Before revision:  In this paper, we use a generic model of a Bayesian game with the solution concept of Bayes Correlated Equilibria of Bergemann and Morris (2016) and show that if the information structure is rich enough and the equilibrium selection rule satisfies a weak invariance property, decomposition-based predictions are identical to equilibrium-based predictions. This result opens up the possibility for applying a decomposition method for counterfactual analysis in various strategic settingswithout specifying and estimating details of the game.
After revision:  In this paper, we use a generic model of a Bayesian game with the solution concept of Bayes Correlated Equilibria of Bergemann and Morris (2016) and show that if the information structure is rich enough and the equilibrium selection rule satisfies a weak invariance property, decomposition-based predictions are identical to equilibrium-based prediction in this setting is interval-identified by the decomposition-based prediction which can be estimated without specifying the details of the game such as payoffs or distribution of unobserved heterogeneity. This approach greatly simplifies the counterfactual analysis in various strategic settingswithout specifying and estimating details of the game.