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Nonparametric methods for inference in the presence of instrumental variables
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
arXiv: Econometrics, 2019
We consider a nonparametric instrumental regression model with continuous endogenous regressor where instruments are fully independent of the error term. This assumption allows us to extend the reach of this model to cases where the instrumental variable is discrete, and therefore to substantially enlarge its potential empirical applications. Under our assumptions, the regression function becomes solution to a nonlinear integral equation. We contribute to existing literature by providing an exhaustive analysis of identification and a simple iterative estimation procedure. Details on the implementation and on the asymptotic properties of this estimation algorithm are given. We conclude the paper with a simulation experiment for a binary instrument and an empirical application to the estimation of the Engel curve for food, where we show that our estimator delivers results that are consistent with existing evidence under several discretizations of the instrumental variable.
Applied Nonparametric Instrumental Variables Estimation
Econometrica, 2011
Instrumental variables are widely used in applied econometrics to achieve identification and carry out estimation and inference in models that contain endogenous explanatory variables. In most applications, the function of interest (e.g., an Engel curve or demand function) is assumed to be known up to finitely many parameters (e.g., a linear model), and instrumental variables are used identify and estimate these parameters. However, linear and other finite-dimensional parametric models make strong assumptions about the population being modeled that are rarely if ever justified by economic theory or other a priori reasoning and can lead to seriously erroneous conclusions if they are incorrect. This paper explores what can be learned when the function of interest is identified through an instrumental variable but is not assumed to be known up to finitely many parameters. The paper explains the differences between parametric and nonparametric estimators that are important for applied research, describes an easily implemented nonparametric instrumental variables estimator, and presents empirical examples in which nonparametric methods lead to substantive conclusions that are quite different from those obtained using standard, parametric estimators.
Estimating Engel curves under unit and item nonresponse
Journal of Applied Econometrics, 2011
This paper estimates food Engel curves using data from the first wave of the Survey on Health, Aging and Retirement in Europe (SHARE). Our statistical model simultaneously takes into account selectivity due to unit and item nonresponse, endogeneity problems, and issues related to flexible specification of the relationship of interest. We estimate both parametric and semiparametric specifications of the model. The parametric specification assumes that the unobservables in the model follow a multivariate Gaussian distribution, while the semiparametric specification avoids distributional assumptions about the unobservables.
Non-parametric Models with Instrumental Variables
2010
This chapter gives a survey of econometric models characterized by a relation between observable and unobservable random elements where these unobservable terms are assumed to be independent of another set of observable variables called instrumental variables. This kind of specification is usefull to address the question of endogeneity or of selection bias for examples. These models are treated non parametrically and, in all the examples we consider, the functional parameter of interest is defined as the solution of a linear or non linear integral equation. The estimation procedure then requires to solve a (generally ill-posed) inverse problem.We illustrate the main questions (construction of the equation, identification, numerical solution, asymptotic properties, selection of the regularization parameter) with the different models we present.
Nonparametric Instrumental Regression
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.
Specification testing in nonparametric instrumental variable estimation
Journal of Econometrics, 2012
In nonparametric instrumental variables estimation, the function being estimated is the solution to an integral equation. A solution may not exist if, for example, the instrument is not valid. This paper discusses the problem of testing the null hypothesis that a solution exists against the alternative that there is no solution. We give necessary and sufficient conditions for existence of a solution and show that uniformly consistent testing of an unrestricted null hypothesis is not possible. Uniformly consistent testing is possible, however, if the nullhypothesis is restricted by assuming that any solution to the integral equation is smooth. Many functions of interest in applied econometrics, including demand functions and Engel curves, are expected to be smooth. The paper presents a statistic for testing the null hypothesis that a smooth solution exists. The test is consistent uniformly over a large class of probability distributions of the observable random variables for which the integral equation has no smooth solution. The finite-sample performance of the test is illustrated through Monte Carlo experiments.
Parameterization and inference for nonparametric regression problems
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2001
We consider local likelihood or local estimating equations, in which a multivariate function Â(.) is estimated but a derived function (.) of Â(.) is of interest. In many applications, when most naturally formulated the derived function is a non-linear function of Â(.). In trying to understand whether the derived non-linear function is constant or linear, a problem arises with this approach: when the function is actually constant or linear, the expectation of the function estimate need not be constant or linear, at least to second order. In such circumstances, the simplest standard methods in nonparametric regression for testing whether a function is constant or linear cannot be applied. We develop a simple general solution which is applicable to nonparametric regression, varying-coef®cient models, nonparametric generalized linear models, etc. We show that, in local linear kernel regression, inference about the derived function (.) is facilitated without a loss of power by reparameterization so that (.) is itself a component of Â(.). Our approach is in contrast with the standard practice of choosing Â(.) for convenience and allowing (.) to be a non-linear function of Â(.). The methods are applied to an important data set in nutritional epidemiology.
A Specification Test for Nonparametric Instrumental Variable Regression
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.
Estimation and Testing an Additive Partially Linear Model in a System of Engel Curves
2006
The form of the Engel curve has long been a subject of discussion in appliedeconometrics and until now there has no been definitive conclusion about its form. In this paperan additive partially linear model is used to estimate semiparametrically the effect of totalexpenditure in the context of the Engel curves. Additionally, we consider the non-parametricinclusion of some regressors which traditionally have a non linear effect such as age andschooling. To that end we compare an additive partially linear model with the fullynonparametric one using recent popular test statistics. We also provide the p-values computedby bootstrap and subsampling schemes for the proposed test statistics. Empirical analysis basedon data drawn from the Spanish Expenditure Survey 1990-91 shows that modelling the effectsof expenditure, age and schooling on budget share deserves a treatment better than that adoptedin simple semiparametric analysis.