Subset Statistics in the linear IV regression model

We show that the limiting distributions of subset generalizations of the weak instrument robust instrumental variable statistics are boundedly similar when the remaining structural parameters are estimated using maximum likelihood. They are bounded from above by the limiting distributions which apply when the remaining structural parameters are well identified and from below by the limiting distributions which hold when the remaining structural parameters are completely unidentified. The lower bound distribution does not depend on nuisance parameters and converges in case of Kleibergen's (2002) Lagrange multiplier statistic to the limiting distribution under the high level assumption when the number of instruments gets large. The power curves of the subset statistics are non-standard since the subset tests converge to identification statistics for distant values of the parameter of interest. The power of a test on a well-identified parameter is therefore low for distant values when one of the remaining structural parameter is weakly identified and is equal to the power of a test for a distant value of one of the remaining structural parameters. All subset results extend to statistics that conduct tests on the parameters of the included exogenous variables.