Conditional moment restrictions and the role of density information in estimated structural models

While incomplete models are desirable due to their robustness to misspecification, they cannot be used to conduct full information exercises i.e. counterfactual experiments and predictions. Moreover, the performance of the corresponding GMM estimators is fragile in small samples. To deal with both issues, we propose the use of an auxiliary conditional model for the observables f(X|Z, ϕ), where the equilibrium conditions E(m(X, ϑ)|Z) = 0 are imposed on f(X|Z, ϕ) using information projections, and (ϑ, ϕ) are estimated jointly. We provide the asymptotic theory for parameter estimates for a general set of conditional projection densities, under correct and local misspecification of f(X|Z, ϕ). In either cases, efficiency gains are significant. We provide simulation evidence for the Mean Squared Error (MSE) both under the case of local and fixed density misspecification and apply the method to the prototypical stochastic growth model. Moreover, we illustrate that given (ϑ, ˆ ϕˆ) it is now feasible to do counterfactual experiments without explicitly solving for the equilibrium law of motion.