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Empirical Likelihood for Random Sets

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  • Karun Adusumilli
  • Taisuke Otsu

Abstract

We extend the method of empirical likelihood to cover hypotheses involving the Aumann expectation of random sets. By exploiting the properties of random sets, we convert the testing problem into one involving a continuum of moment restrictions for which we propose two inferential procedures. The first, which we term marked empirical likelihood, corresponds to constructing a non-parametric likelihood for each moment restriction and assessing the resulting process. The second, termed sieve empirical likelihood, corresponds to constructing a likelihood for a vector of moments with growing dimension. We derive the asymptotic distributions under the null and sequence of local alternatives for both types of tests and prove their consistency. The applicability of these inferential procedures is demonstrated in the context of two examples on the mean of interval observations and best linear predictors for interval outcomes.

Suggested Citation

  • Karun Adusumilli & Taisuke Otsu, 2014. "Empirical Likelihood for Random Sets," STICERD - Econometrics Paper Series 574, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  • Handle: RePEc:cep:stiecm:574
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    File URL: http://sticerd.lse.ac.uk/dps/em/em574.pdf
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    References listed on IDEAS

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    1. Christian Bontemps & Thierry Magnac & Eric Maurin, 2012. "Set Identified Linear Models," Econometrica, Econometric Society, vol. 80(3), pages 1129-1155, May.
    2. Arun Chandrasekhar & Victor Chernozhukov & Francesca Molinari & Paul Schrimpf, 2012. "Inference for best linear approximations to set identified functions," CeMMAP working papers CWP43/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Arie Beresteanu & Francesca Molinari, 2008. "Asymptotic Properties for a Class of Partially Identified Models," Econometrica, Econometric Society, vol. 76(4), pages 763-814, July.
    4. Song Chen & Ingrid Van Keilegom, 2009. "A review on empirical likelihood methods for regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 415-447, November.
    5. Hiroaki Kaido & Andres Santos, 2014. "Asymptotically Efficient Estimation of Models Defined by Convex Moment Inequalities," Econometrica, Econometric Society, vol. 82(1), pages 387-413, January.
    6. de Jong, R.M. & Bierens, H.J., 1994. "On the Limit Behavior of a Chi-Square Type Test if the Number of Conditional Moments Tested Approaches Infinity," Econometric Theory, Cambridge University Press, vol. 10(01), pages 70-90, March.
    7. Ilya Molchanov & Francesca Molinari, 2014. "Applications of Random Set Theory in Econometrics," Annual Review of Economics, Annual Reviews, vol. 6(1), pages 229-251, August.
    8. Einmahl, J.H.J. & McKeague, I.W., 2003. "Empirical likelihood based hypothesis testing," Other publications TiSEM 2ddb34d8-8ae7-46e3-8004-c, Tilburg University, School of Economics and Management.
    9. Kaido, Hiroaki, 2016. "A dual approach to inference for partially identified econometric models," Journal of Econometrics, Elsevier, vol. 192(1), pages 269-290.
    10. Song Xi Chen & Liang Peng & Ying-Li Qin, 2009. "Effects of data dimension on empirical likelihood," Biometrika, Biometrika Trust, vol. 96(3), pages 711-722.
    11. Song Chen & Ingrid Van Keilegom, 2009. "Rejoinder on: A review on empirical likelihood methods for regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 468-474, November.
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    Cited by:

    1. White, Halbert & Kim, Tae-Hwan & Manganelli, Simone, 2015. "VAR for VaR: Measuring tail dependence using multivariate regression quantiles," Journal of Econometrics, Elsevier, vol. 187(1), pages 169-188.
    2. Whang, Yoon-Jae, 2006. "Smoothed Empirical Likelihood Methods For Quantile Regression Models," Econometric Theory, Cambridge University Press, vol. 22(02), pages 173-205, April.
    3. White, Halbert & Kim, Tae-Hwan & Manganelli, Simone, 2010. "VAR for VaR: measuring systemic risk using multivariate regression quantiles," MPRA Paper 35372, University Library of Munich, Germany.
    4. Philip Kostov, 2013. "Empirical likelihood estimation of the spatial quantile regression," Journal of Geographical Systems, Springer, vol. 15(1), pages 51-69, January.
    5. Manganelli, Simone & White, Halbert & Kim, Tae-Hwan, 2008. "Modeling autoregressive conditional skewness and kurtosis with multi-quantile CAViaR," Working Paper Series 957, European Central Bank.

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    • N0 - Economic History - - General

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