Estimating probit models with self-selected treatments

Outcomes research often requires estimating the impact of a binary treatment on a binary outcome in a non-randomized setting, such as the effect of taking a drug on mortality. The data often come from self-selected samples, leading to a spurious correlation between the treatment and outcome when standard binary dependent variable techniques, like logit or probit, are used. Intuition suggests that a two-step procedure (analogous to two-stage least squares) might be sufficient to deal with this problem if variables are available that are correlated with the treatment choice but not the outcome. This paper demonstrates the limitations of such a two-step procedure. We show that such estimators will not generally be consistent. We conduct a Monte Carlo exercise to compare the performance of the two-step probit estimator, the two-stage least squares linear probability model estimator, and the multivariate probit. The results from this exercise argue in favour of using the multivariate probit rather than the two-step or linear probability model estimators, especially when there is more than one treatment, when the average probability of the dependent variable is close to 0 or 1, or when the data generating process is not normal. We demonstrate how these different methods perform in an empirical example examining the effect of private and public insurance coverage on the mortality of HIV+ patients.