We derive bounds on the causal effect of belief-dependent preferences (reciprocity and guilt aversion) on choices in sequential two-player games without exploiting information or data on the (higher-order) beliefs of players. We show how informative bounds can be derived by exploiting a specific invariance property common to those preferences. We illustrate our approach by analyzing data from an experiment conducted in Denmark. Our approach produces tight bounds on the causal effect of reciprocity in the games we consider. These bounds suggest there exists significant reciprocity in our population - a result also substantiated by the participants' answers to a post-experimental questionnaire. On the other hand, our approach yields high implausible estimates of guilt aversion. We contrast our estimated bounds with point estimates obtained using data on self-declared higher-order beliefs, keeping all other aspects of the model unchanged. We find that point estimates fall within our estimated bounds suggesting that elicited higher-order belief data in our experiment is weakly (if at all) affected by a potential endogeneity problem due to e.g. false consensus effects.
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