Specification testing in nonparametric instrumental variable estimation

Joel L. Horowitz*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


In nonparametric instrumental variable 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 null hypothesis 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.

Original languageEnglish (US)
Pages (from-to)383-396
Number of pages14
JournalJournal of Econometrics
Issue number2
StatePublished - Apr 2012


  • Instrumental variable
  • Inverse problem
  • Linear operator
  • Series estimator

ASJC Scopus subject areas

  • Economics and Econometrics


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