Semiparametric and Nonparametric Advances Confronting Non-ideal and Testable Model Assumptions in Causal Inference and Missing Data

Structural identification assumptions are central to the causal inference literature. In practice, it is often crucial to assess their validity or to test implications that follow from them. In many settings, such tests can be framed as evaluating whether a function-valued parameter equals zero. In this paper, we propose a class of generalized projection tests based on series estimators for testing such function-valued parameters. We establish conditions under which the proposed tests are valid and illustrate their applicability through examples from the data fusion and instrumental variables literatures. Our approach accommodates flexible machine learning methods for estimating nuisance parameters. In contrast to existing approaches, the limiting distribution of the proposed test statistics is straightforward to compute under the null hypothesis. We apply our method to test the equality of conditional COVID-19 incidence rates across vaccine arms in the COVID-19 Variant Immunologic Landscape (COVAIL) trial. 

Motivated by the study of heterogeneous returns to education in Brand and Xie (2010), which considers how the effect of completing college on earnings varies with the (unknown) probability of completing college, we analyze the problem of estimating a nonparametric regression function when certain covariates are estimated in a first step. Plug-in estimators that treat the estimated covariates as known generally suffer from first-stage estimation error. To mitigate this issue, we analyze two debiasing approaches within a framework that is agnostic to the choice of first-stage estimation method and consider both local- and sieves-based methods for the second-stage regression. The methods considered are: (i) influence function-based estimators of pathwise differentiable parameters that approximate the target estimand, (ii) a variant of plug-in estimators that directly aim to correct the bias. For each method, we upper bound the estimation risk and characterize conditions under which oracle rates can be approached, highlighting the possible gains in terms of convergence rates relative to the plug-in. Simulation results corroborate the theoretical findings. We apply our methodology to data from the National Longitudinal Survey of Youth 1997 and find evidence that completing college yields the largest reductions in unemployment for individuals least likely to do so, consistent with earlier findings in the literature (Brand and Xie, 2010; Brand, 2023).