Efficient two-sided nonsimilar invariant tests in IV regression with weak instruments

As Nelson and Startz [Nelson, C.R., Startz, R., 1990a. The distribution of the instrumental variable estimator and its t ratio when the instrument is a poor one. Journal of Business 63, S125–S140; Nelson, C.R., Startz, R., 1990b. Some further results on the exact small sample properties of the instrumental variables estimator. Econometrica 58, 967–976] dramatically demonstrated, standard hypothesis tests and confidence intervals in instrumental variables regression are invalid when instruments are weak. Recent work on hypothesis tests for the coefficient on a single included endogenous regressor when instruments may be weak has focused on similar tests. This paper extends that work to nonsimilar tests, of which similar tests are a subset. The power envelope for two-sided invariant (to rotations of the instruments) nonsimilar tests is characterized theoretically, then evaluated numerically for five IVs. The power envelopes for similar and nonsimilar tests differ theoretically, but are found to be very close numerically. The nonsimilar test power envelope is effectively achieved by the Moreira [Moreira, M.J., 2003. A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048] conditional likelihood ratio test, so that test is effectively uniformly most powerful invariant (UMPI). We also provide a new nonsimilar test, P ∗ , which has χ 1 2 critical values, is asymptotically efficient under strong instruments, involves only elementary functions, and is very nearly UMPI.