Parameter Set Inference in a Class of Econometric Models

We provide new methods for inference in econometric models where the parameter of interest is a set. These models arise in many situations where point identification requires strong (and sometimes untestable) assumptions. Every parameter vector in the set of interest represents a feasible economic model that generated the data. Our point of departure is {\it set} $\Theta_I$ that minimizes a given population criterion function $Q(\theta)$. To obtain valid inferences on $\Theta_I$, we characterize first the large sample properties of the sample criterion function $Q_n(\theta)$. These are then used to construct confidence sets for $\Theta_I$ that contain this set with a given prespecified probability. The method we use picks an appropriate level set of the objective function by ``cutting off'' this function at a level that corresponds to an appropriately chosen percentile of a key ``coverage statistic." This is a likelihood ratio type quantity (and reduces to the usual likelihood ratio when the set $\Theta_I$ is a singleton). Our confidence set then is an appropriately chosen level set of the objective function. This general framework is illustrated in two important examples: regressions where data on outcomes is observed in intervals, and method of moments where the criterion function is minimized on a set. A subsampling procedure that is used to obtain the asymptotic critical values for the coverage statistic is provided and shown to be consistent. For the two examples, we also provide indirect bootstrap procedures that are based on resampling a dual statistic that has the same asymptotic distribution as the coverage statistic. We illustrate our methods in an empirical example of returns to schooling using data from the CPS