Bajari et al. (Quantitative Marketing and Economics, 14(4), 271–323, 2016) showed conditions under which the discount factor is identified in a finite horizon optimal stopping problem. We show that these conditions can be cast as a special case of a class of exclusion restrictions which are relevant for a broader scope of applications, and extend the identification result to both finite horizon and infinite horizon optimal stopping problems under more general exclusion restrictions. We also show how a similar approach gives identification of general discount functions in finite horizon optimal stopping problems. The identification results directly suggest estimators of the discount functions that are easy to compute.