  Triangular systems with nonadditively separable unobserved heterogeneity
provide a theoretically appealing framework for the modelling of complex
structural relationships. However, they are not commonly used in practice due
to the need for exogenous variables with large support for identification, the
curse of dimensionality in estimation, and the lack of inferential tools. This
paper introduces two classes of semiparametric nonseparable triangular models
that address these limitations. They are based on distribution and quantile
regression modelling of the reduced form conditional distributions of the
endogenous variables. We show that average, distribution and quantile
structural functions are identified in these systems through a control function
approach that does not require a large support condition. We propose a
computationally attractive three-stage procedure to estimate the structural
functions where the first two stages consist of quantile or distribution
regressions. We provide asymptotic theory and uniform inference methods for
each stage. In particular, we derive functional central limit theorems and
bootstrap functional central limit theorems for the distribution regression
estimators of the structural functions. These results establish the validity of
the bootstrap for three-stage estimators of structural functions, and lead to
simple inference algorithms. We illustrate the implementation and applicability
of all our methods with numerical simulations and an empirical application to
demand analysis.
