Semiparametric Estimation of Structural Functions in Nonseparable Triangular Models

TL;DR

This paper develops semiparametric methods for estimating structural functions in nonseparable triangular models, overcoming identification and computational challenges, and provides inference tools validated through simulations and an empirical demand analysis.

Contribution

It introduces two classes of semiparametric models that identify structural functions without large support conditions and offers a three-stage estimation procedure with asymptotic theory and bootstrap inference.

Findings

Structural functions are identified via a control function approach.

The proposed estimators are asymptotically normal with valid bootstrap inference.

Numerical simulations and an empirical application demonstrate the methods' effectiveness.

Abstract

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.