Heterogenous Coefficients, Discrete Instruments, and Identification of Treatment Effects

Multidimensional heterogeneity and endogeneity are important features of a wide class of econometric models. We consider heterogenous coefficients models where the outcome is a linear combination of known functions of treatment and heterogenous coefficients. We use control variables to obtain identification results for average treatment effects. With discrete instruments in a triangular model we find that average treatment effects cannot be identified when the number of support points is less than or equal to the number of coefficients. A sufficient condition for identification is that the second moment matrix of the treatment functions given the control is nonsingular with probability one. We relate this condition to identification of average treatment effects with multiple treatments.Comment: 15 page

Treatment Effects

This essay discusses the issues of identification and estimation of the average treatment effect and the average effect of treatment on the treated

Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity

This paper investigates identification and inference in a nonparametric structural model with instrumental variables and non-additive errors. We allow for non-additive errors because the unobserved heterogeneity in marginal returns that often motivates concerns about endogeneity of choices requires objective functions that are non-additive in observed and unobserved components. We formulate several independence and monotonicity conditions that are sufficient for identification of a number of objects of interest, including the average conditional response, the average structural function, as well as the full structural response function. For inference we propose a two-step series estimator. The first step consists of estimating the conditional distribution of the endogenous regressor given the instrument. In the second step the estimated conditional distribution function is used as a regressor in a nonlinear control function approach. We establish rates of convergence, asymptotic normality, and give a consistent asymptotic variance estimator.

Automatic Debiased Machine Learning of Causal and Structural Effects

Many causal and structural effects depend on regressions. Examples include average treatment effects, policy effects, average derivatives, regression decompositions, economic average equivalent variation, and parameters of economic structural models. The regressions may be high dimensional. Plugging machine learners into identifying equations can lead to poor inference due to bias and/or model selection. This paper gives automatic debiasing for estimating equations and valid asymptotic inference for the estimators of effects of interest. The debiasing is automatic in that its construction uses the identifying equations without the full form of the bias correction and is performed by machine learning. Novel results include convergence rates for Lasso and Dantzig learners of the bias correction, primitive conditions for asymptotic inference for important examples, and general conditions for GMM. A variety of regression learners and identifying equations are covered. Automatic debiased machine learning (Auto-DML) is applied to estimating the average treatment effect on the treated for the NSW job training data and to estimating demand elasticities from Nielsen scanner data while allowing preferences to be correlated with prices and income

Constrained Conditional Moment Restriction Models

This paper examines a general class of inferential problems in semiparametric and nonparametric models defined by conditional moment restrictions. We construct tests for the hypothesis that at least one element of the identified set satisfies a conjectured (Banach space) "equality" and/or (a Banach lattice) "inequality" constraint. Our procedure is applicable to identified and partially identified models, and is shown to control the level, and under some conditions the size, asymptotically uniformly in an appropriate class of distributions. The critical values are obtained by building a strong approximation to the statistic and then bootstrapping a (conservatively) relaxed form of the statistic. Sufficient conditions are provided, including strong approximations using Koltchinskii's coupling. Leading important special cases encompassed by the framework we study include: (i) Tests of shape restrictions for infinite dimensional parameters; (ii) Confidence regions for functionals that impose shape restrictions on the underlying parameter; (iii) Inference for functionals in semiparametric and nonparametric models defined by conditional moment (in)equalities; and (iv) Uniform inference in possibly nonlinear and severely ill-posed problems

Treatment effects (in Russian)

This essay discusses the issues of identification and estimation of the average treatment effect and the average effect of treatment on the treated.

Inference in Linear Regression Models with Many Covariates and Heteroskedasticity

The linear regression model is widely used in empirical work in Economics, Statistics, and many other disciplines. Researchers often include many covariates in their linear model specification in an attempt to control for confounders. We give inference methods that allow for many covariates and heteroskedasticity. Our results are obtained using high-dimensional approximations, where the number of included covariates are allowed to grow as fast as the sample size. We find that all of the usual versions of Eicker-White heteroskedasticity consistent standard error estimators for linear models are inconsistent under this asymptotics. We then propose a new heteroskedasticity consistent standard error formula that is fully automatic and robust to both (conditional)\ heteroskedasticity of unknown form and the inclusion of possibly many covariates. We apply our findings to three settings: parametric linear models with many covariates, linear panel models with many fixed effects, and semiparametric semi-linear models with many technical regressors. Simulation evidence consistent with our theoretical results is also provided. The proposed methods are also illustrated with an empirical application

Alternative Asymptotics and the Partially Linear Model with Many Regressors

Non-standard distributional approximations have received considerable attention in recent years. They often provide more accurate approximations in small samples, and theoretical improvements in some cases. This paper shows that the seemingly unrelated "many instruments asymptotics" and "small bandwidth asymptotics" share a common structure, where the object determining the limiting distribution is a V-statistic with a remainder that is an asymptotically normal degenerate U-statistic. We illustrate how this general structure can be used to derive new results by obtaining a new asymptotic distribution of a series estimator of the partially linear model when the number of terms in the series approximation possibly grows as fast as the sample size, which we call "many terms asymptotics"

A symptotic Bias for GMM and GEL Estimators with Estimated Nuisance Parameter

This papers studies and compares the asymptotic bias of GMM and generalized empirical likelihood (GEL) estimators in the presence of estimated nuisance parameters. We consider cases in which the nuisance parameter is estimated from independent and identical samples. A simulation experiment is conducted for covariance structure models. Empirical likelihood offers much reduced mean and median bias, root mean squared error and mean absolute error, as compared with two-step GMM and other GEL methods. Both analytical and bootstrap bias-adjusted two-step GMM estima-tors are compared. Analytical bias-adjustment appears to be a serious competitor to bootstrap methods in terms of finite sample bias, root mean squared error and mean absolute error. Finite sample variance seems to be little affected