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Identification and shape restrictions in nonparametric instrumental variables estimation

Authors: Joachim Freyberger and Joel L. Horowitz
Date: 01 July 2013
Type: cemmap Working Paper, CWP31/13
DOI: 10.1920/wp.cem.2013.3113

Abstract

This paper is concerned with inference about an unidentified linear function, L(g), where the function g satisfies the relation Y=g(X)+U; E(U |W)=0. In this relation, Y is the dependent variable, X is a possibly endogenous explanatory variable, W is an instrument for X and U is an unobserved random variable. The data are an independent random sample of (Y, X, W). In much applied research, X and W are discrete, and W has fewer points of support than X. Consequently, neither g nor L(g) is nonparametrically identified. Indeed, L(g) can have any value in (-∞, ∞). In applied research, this problem is typically overcome and point identification is achieved by assuming that g is a linear function of X. However, the assumption of linearity is arbitrary. It is untestable if W is binary, as is the case in many applications. This paper explores the use of shape restrictions, such as monotonicity or convexity, for achieving interval identification of L(g). Economic theory often provides such shape restrictions. This paper shows that they restrict L(g) to an interval whose upper and lower bounds can be obtained by solving linear programming problems. Inference about the identified interval and the functional L(g) can be carried out by using the bootstrap. An empirical application illustrates the usefulness of shape restrictions for carrying out nonparametric inferences about L(g). An extension to nonseparable and quantile IV models is described. Download full version
Now published:
Joachim Freyberger and Joel L. Horowitz November 2015, Identification and shape restrictions in nonparametric instrumental variables estimation, Journal Article, Elsevier
Previous version:
Joachim Freyberger and Joel L. Horowitz June 2012, Identification and shape restrictions in nonparametric instrumental variables estimation, cemmap Working Paper, CWP15/12

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