This paper considers identification and estimation of a nonparametric regression model with an unobserved discrete covariate. The sample consists of a dependent variable and a set of covariates, one of which is discrete and arbitrarily correlates with the unobserved covariate. The observed discrete covariate has the same support as the unobserved covariate, and can be interpreted as a proxy or mismeasure of the unobserved one, but with a nonclassical measurement error that has an unknown distribution. We obtain nonparametric identification of the model given monotonicity of the regression function and a rank condition that is directly testable given the data. Our identification strategy does not require additional sample information, such as instrumental variables or a secondary sample. We then estimate the model via the method of sieve maximum likelihood, and provide root-n asymptotic normality and semiparametric efficiency of smooth functionals of interest. Two small simulations are presented to illustrate the identification and the estimation results.
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