Estimating treatment effectiveness with sample selection

We consider a situation where treatment outcome is observed after two stages of selection; first of participation into the treatment, then in completion of the treatment. Estimates were obtained using two methods. First, three different binary response selection models were estimated sequentially in multiple steps. Second, all three equations were estimated jointly. All methods produce similar parameter estimates. We find evidence of selection effects from completion to outcome that could bias parameter estimates of the outcome equation, but not from participation to outcome, indicating that correcting only for participation may be insufficient to avoid biased estimates in the outcome equation.