Treatment Effects with Unobserved Heterogeneity: A Set Identification Approach

We propose the sharp identifiable bounds of the distribution functions of potential outcomes using a panel with fixed T. We allow for the possibility that the statistical randomization of treatment assignments is not achieved until unobserved heterogeneity is properly controlled for. We use certain stationarity assumptions to obtain the bounds. Dynamics in the treatment decisions is allowed as long as the stationarity assumptions are satisfied. In particular, we present an example where our assumptions are satisfied and the treatment decision of the present time may depend on the treatments and the observed outcomes of the past. As an empirical illustration we study the effect of smoking during pregnancy on infant birth weights. We found that for the group of switchers the birth weight with smoking is first order stochastically dominated by that with non-smoking.