Heterogeneity, Optimality, and Sensitivity in Causal Inference

 Identifying and efficiently estimating causal effects under minimal assumptions is a crucial endeavor across the sciences and society. We address several problems related to identification under violations of causal assumptions and efficient estimation of causal effects. In the first part of the thesis, we study conditional effect estimation under violations of the positivity assumption, which asserts that each subject has a non-zero probability of receiving treatment. We propose conditional effects based on incremental propensity score interventions, which are robust to violations of the positivity assumption, and develop efficient estimators for them. In the second part of the thesis, we focus on functional estimation, efficiency, and inference. We develop efficient estimators for the Expected Conditional Covariance. We use a recently proposed “double cross-fit doubly robust” (DCDR) estimator and establish that it achieves semiparametric efficiency under minimal conditions and minimax optimality in Hölder smoothness classes, and that it can be undersmoothed for slower-than-√ n inference. In the third part of the thesis we study average effect estimation under violations of the no unmeasured confounding assumption, which says that treatment is as-if randomized within covariate strata. For this purpose, we propose novel calibrated sensitivity models, which directly incorporate measured confounding into a sensitivity model, thereby bounding the error due to unmeasured confounding by measured confounding multiplied by a sensitivity parameter. We illustrate how to construct calibrated sensitivity models via several examples, demonstrate their advantages over standard sensitivity analyses and post hoc calibration/benchmarking, and establish methods for estimation and inference.