On ill-posedness of nonparametric instrumental variable regression with convexity constraints

2007

The nonparametric estimation of a regression function x from conditional moment restrictions involving instrumental variables is considered. The rate of convergence of penalized estimators is studied in the case where x is not identified from the conditional moment restriction. We also study the gain of modifying the penalty in the estimation, considering for instance a Sobolev-type of penalty. We analyze

Journal of Statistical Planning and Inference, 2013

We consider the nonparametric regression model with an additive error that is correlated with the explanatory variables. We suppose the existence of instrumental variables that are considered in this model for the identification and the estimation of the regression function. The nonparametric estimation by instrumental variables is an ill-posed linear inverse problem with an unknown but estimable operator. We provide a new estimator of the regression function using an iterative regularization method (the Landweber-Fridman method). The optimal number of iterations and the convergence of the mean square error of the resulting estimator are derived under both mild and severe degrees of ill-posedness. A Monte-Carlo exercise shows the impact of some parameters on the estimator and concludes on the reasonable finite sample performance of the new estimator.

Econometrica, 2011

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

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.

The Econometrics Journal, 2012

We consider the semiparametric regression X t β+φ(Z) where β and φ(·) are unknown slope coefficient vector and function, and where the variables (X, Z) are endogeneous. We propose necessary and sufficient conditions for the identification of the parameters in the presence of instrumental variables. We also focus on the estimation of β. An incorrect parametrization of φ generally leads to an inconsistent estimator of β, whereas consistent nonparametric estimators for β have a slow rate of convergence. An additional complication is that the solution of the equation necessitates the inversion of a compact operator which can be estimated nonparametrically. In general this inversion is not stable, thus the estimation of β is ill-posed. In this paper, a √ n-consistent estimator for β is derived under mild assumptions. One of these assumptions is given by the socalled source condition which we explicit and interpret in the paper. Finally we show that the estimator achieves the semiparametric efficiency bound, even if the model is heteroskedastic.

Journal of Econometrics, 2015

Identification and shape restrictions in nonparametric instrumental variables estimation cemmap working paper, No. CWP31/13

Journal of Econometrics, 2014

In nonparametric instrumental variables estimation, the mapping that identifies the function of interest, g say, is discontinuous and must be regularized (that is, modified) to make consistent estimation possible. The amount of modification is controlled by a regularization parameter. The optimal value of this parameter depends on unknown population characteristics and cannot be calculated in applications. Theoretically justified methods for choosing the regularization parameter empirically in applications are not yet available. This paper presents such a method for use in series estimation, where the regularization parameter is the number of terms in a series approximation to g. The method does not require knowledge of the smoothness of g or of other unknown functions. It adapts to their unknown smoothness. The estimator of g based on the empirically selected regularization parameter converges in probability at a rate that is at least as fast as the asymptotically optimal rate multiplied by 1/ 2 (log) n , where n is the sample size. The asymptotic integrated mean-square error (AIMSE) of the estimator is within a specified factor of the optimal AIMSE.

The Annals of Statistics, 2005

We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables. For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estimators. In the presence of instrumental variables the relation that identifies the regression function also defines an ill-posed inverse problem, the "difficulty" of which depends on eigenvalues of a certain integral operator which is determined by the joint density of endogenous and instrumental variables. We delineate the role played by problem difficulty in determining both the optimal convergence rate and the appropriate choice of smoothing parameter. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2005, Vol. 33, No. 6, 2904-2929. This reprint differs from the original in pagination and typographic detail. 1 2 P. HALL AND J. L. HOROWITZ individuals tend to choose high levels of education, then education is correlated with ability, thereby causing U i to be correlated with at least some components of X i . Suppose, however, that for each i we have available another observed data value, W i , say (an instrumental variable), for which

Working Papers, 2008

This paper proposes and discusses an instrumental variable estimator that can be of particular relevance when many instruments are available. Intuition and recent work (see, eg, Hahn (2002)) suggest that parsimonious devices used in the construction of the final ...

SSRN Electronic Journal, 2008

This technical report contains the proofs of the technical Lemmas B.1-B.8, C.1-C.4, and D.1-D.3 in the paper entitled "A Specification Test for Nonparametric Instrumental Variable Regression" and written by P. Gagliardini and O. Scaillet. Equations labelled as (n) refer to the paper, and Equations labelled as (TR.n) refer to the technical report. To simplify the proofs, we adopt a product kernel in the estimation of the density of (Y, X, Z). We use the generic notation K for both the 3-dimensional product kernel and each of its components.