A powerful subvector Anderson-Rubin test in linear instrumental variables regression with conditional heteroskedastictiy

We introduce a new test for a two-sided hypothesis involving a subset of the structural parameter vector in the linear instrumental variables (IVs) model. Guggenberger, Kleibergen, and Mavroeidis (2019, Quantitative Economics, 10, 487–526; hereafter GKM19) introduce a subvector Anderson–Rubin (AR) test with data-dependent critical values that has asymptotic size equal to nominal size for a parameter space that allows for arbitrary strength or weakness of the IVs and has uniformly nonsmaller power than the projected AR test studied in Guggenberger et al. (2012, Econometrica, 80(6), 2649–2666). However, GKM19 imposes the restrictive assumption of conditional homoskedasticity (CHOM). The main contribution here is to robustify the procedure in GKM19 to arbitrary forms of conditional heteroskedasticity. We first adapt the method in GKM19 to a setup where a certain covariance matrix has an approximate Kronecker product (AKP) structure which nests CHOM. The new test equals this adaptation when the data are consistent with AKP structure as decided by a model selection procedure. Otherwise, the test equals the AR/AR test in Andrews (2017, Identification-Robust Subvector Inference, Cowles Foundation Discussion Papers 3005, Yale University) that is fully robust to conditional heteroskedasticity but less powerful than the adapted method. We show theoretically that the new test has asymptotic size bounded by the nominal size and document improved power relative to the AR/AR test in a wide array of Monte Carlo simulations when the covariance matrix is not too far from AKP.