The Impact of a Hausman Pretest on the Size of Hypothesis Tests

This paper investigates the size properties of a two-stage test in the linear instrumental variables model when in the first stage a Hausman (1978) specification test is used as a pretest of exogeneity of a regressor. In the second stage, a simple hypothesis about a component of the structural parameter vector is tested, using a t-statistic that is based on either the ordinary least squares (OLS) or the two-stage least squares estimator (2SLS) depending on the outcome of the Hausman pretest. The asymptotic size of the two-stage test is derived in a model where weak instruments are ruled out by imposing a lower bound on the strength of the instruments. The asymptotic size is a function of this lower bound and the pretest and second stage nominal sizes. The asymptotic size increases as the lower bound and the pretest size decrease. It equals 1 for empirically relevant choices of the parameter space. It is also shown that, asymptotically, the conditional size of the second stage test, conditional on the pretest not rejecting the null of regressor exogeneity, is 1 even for a large lower bound on the strength of the instruments. The size distortion is caused by a discontinuity of the asymptotic distribution of the test statistic in the correlation parameter between the structural and reduced form error terms. The Hausman pretest does not have sufficient power against correlations that are local to zero while the OLS t-statistic takes on large values for such nonzero correlations. Instead of using the two-stage procedure, the recommendation then is to use a t-statistic based on the 2SLS estimator or, if weak instruments are a concern, the conditional likelihood ratio test by Moreira (2003).