Analytical formulae for accurately sized t-tests in the single instrument case

Abstract

I construct novel analytical expressions for asymptotically valid right-tailed and left-tailed t-tests in the single instrument-single regressor case. The underlying disturbances are allowed to be non-Gaussian and (in a simple extension of the baseline case) heteroskedastic. The critical values for the tests are constructed under the (testable) assumption that the expectation of the F-statistic in the first-stage regression is greater than a lower bound (1+μLB2). The asymptotic sizes of the implied one-tailed (two-tailed) tests are within 2Φ(−μLB)(4Φ(−μLB)) of their nominal values, where Φ is the cdf of a standard normal random variable. The resulting degree of control over asymptotic test sizes is considerably superior to that provided by the standard Stock–Yogo (2005) approach.