Association for Public Policy Analysis & Management

In this paper, we propose a semi-nonparametric estimation procedure for estimating average treatment effects with a binary treatment in the framework of endogenous switching models. We consider a generalized model for the reduced form of the treatment variable that allows for the heterogeneity in terms of an distribution- free, conditional-heteroskedastic error. In the outcome equation, we consider both constant coefficient and heterogeneous coefficient case. We begin with pure-cross section model and extend the estimation methods to the panel data case.

Endogenous switching models have been traditionally estimated using a joint maximum likelihood estimation procedure that requires full specification of the joint distribution of the unobservables in both primary outcome equation and in the reduced form of the treatment variable. This approach not only imposes restrictive distributional assumptions on the model but is also computationally challenging. Murtazashvili and Wooldridge(2015) use control function methods to obtain a computationally simple estimation of switching models. They derive simple multi-step estimation methods for cross-section and panel data linear models with both constant and heterogeneous slopes. They however still maintain distributional assumptions and homoskedastic errors.

In this paper, we extend the control function methods to a distribution-free estimation of the switching models. In the reduced form for the treatment variable, we allow heterogeneity in the model by incorporating multiplicative heteroskedasticity of an unknown form. This permits a general model in the reduced form where the unconditional error distribution is of an unknown form as opposed to the traditional parametric assumptions that translate into a more restrictive probit model for the treatment variable. In the primary outcome model, we first consider the constant coefficients case while still imposing no distributional assumptions for the errors. We obtain the estimating equation that includes the correction terms accounting for the bias caused due to the endogenous switching. Since our treatment variable model allows hetereskedasticity of an unkown form, we obtain the correction terms in the estimating equation to be functions of unkown form.

We propose a two-step non-parametric estimation method to estimate our parameters that lie in an infinite dimensional parameter space. In particular, we use method of sieves to optimize the criterion functions in both the steps. The first step involves a sieve maximum likelihood estimation using a probit link function while the second step involves the sieve estimation of the estimating equation with the correction terms. We extend our estimation methods to the case where the outcome equation has heterogeneous coefficients. This allows for a more structural form of heterogeneity that is highly desirable in the primary equation.

We use the recent developments in the semi-nonparametric sieve estimation literature to derive the asymptotic properties of our estimators. Finally, we illustrate our methods with an empirical application.

 Reference: Murtazashvili, I., & Wooldridge, J. M. (2015). A control function approach to estimating switching regression models with endogenous explanatory variables and endogenous switching. Journal of Econometrics.