Inference on parameter sets in econometric models

This paper provides confidence regions for minima of an econometric criterion function Q([theta]). The minima form a set of parameters, [theta]I, called the identified set. In economic applications, [theta]I represents a class of economic models that are consistent with the data. Our inference procedures are criterion function based and so our confidence regions, which cover [theta]I with a prespecified probability, are appropriate level sets of Qn([theta]), the sample analog of Q([theta]). When [theta]I is a singleton, our confidence sets reduce to the conventional confidence regions based on inverting the likelihood or other criterion functions. We show that our procedure is valid under general yet simple conditions, and we provide feasible resampling procedure for implementing the approach in practice. We then show that these general conditions hold in a wide class of parametric econometric models. In order to verify the conditions, we develop methods of analyzing the asymptotic behavior of econometric criterion functions under set identification and also characterize the rates of convergence of the confidence regions to the identified set. We apply our methods to regressions with in terval data and set identified method of moments problems. We illustrate our methods in an empirical Monte Carlo study based on Current Population Survey data. Keywords: Set estimator, level sets, interval regression, subsampling bootsrap. JEL Classifications: C13, C14, C21, C41, C51, C53.