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Asymptotic properties for a class of partially identified models

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  • 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 sample means and 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 identification region of the model with respect to the Hausdorff distance. We then show that the Hausdorff distance between the estimator and the population identification region, when properly normalized by vn, converges in distribution to the supremum of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate this limiting distribution. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and to calculate the power of the test. We show that these results can be used to construct a confidence collection, that is a collection of sets that, when specified as null hypothesis for the true value of the population identification region, cannot be rejected by our test.

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File URL: http://cemmap.ifs.org.uk/wps/cwp1006.pdf
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Bibliographic Info

Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP10/06.

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Length: 58 pp.
Date of creation: Jun 2006
Date of revision:
Handle: RePEc:ifs:cemmap:10/06

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Related research

Keywords: Partial Identification; Confidence Collections; Set-Valued Random Variables.;

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References

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  1. Thierry Magnac & Eric Maurin, 2004. "Partial Identification in Monotone Binary Models : Discrete Regressors and Interval Data," Working Papers 2004-11, Centre de Recherche en Economie et Statistique.
  2. 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.
  3. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, April.
  4. Donald W. K. Andrews, 1997. "A Conditional Kolmogorov Test," Econometrica, Econometric Society, vol. 65(5), pages 1097-1128, September.
  5. Molinari, Francesca, 2008. "Partial identification of probability distributions with misclassified data," Journal of Econometrics, Elsevier, vol. 144(1), pages 81-117, May.
  6. 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.
  7. 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.
  8. Arthur Lewbel, 2000. "Identification of the Binary Choice Model with Misclassification," Boston College Working Papers in Economics 457, Boston College Department of Economics.
  9. Gleb Koshevoy, 1997. "The Lorenz zonotope and multivariate majorizations," Social Choice and Welfare, Springer, vol. 15(1), pages 1-14.
  10. 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.
  11. Guido Imbens & Charles F. Manski, 2003. "Confidence intervals for partially identified parameters," CeMMAP working papers CWP09/03, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  12. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
  13. Adam Rosen, 2006. "Confidence sets for partially identified parameters that satisfy a finite number of moment inequalities," CeMMAP working papers CWP25/06, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  14. 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.
  15. K. Mosler, 2003. "Central regions and dependency," Econometrics 0309004, EconWPA.
  16. 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.
  17. Manski, C.F., 1990. "The Selection Problem," Working papers 90-12, Wisconsin Madison - Social Systems.
  18. 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, 01.
  19. 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.
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