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The results of most experimental games violate fundamental predictions of game theory. But standard statistical analysis of the results is problematic. A subject who takes account of the observed errors can typically do better by choosing an action other than that predicted by the Nash equilibrium of the complete information game. Thus subjects we label as irrational may be choosing optimally in the given situation. A naive statistical analysis can fail to reject the hypothesis that we observe a Nash equilibrium even as the players can do better by deviating from the Nash equilibrium strategy. We apply structural estimation techniques to a model of the experimental session as a Bayesian game of incomplete information. This approach allows us to consistently estimate and test models of strategic choice. It differs from previous work in several ways. First, we assume that what we observe is an equilibrium and from that determine the informational structure and distribution of player characteristics which support the actions in equilibrium. Second, we develop a numerical technique for computing equilibrium mappings of arbitrary games so that there are not a priori limitations on the types of players which can be considered. Third, we fully develop the statistical model which is implied by the theory. Two applications develop empirical models to examine strategy choices in coordination games and in first price auctions. Both applications reveal advantages of using this approach to analyze data from games. First, we have consistent estimators of parameters which have a clear interpretation within the theory. Second, we have a best test of hypotheses which are also clearly stated within the theoretical framework. Thus, we can distinguish between different plausible explanations and rigorously test whether they are supported by the data.