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Confidence intervals for partially identified parameters

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  • Guido Imbens
  • Charles F. Manski

Abstract

In the last decade a growing body of research has studied inference on partially identified parameters (e.g., Manski, 1990, 2003). In many cases where the parameter of interest is realvalued, the identification region is an interval whose lower and upper bounds may be estimated from sample data. Confidence intervals may be constructed to take account of the sampling variation in estimates of these bounds. Horowitz and Manski (1998, 2000) proposed and applied interval estimates that asymptotically cover the entire identification region with fixed probability. Here we introduce conceptually different interval estimates that asymptotically cover each element in the identification region with fixed probability (but not necessarily every element simultaneously). We show that these two types of interval estimate are different in practice, the latter in general being shorter. The difference in length (in excess of the length of the identification set itself) can be substantial, and in large samples is comparable to the difference of one and twosided confidence intervals. A complication arises from the fact that the simplest version of the proposed interval is discontinuous in the limit case of point identification, leading to coverage rates that are not uniform in important subsets of the parameter space. We develop a modification depending on the width of the identification region that restores uniformity. We show that under some conditions, using the estimated width of the identification region instead of the true width maintains uniformity.

Suggested Citation

  • Guido Imbens & Charles F. Manski, 2003. "Confidence intervals for partially identified parameters," CeMMAP working papers 09/03, Institute for Fiscal Studies.
  • Handle: RePEc:azt:cemmap:09/03
    DOI: 10.1920/wp.cem.2003.0903
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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. Richard Blundell & Amanda Gosling & Hidehiko Ichimura & Costas Meghir, 2007. "Changes in the Distribution of Male and Female Wages Accounting for Employment Composition Using Bounds," Econometrica, Econometric Society, vol. 75(2), pages 323-363, March.
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    4. 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.
    5. Bollinger, Christopher R., 1996. "Bounding mean regressions when a binary regressor is mismeasured," Journal of Econometrics, Elsevier, vol. 73(2), pages 387-399, August.
    6. Meyer, Bruce D & Viscusi, W Kip & Durbin, David L, 1995. "Workers' Compensation and Injury Duration: Evidence from a Natural Experiment," American Economic Review, American Economic Association, vol. 85(3), pages 322-340, June.
    7. Philip A. Haile & Elie Tamer, 2003. "Inference with an Incomplete Model of English Auctions," Journal of Political Economy, University of Chicago Press, vol. 111(1), pages 1-51, February.
    8. Molinari, Francesca, 2008. "Partial identification of probability distributions with misclassified data," Journal of Econometrics, Elsevier, vol. 144(1), pages 81-117, May.
    9. Manski, Charles F, 1990. "Nonparametric Bounds on Treatment Effects," American Economic Review, American Economic Association, vol. 80(2), pages 319-323, May.
    10. 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.
    11. David S. Lee, 2002. "Trimming for Bounds on Treatment Effects with Missing Outcomes," NBER Technical Working Papers 0277, National Bureau of Economic Research, Inc.
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    15. Manski, C.F. & Sandefur, G.D. & Mclanahan, S. & Powers, D., 1990. "Alternative Estimates Of The Effect Of Family Stucture During Adolescence On Hight School Graduation," Working papers 90-31, Wisconsin Madison - Social Systems.
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