We propose some practical solutions for causal effects estimation when compliance to assignments is only partial and some of the standard assumptions do not hold. We follow the potential outcome approach but in contrast to Imbens and Rubin (1997), we require no prior classification of the compliance behavior. When noncompliance is not ignorable, it is known that adjusting for arbitrary covariates can actually increase the estimation bias. We propose an approach where a covariate is adjusted for only when the estimate of the selection bias of the experiment as provided by that covariate is consistent with the data and prior information on the study. Next, we investigate cases when the overlap assumption does not hold and, on the basis of their covariates, some units are excluded from the experiment or equivalently, never comply with their assignments. In that context, we show that a consistent estimation of the causal effect of the treatment is possible based on a regression model estimation of the conditional expectation of the outcome given the covariates. We illustrate the methodology with several examples such as the access to influenza vaccine experiment (McDonald et al (1992) and the PROGRESA experiment (Shultz (2004)).

**Yves Atchade and Leonard Wantchekon**