# Asymptotic Properties for a Class of Partially Identified Models

## Author

Listed:
• Arie Beresteanu
• Francesca Molinari

## Abstract

We propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the Aumann expectation of a properly defined set valued random variable (SVRV). An SVRV is a mapping that associates a set (rather than a real number) with each element of the sample space. Examples of population features in this class include interval-identified scalar parameters, best linear predictors with interval outcome data, and parameters of semiparametric binary models with interval regressor data. We extend the analogy principle to SVRVs and show that the sample analog estimator of the population identification region is given by a transformation of a Minkowski average of SVRVs. Using the results of the mathematics literature on SVRVs, we show that this estimator converges in probability to the population identification region with respect to the Hausdorff distance. We then show that the Hausdorff distance and the directed Hausdorff distance between the population identification region and the estimator, when properly normalized by $\sqrt{n}$ n , converge in distribution to functions of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate these limiting distributions. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and its subsets. We show that these results can be used to construct a confidence collection and a directed confidence collection. Those are (respectively) collection of sets that, when specified as a null hypothesis for the true value (a subset of values) of the population identification region, cannot be rejected by our tests. Copyright Copyright 2008 by The Econometric Society.

## Suggested Citation

• Arie Beresteanu & Francesca Molinari, 2008. "Asymptotic Properties for a Class of Partially Identified Models," Econometrica, Econometric Society, vol. 76(4), pages 763-814, July.
• Handle: RePEc:ecm:emetrp:v:76:y:2008:i:4:p:763-814
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## References listed on IDEAS

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1. Horowitz, Joel & Manski, Charles, 1997. "Nonparametric Analysis of Randomized Experiments With Missing Covariate and Outcome Data," Working Papers 97-16, University of Iowa, Department of Economics.
2. Thierry Magnac & Eric Maurin, 2008. "Partial Identification in Monotone Binary Models: Discrete Regressors and Interval Data," Review of Economic Studies, Oxford University Press, vol. 75(3), pages 835-864.
3. Charles F. Manski & Elie Tamer, 2002. "Inference on Regressions with Interval Data on a Regressor or Outcome," Econometrica, Econometric Society, vol. 70(2), pages 519-546, March.
4. Molinari, Francesca, 2008. "Partial identification of probability distributions with misclassified data," Journal of Econometrics, Elsevier, vol. 144(1), pages 81-117, May.
5. Bo E. Honoré & Elie Tamer, 2002. "Bounds on Parameters in Dynamic Discrete Choice Models," CAM Working Papers 2004-23, University of Copenhagen. Department of Economics. Centre for Applied Microeconometrics, revised Aug 2004.
6. Donald W. K. Andrews, 1997. "A Conditional Kolmogorov Test," Econometrica, Econometric Society, vol. 65(5), pages 1097-1128, September.
7. Lewbel, Arthur, 2000. "Identification Of The Binary Choice Model With Misclassification," Econometric Theory, Cambridge University Press, vol. 16(4), pages 603-609, August.
8. Manski, C.F., 1990. "The Selection Problem," Working papers 90-12, Wisconsin Madison - Social Systems.
9. Gleb Koshevoy, 1997. "The Lorenz zonotope and multivariate majorizations," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 15(1), pages 1-14.
10. E. Tamer & V. Chernozhukov & H. Hong, 2004. "Parameter Set Inference in a Class of Econometric Models," Econometric Society 2004 North American Winter Meetings 382, Econometric Society.
11. Horowitz, Joel L. & Manski, Charles F., 1998. "Censoring of outcomes and regressors due to survey nonresponse: Identification and estimation using weights and imputations," Journal of Econometrics, Elsevier, vol. 84(1), pages 37-58, May.
12. Newey, Whitney K. & McFadden, Daniel, 1986. "Large sample estimation and hypothesis testing," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 36, pages 2111-2245, Elsevier.
13. Karl Mosler, 2003. "Central Regions and Dependency," Methodology and Computing in Applied Probability, Springer, vol. 5(1), pages 5-21, March.
14. Guido W. Imbens & Charles F. Manski, 2004. "Confidence Intervals for Partially Identified Parameters," Econometrica, Econometric Society, vol. 72(6), pages 1845-1857, November.
15. Horowitz, J.L. & Manski, C.F., 1995. "What Can Be Learned About Population Parameters when the Data Are Contaminated," Working Papers 95-18, University of Iowa, Department of Economics.
16. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
17. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, February.
18. Rosen, Adam M., 2008. "Confidence sets for partially identified parameters that satisfy a finite number of moment inequalities," Journal of Econometrics, Elsevier, vol. 146(1), pages 107-117, September.
19. Joseph P. Romano & Azeem M. Shaikh, 2010. "Inference for the Identified Set in Partially Identified Econometric Models," Econometrica, Econometric Society, vol. 78(1), pages 169-211, January.
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### JEL classification:

• C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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