  This paper studies inference of preference parameters in semiparametric
discrete choice models when these parameters are not point-identified and the
identified set is characterized by a class of conditional moment inequalities.
Exploring the semiparametric modeling restrictions, we show that the identified
set can be equivalently formulated by moment inequalities conditional on only
two continuous indexing variables. Such formulation holds regardless of the
covariate dimension, thereby breaking the curse of dimensionality for
nonparametric inference based on the underlying conditional moment
inequalities. We further apply this dimension reducing characterization
approach to the monotone single index model and to a variety of semiparametric
models under which the sign of conditional expectation of a certain
transformation of the outcome is the same as that of the indexing variable.
