This paper proposes a doubly robust two-stage semiparametric difference-in-difference estimator for estimating heterogeneous treatment effects with high-dimensional data. Our new estimator is robust to model miss-specifications and allows for, but does not require, many more regressors than observations. The first stage allows a general set of machine learning methods to be used to estimate the propensity score. In the second stage, we derive the rates of convergence for both the parametric parameter and the unknown function under a partially linear specification for the outcome equation. We also provide bias correction procedures to allow for valid inference for the heterogeneous treatment effects. We evaluate the finite sample performance with extensive simulation studies. Additionally, a real data analysis on the effect of Fair Minimum Wage Act on the unemployment rate is performed as an illustration of our method. An R package for implementing the proposed method is available on Github.
In this review we cover the basics of efficient nonparametric parameter estimation (also called functional estimation), with a focus on parameters that arise in causal inference problems. We review both efficiency bounds (i.e., what is the best possible performance for estimating a given parameter?) and the analysis of particular estimators (i.e., what is this estimator's error, and does it attain the efficiency bound?) under weak assumptions. We emphasize minimax-style efficiency bounds, worked examples, and practical shortcuts for easing derivations. We gloss over most technical details, in …
The research described herewith is to re-visit the classical doubly robust estimation of average treatment effect by conducting a systematic study on the comparisons, in the sense of asymptotic efficiency, among all possible combinations of the estimated propensity score and outcome regression. To this end, we consider all nine combinations under, respectively, parametric, nonparametric and semiparametric structures. The comparisons provide useful information on when and how to efficiently utilize the model structures in practice. Further, when there is model-misspecification, either propensit…
We study a linear random coefficient model where slope parameters may be correlated with some continuous covariates. Such a model specification may occur in empirical research, for instance, when quantifying the effect of a continuous treatment observed at two time periods. We show one can carry identification and estimation without instruments. We propose a semiparametric estimator of average partial effects and of average treatment effects on the treated. We showcase the small sample properties of our estimator in an extensive simulation study. Among other things, we reveal that it compares …
Massachusetts Institute of Technology（麻省理工学院）；Center for Monetary and Financial Studies（货币与金融研究中心）
We propose a semiparametric test to evaluate (i) whether different instruments induce subpopulations of compliers with the same observable characteristics on average, and (ii) whether compliers have observable characteristics that are the same as the full population on average. The test is a flexible robustness check for the external validity of instruments. We use it to reinterpret the difference in LATE estimates that Angrist and Evans (1998) obtain when using different instrumental variables. To justify the test, we characterize the doubly robust moment for Abadie (2003)'s class of complier…
In many semiparametric models, the parameter of interest is identified through conditional expectations, where the conditioning variable involves a single-index that is estimated in the first step. Among the examples are sample selection models and propensity score matching estimators. When the first-step estimator follows cube-root asymptotics, no method of analyzing the asymptotic variance of the second step estimator exists in the literature. This paper provides nontrivial sufficient conditions under which the asymptotic variance is not affected by the first step single index estimator rega…
Trials enroll a large number of subjects in order to attain power, making them expensive and time-consuming. Sample size calculations are often performed with the assumption of an unadjusted analysis, even if the trial analysis plan specifies a more efficient estimator (e.g. ANCOVA). This leads to conservative estimates of required sample sizes and an opportunity for savings. Here we show that a relatively simple formula can be used to estimate the power of any two-arm, single-timepoint trial analyzed with a semiparametric efficient estimator, regardless of the domain of the outcome or kind of…
In this paper we review important aspects of semiparametric theory and empirical processes that arise in causal inference problems. We begin with a brief introduction to the general problem of causal inference, and go on to discuss estimation and inference for causal effects under semiparametric models, which allow parts of the data-generating process to be unrestricted if they are not of particular interest (i.e., nuisance functions). These models are very useful in causal problems because the outcome process is often complex and difficult to model, and there may only be information available…
Although the exposure can be randomly assigned in studies of mediation effects, any form of direct intervention on the mediator is often infeasible. As a result, unmeasured mediator-outcome confounding can seldom be ruled out. We propose semiparametric identification of natural direct and indirect effects in the presence of unmeasured mediator-outcome confounding by leveraging heteroskedasticity restrictions on the observed data law. For inference, we develop semiparametric estimators that remain consistent under partial misspecifications of the observed data model. We illustrate the proposed …
Linear transformation model provides a general framework for analyzing censored survival data with covariates. The proportional hazards and proportional odds models are special cases of the linear transformation model. In biomedical studies, covariates with measurement error may occur in survival data. In this work, we propose a method to obtain estimators of the regression coefficients in the linear transformation model when the covariates are subject to measurement error. In the proposed method, we assume that instrumental variables are available. We develop counting process based estimating…
Causal mediation analysis concerns the pathways through which a treatment affects an outcome. While most of the mediation literature focuses on settings with a single mediator, a flourishing line of research has examined settings involving multiple mediators, under which path-specific effects (PSEs) are often of interest. We consider estimation of PSEs when the treatment effect operates through K(\geq1) causally ordered, possibly multivariate mediators. In this setting, the PSEs for many causal paths are not nonparametrically identified, and we focus on a set of PSEs that are identified under …
The research in this paper gives a systematic investigation on the asymptotic behaviours of four inverse probability weighting (IPW)-based estimators for conditional average treatment effect, with nonparametrically, semiparametrically, parametrically estimated and true propensity score, respectively. To this end, we first pay a particular attention to semiparametric dimension reduction structure such that we can well study the semiparametric-based estimator that can well alleviate the curse of dimensionality and greatly avoid model misspecification. We also derive some further properties of ex…
In this paper we give a brief review of semiparametric theory, using as a running example the common problem of estimating an average causal effect. Semiparametric models allow at least part of the data-generating process to be unspecified and unrestricted, and can often yield robust estimators that nonetheless behave similarly to those based on parametric likelihood assumptions, e.g., fast rates of convergence to normal limiting distributions. We discuss the basics of semiparametric theory, focusing on influence functions.
The instrumental variable method is widely used in the health and social sciences for identification and estimation of causal effects in the presence of potentially unmeasured confounding. In order to improve efficiency, multiple instruments are routinely used, leading to concerns about bias due to possible violation of the instrumental variable assumptions. To address this concern, we introduce a new class of g-estimators that are guaranteed to remain consistent and asymptotically normal for the causal effect of interest provided that a set of at least $γ$ out of $K$ candidate instruments are…
Motivated by recent findings in the field of consumer science, this paper evaluates the causal effect of debit cards on household consumption using population-based data from the Italy Survey on Household Income and Wealth (SHIW). Within the Rubin Causal Model, we focus on the estimand of population average treatment effect for the treated (PATT). We consider three existing estimators, based on regression, mixed matching and regression, propensity score weighting, and propose a new doubly-robust estimator. Semiparametric specification based on power series for the potential outcomes and the pr…
This paper proposes IV-based estimators for the semiparametric distribution regression model in the presence of an endogenous regressor, which are based on an extension of IV probit estimators. We discuss the causal interpretation of the estimators and two methods (monotone rearrangement and isotonic regression) to ensure a monotonically increasing distribution function. Asymptotic properties and simulation evidence are provided. An application to wage equations reveals statistically significant and heterogeneous differences to the inconsistent OLS-based estimator.
This paper analyzes a semiparametric model of network formation in the presence of unobserved agent-specific heterogeneity. The objective is to identify and estimate the preference parameters associated with homophily on observed attributes when the distributions of the unobserved factors are not parametrically specified. This paper offers two main contributions to the literature on network formation. First, it establishes a new point identification result for the vector of parameters that relies on the existence of a special repressor. The identification proof is constructive and characterize…
This paper highlights a tension between semiparametric efficiency and bootstrap consistency in the context of a canonical semiparametric estimation problem, namely the problem of estimating the average density. It is shown that although simple plug-in estimators suffer from bias problems preventing them from achieving semiparametric efficiency under minimal smoothness conditions, the nonparametric bootstrap automatically corrects for this bias and that, as a result, these seemingly inferior estimators achieve bootstrap consistency under minimal smoothness conditions. In contrast, several "debi…
University of Bonn（波恩大学）；CUHK Business School, Chinese University of Hong Kong（香港中文大学商学院）；Graduate School of Economics, University of Osaka（大阪大学经济学研究科）
This article proposes doubly robust estimators for the average treatment effect on the treated (ATT) in difference-in-differences (DID) research designs. In contrast to alternative DID estimators, the proposed estimators are consistent if either (but not necessarily both) a propensity score or outcome regression working models are correctly specified. We also derive the semiparametric efficiency bound for the ATT in DID designs when either panel or repeated cross-section data are available, and show that our proposed estimators attain the semiparametric efficiency bound when the working models…
This paper presents a new perspective on the identification at infinity for the intercept of the sample selection model as identification at the boundary via a transformation of the selection index. This perspective suggests generalizations of estimation at infinity to kernel regression estimation at the boundary and further to local linear estimation at the boundary. The proposed kernel-type estimators with an estimated transformation are proven to be nonparametric-rate consistent and asymptotically normal under mild regularity conditions. A fully data-driven method of selecting the optimal b…
A new and an enriched JPEG algorithm is provided for identifying redundancies in a sequence of irregular noisy data points which also accommodates a reference-free criterion function. Our main contribution is by formulating analytically (instead of approximating) the inverse of the transpose of JPEGwavelet transform without involving matrices which are computationally cumbersome. The algorithm is suitable for the widely-spread situations where the original data distribution is unobservable such as in cases where there is deficient representation of the entire population in the training data (i…
This paper discusses difference-in-differences (DID) estimation when there exist many control variables, potentially more than the sample size. In this case, traditional estimation methods, which require a limited number of variables, do not work. One may consider using statistical or machine learning (ML) methods. However, by the well-known theory of inference of ML methods proposed in Chernozhukov et al. (2018), directly applying ML methods to the conventional semiparametric DID estimators will cause significant bias and make these DID estimators fail to be sqrt{N}-consistent. This article p…
We propose a semiparametric two-stage least square estimator for the heterogeneous treatment effects (HTE). HTE is the solution to certain integral equation which belongs to the class of Fredholm integral equations of the first kind, which is known to be ill-posed problem. Naive semi/nonparametric methods do not provide stable solution to such problems. Then we propose to approximate the function of interest by orthogonal series under the constraint which makes the inverse mapping of integral to be continuous and eliminates the ill-posedness. We illustrate the performance of the proposed estim…
We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, i.e., when it is locally homogeneous of degree zero. When this happens, consistent or equivariant estimation is shown to be impossible. We then show that there exists an underlying regular parameter that fully characterizes the weakly regular parameter. While this parameter is not unique, concepts of sufficiency and minimality help pin down a desirable one. If estimation of minimal sufficient underlying parameters is inefficien…
Wang and Tchetgen Tchetgen (2017) studied identification and estimation of the average treatment effect when some confounders are unmeasured. Under their identification condition, they showed that the semiparametric efficient influence function depends on five unknown functionals. They proposed to parameterize all functionals and estimate the average treatment effect from the efficient influence function by replacing the unknown functionals with estimated functionals. They established that their estimator is consistent when certain functionals are correctly specified and attains the semiparame…
