Bootstrap tests for overidentification in linear regression models

Despite much recent work on the finite-sample properties of estimators and tests for linear regression models with a single endogenous regressor and weak instruments, little attention has been paid to tests for overidentifying restrictions in these circumstances. We study asymptotic tests for overidentification in models estimated by instrumental variables and by limited-information maximum likelihood. We show that all the test statistics, like the ones used for inference on the regression coefficient, are functions of only six quadratic forms in the two endogenous variables of the model. They are closely related to the well-known test statistic of Anderson and Rubin. The distributions of the overidentification statistics are shown to have an ill-defined limit as the strength of the instruments tends to zero along with a parameter related to the correlation between the disturbances of the two equations of the model. Simulation experiments demonstrate that this makes it impossible to perform reliable inference near the point at which the limit is ill-defined. Several bootstrap procedures are proposed. They alleviate the problem and allow reliable inference when the instruments are not too weak.