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This paper develops characterizations of identified sets of structures and structural features for complete and incomplete models involving continuous and/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 significant role. The paper provides the first formal definition 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. One innovation is the use of random sets defined on the space of unobserved variables. This allows identification analysis under mean and quantile independence restrictions on the distributions of unobserved variables conditional on exogenous variables as well as under a full independence restriction. It leads to a novel general characterization of identified sets of structural functions when the sole restriction on the ∗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 benefitted from discussion with participants at seminar and conference presentations given at Boston College, Chicago, Paris School of Economics, Vanderbilt, the April 2013 conference on Mathematical Statistics of Partially Identified Objects at Oberwolfach, the 2013 Cowles summer econometrics conference, Yale, Virginia, Georgetown, Maryland, Johns Hopkins, Toulouse School of Economics, Oxford, Duke, Illinois, Reading, and Surrey. Financial 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) and from the European Research Council (ERC) grant ERC-2009-StG-240910-ROMETA is gratefully acknowledged. †Address: Andrew Chesher, Department of Economics, University College London, Gower Street, London WC1E 6BT, andrew.chesher@ucl.ac.uk. ‡Address: Adam Rosen, Department of Economics, University College London, Gower Street, London WC1E 6BT, adam.rosen@ucl.ac.uk.

Characterizations of identified sets delivered by structural econometric models