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  • Esteban M. Aucejo (a1), Federico A. Bugni (a2) and V. Joseph Hotz (a3)


This paper examines the problem of identification and inference on a conditional moment condition model with missing data, with special focus on the case when the conditioning covariates are missing. We impose no assumption on the distribution of the missing data and we confront the missing data problem by using a worst case scenario approach.

We characterize the sharp identified set and argue that this set is usually too complex to compute or to use for inference. Given this difficulty, we consider the construction of outer identified sets (i.e. supersets of the identified set) that are easier to compute and can still characterize the parameter of interest. Two different outer identification strategies are proposed. Both of these strategies are shown to have nontrivial identifying power and are relatively easy to use and combine for inferential purposes.


Corresponding author

*Address correspondence to Federico A. Bugni, Department of Economics, Duke University, 213 Social Sciences, Box 90097, Durham, NC, 27708; e-mail:


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Abrevaya, J. (2001) The effects of demographics and maternal behavior on the distribution of birth outcomes. Empirical Economics 26, 247257.
Andrews, D.W.K., Berry, S., & Jia-Barwick, P. (2004) Confidence Regions for Parameters in Discrete Games with Multiple Equilibria with an Application to Discount Chain Store Location. Mimeo, Yale University and M.I.T.
Andrews, D.W.K. & Guggenberger, P. (2009) Validity of subsampling and “Plug-in Asymptotic” inference for parameters defined by moment inequalities. Econometric Theory 25, 669709.
Andrews, D.W.K. & Jia-Barwick, P. (2012) Inference for parameters defined by moment inequalities: A recommended moment selection procedure. Econometrica 80, 28052826.
Andrews, D.W.K. & Shi, X. (2013) Inference based on conditional moment inequalities. Econometrica 81, 609666.
Andrews, D.W.K. & Soares, G. (2010) Inference for parameters defined by moment inequalities using generalized moment selection. Econometrica 78, 119157.
Arcidiacono, P., Aucejo, E., Coate, P., & Hotz, V.J. (2012) Affirmative Action and University Fit: Evidence from Proposition 209. NBER Working paper #18523.
Armstrong, T.B. (2012) Asymptotically Exact Inference in Conditional Moment Inequality Models. Mimeo, Yale University.
Armstrong, T.B. (2014) Weighted KS statistics for inference on conditional moment inequalities. Journal of Econometrics 181, 92116.
Armstrong, T.B. & Chan, H.P. (2012) Multiscale Adaptive Inference on Conditional Moment Inequalities. Mimeo, Yale University and National University of Singapore.
Aucejo, E.M., Bugni, F.A., & Hotz, V.J. (2015a) The determinants of graduation outcomes at the University of California before and after Proposition 209. Mimeo, London School of Economics and Political Science and Duke University.
Aucejo, E.M., Bugni, F.A., & Hotz, V.J. (2015b) Online Supplemental Material to “Identification and Inference on Regressions with Missing Covariate Data”. Mimeo, London School of Economics and Political Science and Duke University.
Beresteanu, A. & Molinari, F. (2008) Asymptotic properties for a class of partially identified models. Econometrica 76, 763814.
Beresteanu, A., Molinari, F., & Molchanov, I. (2011) Sharp identification regions in models with convex predictions. Econometrica 79, 17851821.
Bontemps, C., Magnac, T., & Maurin, E. (2012) Set identified linear models. Econometrica 80, 11291155.
Bugni, F.A. (2010) Bootstrap inference in partially identified models defined by moment inequalities: Coverage of the identified set. Econometrica 78, 735753.
Bugni, F.A. (2015) A comparison of inferential methods in partially identified models in terms of error in coverage probability (formerly circulated as “Bootstrap Inference in Partially Identified Models Defined by Moment Inequalities: Coverage of the Elements of the Identified Set”). Econometric Theory FirstView, 156.
Bugni, F.A., Canay, I.A., & Shi, X. (2015) Specification tests for partially identified models defined by moment inequalities. Journal of Econometrics 185, 259282.
Canay, I.A. (2010) E.L. inference for partially identified models: Large deviations optimality and bootstrap validity. Journal of Econometrics 156, 408425.
Chernozhukov, V., Hong, H., & Tamer, E. (2007) Estimation and confidence regions for parameter sets in econometric models. Econometrica 75, 12431284.
Chernozhukov, V., Lee, S., & Rosen, A.M. (2013) Intersection bounds: Estimation and inference. Econometrica 81, 667737.
Chesher, A. & Rosen, A. (2014a) Generalized Instrumental Variable Models. Mimeo, CeMMAP working paper CWP04/14.
Chesher, A. & Rosen, A. (2014b) An instrumental variable random coefficients model for binary outcomes. Econometrics Journal 17, S1S19.
Chesher, A., Rosen, A., & Smolinski, K. (2013) An instrumental variable model for multiple discrete choice. Quantitative Economics 4, 157196.
Chetverikov, D. (2013) Adaptive test of conditional moment inequalities. Unpublished manuscript.
Domínguez, M. & Lobato, I.N. (2004) Consistent estimation of models defined by conditional moment restrictions. Econometrica 72, 16011615.
Galichon, A. & Henry, M. (2006) Inference in Incomplete Models. Mimeo, Ecole Polytechnique, Paris - Department of Economic Sciences and Pennsylvania State University.
Galichon, A. & Henry, M. (2011) Set identification in models with multiple equilibria. Journal of Econometrics 78, 12641298.
Galichon, A. & Henry, M. (2013) Dilation bootstrap: A methodology for constructing confidence regions with partially identified models. Journal of Econometrics 177, 109115.
Horowitz, J.L. & Manski, C.F. (1995) Identification and robustness with contaminated and corrupted data. Econometrica 63, 281302.
Horowitz, J.L. & Manski, C.F. (1998) Censoring of outcomes and regressors due to survey nonresponse: Identification and estimation using weights and imputations. Journal of Econometrics 84, 3758.
Horowitz, J.L. & Manski, C.F. (2000) Nonparametric analysis of randomized experiments with missing covariate and outcome data. Journal of the American Statistical Association 95, 7788.
Horowitz, J.L. & Manski, C.F. (2006) Identification and estimation of statistical functionals using incomplete data. Journal of Econometrics 132, 445459.
Horowitz, J.L., Manski, C.F., Ponomareva, M., & Stoye, J. (2003) Computation of bounds on population parameters when the data are incomplete. Reliable Computing 9, 419440.
Imbens, G. & Manski, C.F. (2004) Confidence intervals for partially identified parameters. Econometrica 72, 18451857.
Kim, K. (2008) Set Estimation and Inference with Models Characterized by Conditional Moment Inequalities. Mimeo, Michigan State University.
Lewbel, A. (2002) Estimation of average treatment effects with misclassification. Econometrica 75, 537551.
Lundberg, S. (1988) Labor supply of husbands and wives: A simultaneous equations approach. The Review of Economics and Statistics 70, 224235.
Mahajan, A. (2006) Identification and estimation of regression models with missclassification. Econometrica 74, 631665.
Manski, C.F. (2003) Partial Identification of Probability Distributions. Springer-Verlag.
Manski, C.F. & Tamer, E. (2002) Inference of regressions with interval data on a regressor or outcome. Econometrica 70, 519546.
Molinari, F. (2008) Partial identification of probability distributions with misclassified data. Journal of Econometrics 144, 81117.
Pakes, A., Porter, J., Ho, K., & Ishii, J. (2014) Moment inequalities and their application. Econometrica 83, 315334.
Pollard, D. (1990) Empirical Processes: Theory and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics, vol. 2.
Ponomareva, M. (2010) Inference in Models Defined by Conditional Moment Inequalities with Continuous Covariates. Mimeo, Northern Illinois University.
Romano, J.P. & Shaikh, A.M. (2008) Inference for identifiable parameters in partially identified econometric models. Journal of Statistical Planning and Inference 138, 27862807.
Romano, J.P. & Shaikh, A.M. (2010) Inference for the identified set in partially identified econometric models. Econometrica 78, 169211.
Rosen, A.M. (2008) Confidence sets for partially identified parameters that satisfy a finite number of moment inequalities. Journal of Econometrics 146, 107117.
Stoye, J. (2009) More on confidence intervals for partially identified parameters. Econometrica 77, 2991315.
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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
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