Condence Sets for Some Partially Identied Parameters

In this paper, we re-visit the inference problem for interval identi…ed parameters originally studied in Imbens and Manski (2004) and later extended in Stoye (2007). We establish a new con…dence interval that is asymptotically valid under the same assumptions as in Stoye (2007). Like the con…dence interval of Stoye (2007), our new con…dence interval extends that of Imbens and Manski (2004) to allow for the lack of a super-e¢ cient estimator of the length of the identi…ed interval. In addition, it shares the natural nesting property of the original con…dence interval of Imbens and Manski (2004). A simulation study is conducted to examine the …nite sample performance of our new con…dence interval and that of Stoye (2007). Finally we extend our con…dence interval for interval identi…ed parameters to parameters de…ned by moment equalities/inequalities. We thank Patrik Guggenberger, Chuck Manski, Frank Schorfheide, Kevin Song, Jörg Stoye, and Jisong Wu for helpful discussions. We are espsecially grateful to Patrik Guggenberger, Frank Schorfheide, and Jörg Stoye for sending us their papers, to Gustavo Soares for sending us his dissertation, and to Francesca Molinari for providing the data set used in the simulation study in this paper.