Hidehiko Ichimura

Matching As An Econometric Evaluation Estimator: Evidence from Evaluating a Job Training Programme

This paper considers whether it is possible to devise a nonexperimental procedure for evaluating a prototypical job training programme. Using rich nonexperimental data, we examine the performance of a two-stage evaluation methodology that (a) estimates the probability that a person participates in a programme and (b) uses the estimated probability in extensions of the classical method of matching. We decompose the conventional measure of programme evaluation bias into several components and find that bias due to selection on unobservables, commonly called selection bias in econometrics, is empirically less important than other components, although it is still a sizeable fraction of the estimated programme impact. Matching methods applied to comparison groups located in the same labour markets as participants and administered the same questionnaire eliminate much of the bias as conventionally measured, but the remaining bias is a considerable fraction of experimentally-determined programme impact estimates. We test and reject the identifying assumptions that justify the classical method of matching. We present a nonparametric conditional difference-in-differences extension of the method of matching that is consistent with the classical index-sufficient sample selection model and is not rejected by our tests of identifying assumptions. This estimator is effective in eliminating bias, especially when it is due to temporally-invariant omitted variables.

Matching as an Econometric Evaluation Estimator

This paper develops the method of matching as an econometric evaluation estimator. A rigorous distribution theory for kernel-based matching is presented. The method of matching is extended to more general conditions than the ones assumed in the statistical literature on the topic. We focus on the method of propensity score matching and show that it is not necessarily better, in the sense of reducing the variance of the resulting estimator, to use the propensity score method even if propensity score is known. We extend the statistical literature on the propensity score by considering the case when it is estimated both parametrically and nonparametrically. We examine the benefits of separability and exclusion restrictions in improving the efficiency of the estimator. Our methods also apply to the econometric selection bias estimator.

Characterizing Selection Bias Using Experimental Data

This paper develops and applies semiparametric econometric methods to estimate the form of selection bias that arises from using nonexperimental comparison groups to evaluate social programs and to test the identifying assumptions that justify three widely-used classes of estimators and our extensions of them: (a) the method of matching; (b) the classical econometric selection model which represents the bias solely as a function of the probability of participation; and (c) the method of difference-in-differences. Using data from an experiment on a prototypical social program combined with unusually rich data from a nonexperimental comparison group, we reject the assumptions justifying matching and our extensions of that method but find evidence in support of the index-sufficient selection bias model and the assumptions that justify application of a conditional semiparametric version of the method of difference-in-difference. Fa comparable people and to appropriately weight participants and nonparticipants a sources of selection bias as conveniently measured. We present a rigorous defin bias and find that in our data it is a small component of conventially meausred it is still substantial when compared with experimentally-estimated program impa matching participants to comparison group members in the same labor market, givi same questionnaire, and making sure they have comparable characteristics substan the performance of any econometric program evaluation estimator. We show how t analysis to estimate the impact of treatment on the treated using ordinary obser

Semiparametric least squares (SLS) and weighted SLS estimation of single-index models

Abstract For the class of single-index models, I construct a semiparametric estimator of coefficients up to a multiplicative constant that exhibits 1 √ n -consistency and asymptotic normality. This class of models includes censored and truncated Tobit models, binary choice models, and duration models with unobserved individual heterogeneity and random censoring. I also investigate a weighting scheme that achieves the semiparametric efficiency bound.

Identification and estimation of polynomial errors-in-variables models

Abstract Methods of estimation of regression coefficients are proposed when the regression function includes a polynomial in a ‘true’ regressor which is measured with error. Two sources of additional information concerning the unobservable regressor are considered: either an additional indicator of the regressor (itself measured with error) or instrumental variables which characterize the systematic variation in the true regressor. In both cases, estimators are constructed by relating moments involving the unobserved variables to moments of observables; these relations lead to recursion formulae for computation of the regression coefficients and nuisance parameters (e.g., moments of the measurement error). Consistency and asymptotic normality of the estimated coefficients is demonstrated, and consistent estimators of the asymptotic covariant matrices are provided.

Maximum Likelihood Estimation of a Binary Choice Model with Random Coefficients of Unknown Distribution

Abstract We consider a binary response model yi=1{xi′βi+ei⩾0} with xi independent of the unobservables (β i , e i ) . No finite-dimensional parametric restrictions are imposed on F0, the joint distribution of (β i , e i ) . A nonparametric maximum likelihood estimator for F0 is shown to be consistent. We analyze some conditions under which F0 is or is not identified. In particular, we show that if the support of F0 is a subset of any half of the unit hypersphere, then F0 is identified relative to all distributions on the unit hypersphere. We also provide some Monte Carlo evidence on the small sample performance of our estimator.

Chapter 74 Implementing Nonparametric and Semiparametric Estimators

This chapter reviews recent advances in nonparametric and semiparametric estimation, with an emphasis on applicability to empirical research and on resolving issues that arise in implementation. It considers techniques for estimating densities, conditional mean functions, derivatives of functions and conditional quantiles in a flexible way that imposes minimal functional form assumptions. The chapter begins by illustrating how flexible modeling methods have been applied in empirical research, drawing on recent examples of applications from labor economics, consumer demand estimation and treatment effects models. Then, key concepts in semiparametric and nonparametric modeling are introduced that do not have counterparts in parametric modeling, such as the so-called curse of dimensionality, the notion of models with an infinite number of parameters, the criteria used to define optimal convergence rates, and “dimension-free” estimators. After defining these new concepts, a large literature on nonparametric estimation is reviewed and a unifying framework presented for thinking about how different approaches relate to one another. Local polynomial estimators are discussed in detail and their distribution theory is developed. The chapter then shows how nonparametric estimators form the building blocks for many semiparametric estimators, such as estimators for average derivatives, index models, partially linear models, and additively separable models. Semiparametric methods offer a middle ground between fully nonparametric and parametric approaches. Their main advantage is that they typically achieve faster rates of convergence than fully nonparametric approaches. In many cases, they converge at the parametric rate. The second part of the chapter considers in detail two issues that are central with regard to implementing flexible modeling methods: how to select the values of smoothing parameters in an optimal way and how to implement “trimming” procedures. It also reviews newly developed techniques for deriving the distribution theory of semiparametric estimators. The chapter concludes with an overview of approximation methods that speed up the computation of nonparametric estimates and make flexible estimation feasible even in very large size samples.

Treatment Evaluation With Selective Participation and Ineligibles

Matching methods for treatment evaluation based on a conditional independence assumption do not balance selective unobserved differences between treated and nontreated. We derive a simple correction term if there is an instrument that shifts the treatment probability to zero in specific cases. Policies with eligibility restrictions, where treatment is impossible if some variable exceeds a certain value, provide a natural application. In an empirical analysis, we exploit the age eligibility restriction in the Swedish Youth Practice subsidized work program for young unemployed, where compliance is imperfect among the young. Adjusting the matching estimator for selectivity changes the results toward making subsidized work detrimental in moving individuals into employment.

Simple Estimators for Invertible Index Models

ABSTRACT This article considers estimation of the unknown linear index coefficients of a model in which a number of nonparametrically identified reduced form parameters are assumed to be smooth and invertible function of one or more linear indices. The results extend the previous literature by allowing the number of reduced form parameters to exceed the number of indices (i.e., the indices are “overdetermined” by the reduced form parameters. The estimator of the unknown index coefficients (up to scale) is the eigenvector of a matrix (defined in terms of a first-step nonparametric estimator of the reduced form parameters) corresponding to its smallest (in magnitude) eigenvalue. Under suitable conditions, the proposed estimator is shown to be root-n-consistent and asymptotically normal, and under additional restrictions an efficient choice of a “weight matrix” is derived in the overdetermined case.

Rejoinder for “Simple Estimators for Invertible Index Models”

We are grateful to the discussants for their generous and insightful comments on our article, and to the co-editor, Shakeeb Khan, for the opportunity to contribute to this forum and to finish up a decades-long collaborative effort. The discussants’ comments mostly express some well-justified concerns about the strength and generality of the assumptions we impose in our “matching” approach to estimation of index coefficients, concerns which, we believe, are equally applicable to much of the literature on semiparametric index restrictions. Our estimation method relies on three key assumptions—the (linear) index structure of the reduced form parameters, invertibility of the reduced form parameters in the vector of indices, and the variability of the regressors given the reduced form parameters (or, equivalently, given the indices). Given these assumptions, we construct an estimator of a matrix Ŝ, which is essentially an average conditional variance matrix of the regressors given the reduced form parameters, and show how it has exactly one zero eigenvalue (with corresponding eigenvector proportional to the index coefficients θ0). There are many ways one or more of the key assumptions can fail in practice, and the discussants note three possibilities, namely, heteroscedasticity, partial identification of the reduced form parameters, and discrete regressors, each of which can yield an estimated matrix Ŝ with many or no zero eigenvalues in the limit.