Abstract:
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.
A two-stage approach is adopted to estimate the relevant parameters. In the "rst stage
the conditional quantile function is estimated nonparametrically by the local polynomial
estimator discussed in Chaudhuri (Journal of Multivariate Analysis 39 (1991a) 246}269;
Annals of Statistics 19 (1991b) 760}777) and Cavanagh (1996, Preprint). In the second
stage, the monotonicity of the quantile function is exploited to estimate the parameters of
interest 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.
