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Identification and shape restrictions in nonparametric instrumental variables estimation

  • Joachim Freyberger

    (Institute for Fiscal Studies)

  • Joel Horowitz

    ()

    (Institute for Fiscal Studies and Northwestern University)

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    This paper is concerned with inference about an unidentified linear functional, 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 by using the bootstrap. An empirical application illustrates the usefulness of shape restrictions for carrying out nonparametric inference about L(g).

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    File URL: http://www.cemmap.ac.uk/wps/cwp151212.pdf
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    Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP15/12.

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    Date of creation: Jun 2012
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    Handle: RePEc:ifs:cemmap:15/12
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    1. Charles F. Manski & John V. Pepper, 2009. "More on monotone instrumental variables," Econometrics Journal, Royal Economic Society, vol. 12(s1), pages S200-S216, 01.
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    10. Guido Imbens & Charles F. Manski, 2003. "Confidence intervals for partially identified parameters," CeMMAP working papers CWP09/03, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    11. Jorg Stoye, 2009. "More on Confidence Intervals for Partially Identified Parameters," Econometrica, Econometric Society, vol. 77(4), pages 1299-1315, 07.
    12. Ivan Canay & Andres Santos & Azeem Shaikh, 2012. "On the testability of identification in some nonparametric models with endogeneity," CeMMAP working papers CWP18/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    13. Andres Santos, 2012. "Inference in Nonparametric Instrumental Variables With Partial Identification," Econometrica, Econometric Society, vol. 80(1), pages 213-275, 01.
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