Inference on subsets of parameters in linear IV without assuming identification

We show that the (conditional) limiting distributions of the subset extensions of the weak instrument/identification robust instrumental variable statistics are bounded from above by the (conditional) limiting distributions that apply when the remaining structural parameters are well-identified. The identification robust subset statistics are therefore size correct in large samples and their projection based counterparts are conservative. The power curves of tests based on the robust subset statistics are non-standard since they resemble identification statistics at distant values of the parameter of interest. Hence, the power of a test on a well-identified structural parameter is low at distant values when one of the remaining structural parameters is weakly identified. It is identical to the power of a test for a distant value of any of the other structural parameters. All results apply as
well to tests on the parameters of the included exogenous variables.