Estimating Derivatives in Nonseparable Models With Limited Dependent Variables


  • Joseph G. Altonji,

    1. Dept. of Economics, Yale University, 37 Hillhouse Avenue, New Haven, CT 06520, U.S.A.;
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  • Hidehiko Ichimura,

    1. Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033 Japan;
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  • Taisuke Otsu

    1. Cowles Foundation and Dept. of Economics, Yale University, P.O. Box 208281, New Haven, CT 06520, U.S.A.;
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    • We thank a co-editor and three referees for their insightful comments. Our research was supported by National Science Foundation Grant SBR-9512009, the Institute for Policy Research, Northwestern University, the Economic Growth Center and the Cowles Foundation, and Yale University (Altonji), JSPS Basic Research (B) Grant 18330040 (Ichimura), and National Science Foundation Grant SES-0720961 (Otsu).


We present a simple way to estimate the effects of changes in a vector of observable variables X on a limited dependent variable Y when Y is a general nonseparable function of X and unobservables, and X is independent of the unobservables. We treat models in which Y is censored from above, below, or both. The basic idea is to first estimate the derivative of the conditional mean of Y given X at x with respect to x on the uncensored sample without correcting for the effect of x on the censored population. We then correct the derivative for the effects of the selection bias. We discuss nonparametric and semiparametric estimators for the derivative. We also discuss the cases of discrete regressors and of endogenous regressors in both cross section and panel data contexts.