In empirical research, measuring correctly the benefits of welfare interventions is incredibly relevant for policymakers as well as academic researchers. Unfortunately, the endogenous program participation is often misreported in survey data and standard instrumental variable techniques are not sufficient to point identify and consistently estimate the effects of interest. In this paper, we focus on the weighted average of local average treatment effects (LATE) and (i) derive a simple relationship between the causal and the identifiable parameter that can be recovered from the observed data, (ii) provide an instrumental variable method to partially identify the heterogeneous treatment effects, (iii) formalize a strategy to combine administrative data on the misclassification probabilities of treated individuals to further tighten the bounds. Finally, we use our method to reassess the benefits of participating to the 401(k) pension plan on savings.