Information Structure and Statistical Information in Discrete Response Models

Discrete response models are of high interest in economics and econometrics as they encompass treatment eff ects, social interaction and peer e ffect models, and discrete games. We study the impact of the structure of information sets of economic agents on the Fisher information of (strategic) interaction parameters in such models. While in complete information models the information sets of participating economic agents coincide, in incomplete information models each agent has a type, which we model as a payo ff shock, that is not observed by other agents. We allow for the presence of a payoff component that is common knowledge to economic agents but is not observed by the econometrician (representing unobserved heterogeneity) and have the agents' payo ffs in the incomplete information model approach their payoff in the complete information model as the heterogeneity term approaches 0. We find that in the complete information models, there is zero Fisher information for interaction parameters, implying that estimation and inference become nonstandard. In contrast, positive Fisher information can be attained in the incomplete information models with any non-zero variance of player types, and for those we can also fi nd the semiparametric efficiency bound with unknown distribution of unobserved heterogeneity. The contrast in Fisher information is illustrated in two important cases: treatment eff ect models, which we model as a triangular system of equations, and static game models. In static game models we show this result is not due to equilibrium refinement with an increase in incomplete information, as our model has a fixed equilibrium selection mechanism. We find that the key factor in these models is the relative tail behavior of the unobserved component in the economic agents' payoff s and that of the observable covariates