On a general procedure for constructing confidence sets under partially identified models

Abstract

Partially identified models commonly arise in enormous fields, including but not limited to economics, econometrics and medical science. A model is said to be partially identified when the sampling process and maintained assumptions fail to uniquely reveal the parameter of interest as a singleton. Instead, set-valued objects, referred to as identified sets in the related literature, are derived in general as original targets, based on which estimation and inference procedures are henceforth performed.

Under partial identification settings, a number of algorithms have been developed for statistical inferences with respect to identified sets, with associated asymptotic properties extensively studied in literature. Fruitful results are obtained for special models such as those defined using moment inequalities, yet the problem remains inconvenient outside such restrictive contexts. In Chapter 2 we propose a general confidence procedure which not only generalizes existing algorithms but finds applications to settings as yet unexplored. The main thrust of our proposed procedure lies in the construction of a family of expanding sets labeled by a continuous expansion index. Confidence sets are then built by estimating the expansion index which yields the desired coverage level. Wide choices of expansion indices and their corresponding expanding sets render our procedure flexible and applicable under a rich class of partially identified models.

In real implementations, sharp characterization of the identified set is prohibitively complex under various circumstances, making the set-valued object of interest not readily amenable to conventional estimation and inference procedures. In Chapter 3 we introduce as a remedy an outer region strategy, that easy-to-compute supersets are created as surrogates for original targets, at the cost of a loss in the associated identifying power. To fix ideas, we narrow our attention to extremum estimation problems under different types of partial identification settings. Proposed supersets are characterized by a family of lower level sets, based on which we adapt the criterion function approach, originally proposed for partially identified models subject to moment restrictions, to establish a new confidence procedure for the problem under consideration. Though formulated in separate chapters, the general confidence procedure and the outer region approach complement each other, and combine to yield conservative-yet-computationally tractable inferences. We defer a detailed discussion on this matter at the end of Chapter 3.

Performances of the aforesaid schemes are illustrated and evaluated through simulation studies, in Chapters 2 and 3.