Higher-Order Debiased Estimators for General Treatment Models

We have witnessed tremendous progress in developing the foundation for econometrics and causal inference in the past decades. The most popular paradigm is the classical (first-order) semiparametric theory, in which a key building block is the (first-order) influence function. However, it is well known that estimators based on influence functions can be sub-optimal in various settings. To address this issue, higher-order influence functions (HOIF) are developed. However, most existing results in this regard focus on treatment effect parameters defined in explicit forms, such as average treatment effects (ATE). Economists are often confronted with tasks of inferring more complex parameters, such as quantile treatment effects (QTE) or effects of complicated treatment regimes/policy. These more complex parameters can often only be implicitly defined as the solution to nonlinear estimating equations, or M/Z-estimation problems. Given the foundational role of HOIF for estimating explicit parameters such as ATE, a modest step toward enriching the foundation of econometrics and causal inference is to develop higher-order estimators of those more complex parameters. Here, we consider parameters of a class of non-separable structural models in the econometrics literature. Statistical properties of the corresponding higher-order estimators are derived using recent advances in U-processes. Our proposed higher-order estimators relax complexity-reducing assumptions, quantified by Hölder smoothness, imposed on the nuisance parameters compared to existing estimators for many important examples, including QTE and quantile dose-response functions, among others.