates the ‘plug in’ procedure.