Characterizations of identified sets delivered by structural econometric models

Abstract

This paper develops characterizations of identi…ed sets of structures and structural features for complete and incomplete models involving continuous or discrete variables. Multiple values of unobserved variables can be associated with particular combinations of observed variables. This can arise when there are multiple sources of heterogeneity, censored or discrete endogenous variables, or inequality restrictions on functions of observed and unobserved variables. The models generalize the class of incomplete instrumental variable (IV) models in which unobserved variables are single-valued functions of observed variables. Thus the models are referred to as Generalized IV (GIV) models, but there are important cases in which instrumental variable restrictions play no signi…cant role. Building on a de…nition of observational equivalence for incomplete models the development uses results from random set theory which guarantee that the characterizations deliver sharp bounds, thereby dispensing with the need for case-by-case proofs of sharpness. The use of random sets de…ned on the space of unobserved variables allows identi…cation analysis under mean and quantile independence restrictions on the distributions of unobserved variables conditional on exogenous variables as well as under a full independence This paper is a revised version of a paper initially circulated as CeMMAP working paper CWP43/13 under the title "Generalized Instrumental Variable Models." We thank Federico Bugni, Matt Masten, and Francesca Molinari for detailed comments and discussion. We have bene…tted from discussion with participants at seminar and conference presentations given at uVa Amsterdam, Boston College, Boston University, Cornell, Chicago, Paris School of Economics, Vanderbilt, the April 2013 conference on Mathematical Statistics of Partially Identi…ed Objects at Oberwolfach, the 2013 Cowles summer econometrics conference, the 2014 Asia meeting of the Econometric Society in Taipei, the 2014 China meeting of the Econometric Society in Xiamen, the Harvard-MIT econometrics workshop, Yale, Virginia, Georgetown, Maryland, Johns Hopkins, Toulouse School of Economics, Oxford, Duke, UIUC, Reading, NUS Singapore, and Surrey. Both authors gratefully acknowedlge …nancial support from the UK Economic and Social Research Council through a grant (RES-589-28-0001) to the ESRC Centre for Microdata Methods and Practice (CeMMAP). Adam Rosen gratefully acknowledges …nancial support from a British Academy mid-career Fellowship, and from the European Research Council (ERC) grants ERC-2009-StG-240910-ROMETA and ERC-2012-StG-312474. y Address: Andrew Chesher, Department of Economics, University College London, Gower Street, London WC1E 6BT, andrew.chesher@ucl.ac.uk.z Address: Adam Rosen, Department of Economics, University College London, Gower Street, London WC1E 6BT, adam.rosen@ucl.ac.uk. 1 restriction. The results are used to develop sharp bounds on the distribution of valuations in an incomplete model of English auctions, improving on the pointw...

Set Identification, Moment Restrictions, and Inference

“…Polytomous discrete models are treated by Chesher and Smolinski (2012) and Chesher and Rosen (2015a) generalize instrumental variable models. Binary models with a "very exogenous" regressor, whose observations are interval censored, are analyzed by Magnac and Maurin (2008).…”

“…Best linear prediction, or more generally mean independence restrictions, are usually easier to deal with Aumann expectations. In contrast, games of complete or incomplete information, or independence restrictions, are easier to deal with capacity functionals (see also Chesher and Rosen, 2015a).…”

Applied Welfare Analysis for Discrete Choice with Interval-Data on Income

“…Substantively similar ideas are discussed inManski (1994) and, at a somewhat abstract level, in the generalized instrumental variable model inChesher and Rosen (2017).…”