Nonparametric Instrumental Variable Estimators of Structural Quantile Effects

2009

We study Tikhonov Regularized estimation of quantile structural effects implied by a nonseparable model. The nonparametric instrumental variable estimator is based on a minimum distance principle. We show that the minimum distance problem without regularization is locally ill-posed, and consider penalization by the norms of the parameter and its derivative. We derive the asymptotic Mean Integrated Square Error, the rate of convergence and the pointwise asymptotic normality under a regularization parameter depending on sample size. We illustrate our theoretical findings and the small sample properties with simulation results in two numerical examples. We also discuss a data driven selection procedure of the regularization parameter via a spectral representation of the MISE. Finally, we provide an empirical application to estimation of Engel curves.

Econometrica, 2012

We study Tikhonov Regularized estimation of quantile structural effects implied by a nonseparable model. The nonparametric instrumental variable estimator is based on a minimum distance principle. We show that the minimum distance problem without regularization is locally ill-posed, and consider penalization by the norms of the parameter and its derivative. We derive the asymptotic Mean Integrated Square Error, the rate of convergence and the pointwise asymptotic normality under a regularization parameter depending on sample size. We illustrate our theoretical findings and the small sample properties with simulation results in two numerical examples. We also discuss a data driven selection procedure of the regularization parameter via a spectral representation of the MISE. Finally, we provide an empirical application to estimation of Engel curves.

Econometrica, 2007

We consider nonparametric estimation of a regression function that is identified by requiring a specified quantile of the regression "error" conditional on an instrumental variable to be zero. The resulting estimating equation is a nonlinear integral equation of the first kind, which generates an ill-posed-inverse problem. The integral operator and distribution of the instrumental variable are unknown and must be estimated nonparametrically. We show that the estimator is mean-square consistent, derive its rate of convergence in probability, and give conditions under which this rate is optimal in a minimax sense. The results of Monte Carlo experiments show that the estimator behaves well in finite samples. JEL Codes: C13, C31

2011

These supplementary materials contain the proofs of Lemmas A.1-A.2 in Section 1 (illposedness) and Lemma A.3 in Section 2 (consistency). Section 2 also establishes existence of the Q-TiR estimator. Section 3 provides the Frechet derivative of operator A and a characterization of its adjoints. Lemmas A.4-A.11 are proved in Section 4 (asymptotic distribution) and Lemmas A.12-A.15 in Section 5 (estimation of the asymptotic variance). In Section 6 we characterize the asymptotic MISE. In Section 7 we provide an example of a NIVQR model and derive the spectrum of AA (Remark 1), the spectrum of A * A (Remark 2), and the asymptotic behavior of the variance function (Remark 3). To streamline the presentation we gather the proofs of the secondary technical Lemmas B.1-B.15 and C.1-C.10 at the end of this technical report (Section 8). Equations labelled as (n) refer to the paper, and Equations labelled as (SM.n) refer to the supplementary materials. To simplify the proofs, we adopt a product kernel in the estimation of the density of (X, Y, Z) in R d. We use the generic notation K for both the d-dimensional product kernel and each of its components. We take C as a generic constant.

Journal of Econometrics, 2012

We study a Tikhonov Regularized (TiR) estimator of a functional parameter identified by conditional moment restrictions in a linear model with both exogenous and endogenous regressors. The nonparametric instrumental variable estimator is based on a minimum distance principle with penalization by the norms of the parameter and its derivatives. After showing its consistency in the Sobolev norm and uniform consistency under an embedding condition, we derive the expression of the asymptotic Mean Integrated Square Error and the rate of convergence. The optimal value of the regularization parameter is characterized in two examples. We illustrate our theoretical findings and the small sample properties with simulation results. Finally, we provide an empirical application to estimation of an Engel curve, and discuss a data driven selection procedure for the regularization parameter.

2005

Lee (2003) develops a n-consistent estimator of the parametric component of a partially linear quantile regression model, which is used to obtain his one-step semiparametric efficient estimator. As a result, how well the efficient estimator performs depends on the quality of the initial n-consistent estimator. In this paper, we aim to improve the small sample performance of the one-step efficient

We consider asymptotic and finite sample confidence bounds in instrumental variables quantile regressions of wages on schooling with relatively weak instruments. We find practically important differences between the asymptotic and finite sample interval estimates.

2005

This paper is concerned with estimating the additive components of a nonparametric additive quantile regression model. We develop an estimator that is asymptotically normally distributed with a rate of convergence in probability of n −r/(2r+1) when the additive components are r-times continuously differentiable for some r ≥ 2. This result holds regardless of the dimension of the covariates and, therefore, the new estimator has no curse of dimensionality. In addition, the estimator has an oracle property and is easily extended to a generalized additive quantile regression model with a link function. The numerical performance and usefulness of the estimator are illustrated by Monte Carlo experiments and an empirical example. * We thank Andrew Chesher, Hidehiko Ichimura, Roger Koenker, an editor, an associate editor, and two anonymous referees for helpful comments and suggestions. One referee kindly pointed out some mistakes in the proofs of an earlier draft and suggested corrections. Special thanks go to Andrew Chesher for encouraging us to work on this project. A preliminary version of the paper was presented at the

Econometrics Journal, 2023

This paper develops a first-stage linear regression representation for the instrumental variables (IV) quantile regression (QR) model. The quantile first-stage is analogous to the least squares case, i.e., a linear projection of the endogenous variables on the instruments and other exogenous covariates, with the difference that the QR case is a weighted projection. The weights are given by the conditional density function of the innovation term in the QR structural model, conditional on the endogeneous and exogenous covariates, and the instruments as well, at a given quantile. We also show that the required Jacobian identification conditions for IVQR models are embedded in the quantile first-stage. We then suggest inference procedures to evaluate the adequacy of instruments by evaluating their statistical significance using the first-stage result. The test is developed in an overidentification context, since consistent estimation of the weights for implementation of the first-stage requires at least one valid instrument to be available. Monte Carlo experiments provide numerical evidence that the proposed tests work as expected in terms of empirical size and power in finite samples. An empirical application illustrates that checking for the statistical significance of the instruments at different quantiles is important. The proposed procedures may be specially useful in QR since the instruments may be relevant at some quantiles but not at others.

Econometric Theory, 2009

We propose an efficient semiparametric estimator for the coefficients of a multivariate linear regression model -with a conditional quantile restriction for each equation -in which the conditional distributions of errors given regressors are unknown. The procedure can be used to estimate multiple conditional quantiles of the same regression relationship. The proposed estimator is asymptotically as efficient as if the true optimal instruments were known. Simulation results suggest that the estimation procedure works well in practice and dominates an equation-by-equation efficiency correction if the errors are dependent conditional on the regressors.