Split-Sample Score Tests in Linear Instrumental Variables Regression

In this paper we design two split-sample tests for subsets of structural coefficients in a linear Instrumental Variables (IV) regression. Sample splitting serves two purposes – 1) validity of the resultant tests does not depend on the identifiability of the coefficients being tested and 2) it combines information from two unrelated samples one of which need not contain information on the dependent variable. The tests are performed on sub-sample one using the regression coefficients obtained from running the so-called first stage regression on subsample two (sample not containing information on the dependent variable). The first test uses the unbiased split-sample IV estimator of the remaining structural coefficients constrained by the hypothesized value of the structural coefficients of interest [see Angrist and Krueger (1995)]. We call this the USSIV score test. The USSIV score test is asymptotically equivalent to the standard score test based on sub-sample one when the standard regularity conditions are satisfied. However, the USSIV score test can be over-sized if the remaining structural coefficients are not identified. This motivates another test based on Robins (2004), which we call the Robins-test. The Robins-test is never oversized and if the remaining structural coefficients are identified, the Robins-test is asymptotically equivalent to USSIV score test against square-root-n local alternatives.