Finite sample properties of the GMM Anderson–Rubin test

In the construction of the GMM version of the Anderson and Rubin (AR) test statistic there is the choice to use either uncentered or centered moment conditions to form the weighting matrix. We show that, when the number of moment conditions is moderately large, the centered GMM-AR test is oversized. At the same time, the uncentered version becomes conservative at conventional significance levels. Using an asymptotic expansion, we point to a missing degrees-of-freedom correction in the centered version of the GMM-AR test, which implicitly incorporates an Edgeworth correction. Monte Carlo experiments corroborate our theoretical findings and illustrate the accuracy of the degrees-of-freedom corrected, centered GMM-AR statistic in finite samples.