Second-order refinements for [formula omitted]-ratios with many instruments

Abstract

This paper studies second-order properties of the many instruments robust t-ratios based on the limited information maximum likelihood and Fuller estimators for instrumental variable regression models with homoskedastic errors under the many instruments asymptotics, where the number of instruments may increase proportionally with the sample size n, and proposes second-order refinements to the t-ratios to improve the size and power properties. Based on asymptotic expansions of the null and non-null distributions of the t-ratios derived under the many instruments asymptotics, we show that the second-order terms of those expansions may have non-trivial impacts on the size as well as the power properties. Furthermore, we propose adjusted t-ratios whose approximation errors for the null rejection probabilities are of order O(n−1) in contrast to the ones for the unadjusted t-ratios of order O(n−1/2), and show that these adjustments induce some desirable power properties in terms of the local maximinity. Although these results are derived under homoskedastic errors, we also establish a stochastic expansion for a heteroskedasticity robust t-ratio, and propose an analogous adjustment under slight deviations from homoskedasticity.