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This paper considers two-sided tests for the parameter of an endogenousvariable in an instrumental variable (IV) model with heteroskedastic andautocorrelated errors. We develop the finite-sample theory of weighted-averagepower (WAP) tests with normal errors and a known long-run variance. Weintroduce two weights which are invariant to orthogonal transformations of theinstruments; e.g., changing the order in which the instruments appear. Whiletests using the MM1 weight can be severely biased, optimal tests based on theMM2 weight are naturally two-sided when errors are homoskedastic.  We propose two boundary conditions that yield two-sided tests whether errorsare homoskedastic or not. The locally unbiased (LU) condition is related to thepower around the null hypothesis and is a weaker requirement than unbiasedness.The strongly unbiased (SU) condition is more restrictive than LU, but theassociated WAP tests are easier to implement. Several tests are SU in finitesamples or asymptotically, including tests robust to weak IV (such as theAnderson-Rubin, score, conditional quasi-likelihood ratio, and I. Andrews'(2015) PI-CLC tests) and two-sided tests which are optimal when the sample sizeis large and instruments are strong.  We refer to the WAP-SU tests based on our weights as MM1-SU and MM2-SU tests.Dropping the restrictive assumptions of normality and known variance, thetheory is shown to remain valid at the cost of asymptotic approximations. TheMM2-SU test is optimal under the strong IV asymptotics, and outperforms otherexisting tests under the weak IV asymptotics.

Optimal Two-Sided Tests for Instrumental Variables Regression with Heteroskedastic and Autocorrelated Errors