A Bayesian switching model for causal inference with constraints and nonlinear functions

In this article, we generalize the switching model to allow nonlinear regression functions with constraints. We study the associated inference problems in the Bayesian framework and develop efficient Gibbs samplers for the MCMC simulation. We first analyze the linear switching model with linear constraints on the regression coefficients. We next generalize the model to the nonlinear framework. We investigate three methods: piecewise linear approximation, orthogonal polynomial expansion and B-spline approximation. By going beyond the linear model, we extend the boundary of the model's applicability, in particular, when the outcome equations and decision equation are of unknown function form. We further develop a general framework to incorporate various constraints to the model, such as monotonicity and convexity on both outcome and selection equations. These constraints are in general described by a system of linear inequalities on the regression coefficients. There are virtually no restrictions on the number of inequalities and the linear system can even be singular. These constraints are further incorporated into the Bayesian framework through conditional posterior distributions of the parameters that are truncated to certain ranges determined by other variables. For each method we provide analysis on simulated data to illustrate the model application. We further apply these models to analyze union wage premium from labor economics. Results suggest that providing appropriate constraints can improve the efficiency of parameter estimation, provide better function estimates and smooth the nonparametric estimated curve.