Empirical applications of instrumental variables (IV) regressions often involve both low-dimensional and high-dimensional sets of instruments. In Angrist and Krueger (1991), the quarter-of-birth instruments generate a much larger set that includes many interaction-based technical instruments; in shift-share designs considered in Goldsmith-Pinkham, Sorkin, and Swift (2020), a single IV aggregates many underlying base instruments. To improve inference in such settings, this paper proposes a combination test that integrates low- and high-dimensional IV specifications. The test linearly combines a cluster-robust Wald statistic based on low-dimensional IVs with leave-cluster-out Lagrange multiplier (LM) and Anderson-Rubin (AR) statistics constructed from the many IVs. We establish the joint asymptotic normality of these statistics and show that the proposed test is asymptotically optimal within a broad class of tests. The procedure yields costless efficiency improvements, automatically adapts to weak identification of the many instruments, and provides a practical rule of thumb for assessing these gains.
We study inference for quadratic forms of regression coefficients with many covariates. We develop a unifying framework that allows for clustering with diverging cluster sizes, and covers design-based inference on treatment effects, model-based inference on variance components, testing many linear restrictions, and instrumental variables models with many instruments and controls. We show that, in the presence of many controls, existing methods do not yield valid inference in the design-based setting. We develop a novel jackknife variance estimator that yields valid, but conservative inference, and a leave-three-cluster-out estimator that is more computationally involved but is asymptotically exact, up to a nonnegative heterogeneity term that cannot be avoided. For the model-based applications, we propose a different jackknife variance estimator that leads to conservative inference in general, but is asymptotically exact if the number of controls grows more slowly than the sample size. A leave-three-cluster-out variance estimator is asymptotically exact. We illustrate our methods in several applications.
We propose a heteroskedasticity-robust and dimension-adaptive test for multiple linear restrictions in sparse high-dimensional linear regression models. It combines desparsified LASSO estimation with a ridge-regularized quadratic form statistic and critical values that adapt to the restriction dimension. Under stated regularity conditions, our approach is asymptotically valid under a fixed or diverging number of restrictions without specifying the asymptotic regime in advance. Our random-design theory allows independent non-identically distributed covariate rows with marginal sub-Weibull tails and heteroskedastic errors satisfying finite polynomial moments. Monte Carlo simulations demonstrate improved finite-sample size control relative to conventional Wald tests, particularly as the number of restrictions increases.