Abstract. In the last 20 years, the bounding approach for the average treatment effect (ATE) has been developing on the theoretical side, however, empirical work has lagged far behind theory in this area. One main reason is that, in practice, traditional bounding methods fall into two extreme cases: (i) On the one hand, the bounds are too wide to be informative and this happens, in general, when the instrumental variable (IV) has little variation; (ii) while on the other hand, the bounds cross, in which case the researcher learns nothing about the parameter of interest other than that the IV restrictions are rejected. This usually happens when the IV has a rich support and the IV restriction imposed in the model — full, quantile or mean independence— is too stringent, as illustrated in Ginther (2000). In this paper, we provide sharp bounds on the ATE using only a finite set of unconditional moment restrictions, which is a weaker version of mean independence. We revisit Ginther’s (2000) return to schooling application using our bounding approach and derive informative bounds on the average returns to schooling in US.