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Semiparametric regression analysis with missing response at random

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Author Info

  • Qihua Wang
  • Oliver Linton

    ()
    (Institute for Fiscal Studies and London School of Economics)

  • Wolfgang Hardle

Abstract

We develop inference tools in a semiparametric partially linear regression model with missing response data. A class of estimators is defined that includes as special cases: a semiparametric regression imputation estimator, a marginal average estimator and a (marginal) propensity score weighted estimator. We show that any of our class of estimators is asymptotically normal. The three special estimators have the same asymptotic variance. They achieve the semiparametric efficiency bound in the homoskedastic Gaussian case. We show that the Jackknife method can be used to consistently estimate the asymptotic variance. Our model and estimators are defined with a view to avoid the curse of dimensionality, that severely limits the applicability of existing methods. The empirical likelihood method is developed. It is shown that when missing responses are imputed using the semiparametric regression method the empirical log-likelihood is asymptotically a scaled chi-square variable. An adjusted empirical log-likelihood ratio, which is asymptotically standard chi-square, is obtained. Also, a bootstrap empirical log-likelihood ratio is derived and its distribution is used to approximate that of the imputed empirical log-likelihood ratio. A simulation study is conducted to compare the adjusted and bootstrap empirical likelihood with the normal approximation based method in terms of coverage accuracies and average lengths of confidence intervals. Based on biases and standard errors, a comparison is also made by simulation between the proposed estimators and the related estimators.

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File URL: http://cemmap.ifs.org.uk/wps/cwp0311.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 CWP11/03.

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Length: 30 pp.
Date of creation: Apr 2003
Date of revision:
Handle: RePEc:ifs:cemmap:11/03

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Postal: The Institute for Fiscal Studies 7 Ridgmount Street LONDON WC1E 7AE
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Web page: http://cemmap.ifs.org.uk
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References

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  1. Guido Imbens, 2000. "Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score," Econometric Society World Congress 2000 Contributed Papers 1166, Econometric Society.
  2. Jinyong Hahn, 1998. "On the Role of the Propensity Score in Efficient Semiparametric Estimation of Average Treatment Effects," Econometrica, Econometric Society, vol. 66(2), pages 315-332, March.
  3. Linton, Oliver, 1995. "Second Order Approximation in the Partially Linear Regression Model," Econometrica, Econometric Society, vol. 63(5), pages 1079-1112, September.
  4. Newey, W.K. & Powell, J.L. & Walker, J.R., 1990. "Semiparametric Estimation Of Selection Models: Some Empirical Results," Working papers 9001, Wisconsin Madison - Social Systems.
  5. Rice, John, 1986. "Convergence rates for partially splined models," Statistics & Probability Letters, Elsevier, vol. 4(4), pages 203-208, June.
  6. Heckman, James J & Ichimura, Hidehiko & Todd, Petra, 1998. "Matching as an Econometric Evaluation Estimator," Review of Economic Studies, Wiley Blackwell, vol. 65(2), pages 261-94, April.
  7. Qihua Wang, 2002. "Empirical Likelihood-based Inference in Linear Models with Missing Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 29(3), pages 563-576.
  8. Robinson, Peter M, 1988. "Root- N-Consistent Semiparametric Regression," Econometrica, Econometric Society, vol. 56(4), pages 931-54, July.
  9. Yuichi Kitamura & Michael Stutzer, 1997. "An Information-Theoretic Alternative to Generalized Method of Moments Estimation," Econometrica, Econometric Society, vol. 65(4), pages 861-874, July.
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Citations

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Cited by:
  1. Chen, Songxi, 2012. "Estimation in semiparametric models with missing data," MPRA Paper 46216, University Library of Munich, Germany.
  2. Qihua Wang & Gregg Dinse & Chunling Liu, 2012. "Hazard function estimation with cause-of-death data missing at random," Annals of the Institute of Statistical Mathematics, Springer, vol. 64(2), pages 415-438, April.
  3. Bryan S. Graham & Cristine Campos De Xavier Pinto & Daniel Egel, 2012. "Inverse Probability Tilting for Moment Condition Models with Missing Data," Review of Economic Studies, Oxford University Press, vol. 79(3), pages 1053-1079.
  4. Zhao, Hui & Zhao, Pu-Ying & Tang, Nian-Sheng, 2013. "Empirical likelihood inference for mean functionals with nonignorably missing response data," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 101-116.
  5. Guo, Xu & Wang, Tao & Xu, Wangli & Zhu, Lixing, 2014. "Dimension reduction with missing response at random," Computational Statistics & Data Analysis, Elsevier, vol. 69(C), pages 228-242.
  6. Lai, Peng & Wang, Qihua, 2014. "Semiparametric efficient estimation for partially linear single-index models with responses missing at random," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 33-50.
  7. Xiaohong Chen & Han Hong & Alessandro Tarozzi, 2008. "Semiparametric Efficiency in GMM Models of Nonclassical Measurement Errors, Missing Data and Treatment Effects," Cowles Foundation Discussion Papers 1644, Cowles Foundation for Research in Economics, Yale University.
  8. Liang, Hua & Su, Haiyan & Zou, Guohua, 2008. "Confidence intervals for a common mean with missing data with applications in an AIDS study," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 546-553, December.
  9. Song Chen & Ingrid Van Keilegom, 2013. "Estimation in semiparametric models with missing data," Annals of the Institute of Statistical Mathematics, Springer, vol. 65(4), pages 785-805, August.
  10. Inkmann, J., 2005. "Inverse Probability Weighted Generalised Empirical Likelihood Estimators: Firm Size and R&D Revisited," Discussion Paper 2005-131, Tilburg University, Center for Economic Research.
  11. Bianco, Ana & Boente, Graciela & González-Manteiga, Wenceslao & Pérez-González, Ana, 2010. "Estimation of the marginal location under a partially linear model with missing responses," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 546-564, February.
  12. Wang, Qihua & Sun, Zhihua, 2007. "Estimation in partially linear models with missing responses at random," Journal of Multivariate Analysis, Elsevier, vol. 98(7), pages 1470-1493, August.
  13. Wangli Xu & Xu Guo, 2013. "Nonparametric checks for varying coefficient models with missing response at random," Metrika, Springer, vol. 76(4), pages 459-482, May.
  14. Ana Bianco & Graciela Boente & Wenceslao González-Manteiga & Ana Pérez-González, 2011. "Asymptotic behavior of robust estimators in partially linear models with missing responses: the effect of estimating the missing probability on the simplified marginal estimators," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 20(3), pages 524-548, November.
  15. Qi-Hua Wang, 2009. "Statistical estimation in partial linear models with covariate data missing at random," Annals of the Institute of Statistical Mathematics, Springer, vol. 61(1), pages 47-84, March.
  16. Qihua Wang & Tao Zhang & Wolfgang Karl Härdle, 2014. "An Extended Single Index Model with Missing Response at Random," SFB 649 Discussion Papers SFB649DP2014-003, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  17. Wangli Xu & Lixing Zhu, 2013. "Testing the adequacy of varying coefficient models with missing responses at random," Metrika, Springer, vol. 76(1), pages 53-69, January.
  18. Nian-Sheng Tang & Pu-Ying Zhao, 2013. "Empirical likelihood semiparametric nonlinear regression analysis for longitudinal data with responses missing at random," Annals of the Institute of Statistical Mathematics, Springer, vol. 65(4), pages 639-665, August.
  19. Francesco Bravo, 2013. "Partially linear varying coefficient models with missing at random responses," Annals of the Institute of Statistical Mathematics, Springer, vol. 65(4), pages 721-762, August.
  20. A. Pérez-González & J. Vilar-Fernández & W. González-Manteiga, 2009. "Asymptotic properties of local polynomial regression with missing data and correlated errors," Annals of the Institute of Statistical Mathematics, Springer, vol. 61(1), pages 85-109, March.
  21. Sun, Zhihua & Wang, Qihua & Dai, Pengjie, 2009. "Model checking for partially linear models with missing responses at random," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 636-651, April.
  22. Xue, Liugen & Xue, Dong, 2011. "Empirical likelihood for semiparametric regression model with missing response data," Journal of Multivariate Analysis, Elsevier, vol. 102(4), pages 723-740, April.
  23. Mojirsheibani, Majid & Montazeri, Zahra, 2007. "On nonparametric classification with missing covariates," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 1051-1071, May.

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