Two Stage Rank Estimation of Quantile Index Models

This paper estimates a class of models which satisfy a monotonicity condition on the conditional quantile function of the response variable. This class includes as a special case the monotonic transformation model with the error term satisfying a conditional quantile restriction, thus allowing for very general forms of conditional heteroscedasticity. Furthermore, the monotonicity condition enables the inclusion of many popular limited dependent variable models which would be ruled out by imposing a smoothness condition. A two stage, \minimum distance" approach is adopted to estimate the relevant parameters. In the ̄rst stage the (in ̄nite dimensional) reduced form parameters are estimated nonparametrically by the local polynomial estimator discussed in Chaudhuri(1991a,b) and Cavanagh(1996). In the second stage, the monotonicity of the quantile function is exploited to minimize the distance between the reduced and structural forms by maximizing a rank based objective function. The proposed estimator is shown to have desirable asymptotic properties and can then also be used for dimensionality reduction or to estimate the unknown structural function in the context of a transformation model. JEL Classi ̄cation: C13,C14,C24,C41