Estimation and Confidence Regions for Parameter Sets in Econometric Models*

This paper provides estimators and confidence regions for minima of an econometric criterion function Q(θ). The minima form a set of parameters, ΘI , called here the identified set. In economic applications, ΘI represents a set of economic models that in population pass the set of testable restrictions embodied in Q(θ). When ΘI is a singleton, the confidence sets reduce to the conventional confidence regions based on inverting the likelihood or other criterion function. The procedure is valid under general yet simple conditions, and a feasible resampling procedure is provided to implement the approach in practice. These general conditions are shown to hold in a class of moment condition models. In order to verify the conditions, we develop methods of analyzing the asymptotic behavior of econometric criterion functions in these settings and also characterize the rates of convergence of the confidence regions to the identified set.