Unemployment duration and re-employment wages: a control function approach

In the context of instrumental variables (IV) approach, the control function has been widely used in the Applied Econometrics literature. The main objective is the same: to find (at least) one instrumental variable which explains the variation in the endogenous explanatory variable (EEV) of the structural equation. Once this goal is accomplished, the researcher should regress the EEV on the exogenous variables excluded from the structural equation (instrumental variables). From this regression, usually denoted as first stage, one should obtain the generalized residuals and plug them into the structural equation (second stage). These residuals will then serve as a control function to transform the EEV into an appropriate exogenous variable. The main advantage of this method is that, unlike the two-stage least squares approach (2SLS), it can be applied to non-linear models (Wooldridge, 2015). Such situation arises when the outcome variable of the structural equation is discrete, truncated, or censored, for example. The estimation of a non-linear model, as opposed to the typical ordinary least squares regression (OLS), can also be required in the first stage. In this paper I provide an application to the later by estimating an accelerated failure model to explain the unemployment duration (my EEV). In order to apply the control function to non-linear models, Stata only offers the command etregress at the moment, which allows for a binary treatment variable. To complement this option, I hereby propose a user-written program which allows for a censored treatment variable. As the program is directed to duration models, the user will be able to choose the type of survival analysis to perform in the first stage. Due to the separate estimation of each stage the program calculates bootstrapped standard errors for the second stage.