Working Paper

Identification and shape restrictions in nonparametric instrumental variables estimation


Joachim Freyberger, Joel L. Horowitz

Published Date

1 July 2013


Working Paper (CWP31/13)

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.