In applied problems it is common to specify a model for the conditional mean of a response given a set of regressors. A subset of the regressors may be missing for some study subjects either by design or happenstance. In this article we propose a new class of semiparametric estimators, based on inverse probability weighted estimating equations, that are consistent for parameter vector α0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a representation for the efficient score...

https://www.jstor.org/stable/pdfplus/2290910.pdf

Estimation of Regression Coefficients When Some Regressors are not Always Observed

Summary
We revisit the classic semi-parametric problem of inference on a low-dimensional parameter θ0 in the presence of high-dimensional nuisance parameters η0. We depart from the classical setting by allowing for η0 to be so high-dimensional that the traditional assumptions (e.g. Donsker properties) that limit complexity of the parameter space for this object break down. To estimate η0, we consider the use of statistical or machine learning (ML) methods, which are particularly well suited to estimation in modern, very high-dimensional cases. ML methods perform well by employing regularization to reduce variance and trading off regularization bias with overfitting in practice. However, both regularization bias and overfitting in estimating η0 cause a heavy bias in estimators of θ0 that are obtained by naively plugging ML estimators of η0 into estimating equations for θ0. This bias results in the naive estimator failing to be N−1/2 consistent, where N is the sample size. We show that the impact of regularization bias and overfitting on estimation of the parameter of interest θ0 can be removed by using two simple, yet critical, ingredients: (1) using Neyman-orthogonal moments/scores that have reduced sensitivity with respect to nuisance parameters to estimate θ0; (2) making use of cross-fitting, which provides an efficient form of data-splitting. We call the resulting set of methods double or debiased ML (DML). We verify that DML delivers point estimators that concentrate in an N−1/2-neighbourhood of the true parameter values and are approximately unbiased and normally distributed, which allows construction of valid confidence statements. The generic statistical theory of DML is elementary and simultaneously relies on only weak theoretical requirements, which will admit the use of a broad array of modern ML methods for estimating the nuisance parameters, such as random forests, lasso, ridge, deep neural nets, boosted trees, and various hybrids and ensembles of these methods. We illustrate the general theory by applying it to provide theoretical properties of the following: DML applied to learn the main regression parameter in a partially linear regression model; DML applied to learn the coefficient on an endogenous variable in a partially linear instrumental variables model; DML applied to learn the average treatment effect and the average treatment effect on the treated under unconfoundedness; DML applied to learn the local average treatment effect in an instrumental variables setting. In addition to these theoretical applications, we also illustrate the use of DML in three empirical examples.

https://www.ifs.org.uk/uploads/cemmap/wps/CWP281717.pdf

Double/debiased machine learning for treatment and structural parameters

We consider inference about the parameter β* indexing the conditional mean of a vector of correlated outcomes given a vector of explanatory variables when some of the outcomes are missing in a subsample of the study and the probability of response depends on both observed and unobserved data values; that is, nonresponse is nonignorable. We propose a class of augmented inverse probability of response weighted estimators that are consistent and asymptotically normal (CAN) for estimating β* when the response probabilities can be parametrically modeled and a CAN estimator exists. The proposed estimators do not require full specification of a parametric likelihood, and their computation does not require numerical integration. Our estimators can be viewed as an extension of generalized estimating equation estimators that allows for nonignorable nonresponse. We show that our class essentially consists of all CAN estimators of β*. We also show that the asymptotic variance of the optimal estimator in our ...

https://www.jstor.org/stable/info/2670049

Semiparametric Regression for Repeated Outcomes With Nonignorable Nonresponse

The focus of this paper is the nonparametric estimation of an instrumental regression function f defined by conditional moment restrictions that stem from a structural econometric model E[Y − f (Z) | W] = 0, and involve endogenous variables Y and Z and instruments W. The function f is the solution of an ill-posed inverse problem and we propose an estimation procedure based on Tikhonov regularization. The paper analyzes identification and overidentification of this model, and presents asymptotic properties of the estimated nonparametric instrumental regression function.

Nonparametric Instrumental Regression

or panel samples to improve accuracy of estimation. In this paper we show how to do this efficiently. We show that the gains from use of marginal information can be substantial. We also discuss how to test the compatibility of sample and marginal information.

Combining Micro and Macro Data in Microeconometric Models

The purpose of this note is to show how semiparametric estimators with a small bias property can be constructed. The small bias property (SBP) of a semiparametric estimator is that its bias converges to zero faster than the pointwise and integrated bias of the nonparametric estimator on which it is based. We show that semiparametric estimators based on twicing kernels have the SBP. We also show that semiparametric estimators where nonparametric kernel estimation does not affect the asymptotic variance have the SBP. In addition we discuss an interpretation of series and sieve estimators as idempotent transformations of the empirical distribution that helps explain the known result that they lead to the SBP. In Monte Carlo experiments we find that estimators with the SBP have mean-square error that is smaller and less sensitive to bandwidth than those that do not have the SBP.

Twicing Kernels and a Small Bias Property of Semiparametric Estimators

SUMMARY Pepe and Fleming, and Carroll and Wand have recently proposed estimators in a parametric model for the density of a random variable Y conditional on a vector of covariates (X, V) when data on one of the regressors X is missing for some study subjects. We propose a new class of estimators that remains consistent and asymptotically normal even when the probability that X is missing depends on the observed V and Y, includes an estimator whose asymptotic variance attains the semiparametric variance bound for the model and, when the data are missing completely at random, includes an estimator that is asymptotically equivalent to the inefficient estimators proposed by Pepe and Fleming and by Carroll and Wand. The optimal estimator in our class depends on the unknown probability law generating the data. When the vector V of non-missing regressors has at most two continuous components, we propose an adaptive semiparametric efficient estimator and compare the performance of the proposed semiparametric efficient estimator with the estimators proposed by Pepe and Fleming and Carroll and Wand in a small simulation study. When V has many continuous components, we propose an alternative class of adaptive estimators that should have high efficiency.

Fushing Hsieh

Papers

Twicing Kernels and a Small Bias Property of Semiparametric Estimators

Semiparametric efficient estimation of a conditional density with missing or mismeasured covariates

Undersmoothing and bias corrected functional estimation