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Semiparametric Estimation of a Regression Model with an Unknown Transformation of the Dependent Variable

Joel L. Horowitz

Abstract

This paper presents a method for estimating the model $\Lambda(Y) = \beta'X + U$, where $Y$ is a scalar, $\Lambda$ is an unknown increasing function, $X$ is a vector of explanatory variables, $\beta$ is a vector of unknown parameters, and $U$ has unknown cumulative distribution function $F$. It is not assumed that $\Lambda$ and $F$ belong to known parametric families; they are estimated nonparametrically. This model generalizes a large number of widely used models that make stronger a priori assumptions about $\Lambda$ and/or $F$. The paper develops $n^{1/2}$-consistent, asymptotically normal estimators of $\Lambda, F$, and quantiles of the conditional distribution of $Y$. Estimators of $\beta$ that are $n^{1/2}$-consistent and asymptotically normal already exist. The results of Monte Carlo experiments indicate that the new estimators work reasonably well in samples of size 100.