Minimax-inspired Semiparametric Estimation and Causal Inference

Abstract

This thesis focuses on estimation and inference for a large class of semiparametric estimands: the class of continuous functionals of regression functions. This class includes a number of estimands derived from causal inference problems, among then the average treatment effect for a binary treatment when treatment assignment is unconfounded and many of its generalizations for non-binary treatments and individualized treatment policies. Chapter 2, based on work with Stefan Wager, introduces the augmented minimax linear es- timator (AMLE), a general approach to the problem of estimating a continuous linear functional of a regression function. In this approach, we estimate the regression function, then subtract from a simple plug-in estimator of the functional a weighted combination of the estimated regression function’s residuals. For this, we use weights chosen to minimize the maximum of the mean squared error of the resulting estimator over regression functions in a chosen neighborhood of our estimated regression function. These weights are shown to be a universally consistent estimator our linear functional’s Riesz representer, the use of which would result in an exact bias correction for our plug- in estimator. While this convergence can be slow, especially when the Riesz representer is highly nonsmooth, the action of these weights on functions in the aforementioned neighborhood imitates that of the Riesz representer accurately even when they are slow to converge in other respects. As a result, we show that under no regularity conditions on the Riesz representer and minimal regularity conditions on the regression function, the proposed estimator is semiparametrically efficient. In simulation, it is shown to perform very well in the context of estimating the average partial effect in the conditional linear model, a simultaneous generalization of the average treatment effect to address continuous-valued treatments and of the partial linear model to address treatment effect heterogeneity. Chapter 3, based on work with Arian Maleki and José Zubizarreta, studies the minimax linear estimator, a simplified version of the AMLE in which the estimated regression function is taken to be zero, for a class of estimands generalizing the mean with outcomes missing at random. We show semiparametric efficiency under conditions that are only slightly stronger than those required for the AMLE. In addition, we bound the deviation of our estimator’s error from the averaged efficient influence function, characterizing the degree to which the first order asymptotic characterization of semiparametric efficiency is meaningful in finite samples. In simulation, this estimator is shown to perform well relative to alternatives in high-noise, small-sample settings with limited overlap between the covariate distribution of missing and nonmissing units, a setting that is challenging for approaches reliant on accurate estimation of either or both of the regression function and the propensity score. Chapter 4 discusses an approach to rounding linear estimators for the targeted average treatment effect into matching estimators. The targeted average treatment effect is a generalization of the average treatment effect and the average treatment effect on the treated units