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Semiparametric Regression Analysis under Imputation for Missing Response Data

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  • Wolfgang Haerdle
  • Oliver Linton
  • Qihua Wang

Abstract

We develop inference tools in a semiparametric regression model with missing response data. A semiparametric regression imputation estimator, a marginal average estimator and a (marginal) propensity score weighted estimator are defined. All the estimators are proved to be asymptotically normal, with 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, and 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. Furthermore, a real data analysis is given to illustrate our methods.

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Paper provided by Suntory and Toyota International Centres for Economics and Related Disciplines, LSE in its series STICERD - Econometrics Paper Series with number /2003/454.

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Date of creation: May 2003
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Handle: RePEc:cep:stiecm:/2003/454

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Web page: http://sticerd.lse.ac.uk/_new/publications/default.asp

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Keywords: Asymptotic normality; empirical likelihood; semiparametric imputation.;

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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. Enno Mammen & Oliver Linton & J Nielsen, 2000. "The existence and asymptotic properties of a backfitting projection algorithm under weak conditions," LSE Research Online Documents on Economics 2315, London School of Economics and Political Science, LSE Library.
  3. Oliver Linton, 1993. "Second Order Approximation in the Partially Linear Regression Model," Cowles Foundation Discussion Papers 1065, Cowles Foundation for Research in Economics, Yale University.
  4. 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.
  5. Rice, John, 1986. "Convergence rates for partially splined models," Statistics & Probability Letters, Elsevier, vol. 4(4), pages 203-208, June.
  6. 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.
  7. repec:att:wimass:9001 is not listed on IDEAS
  8. Newey, Whitney K & Powell, James L & Walker, James R, 1990. "Semiparametric Estimation of Selection Models: Some Empirical Results," American Economic Review, American Economic Association, vol. 80(2), pages 324-28, May.
  9. Hardle, Wolfgang & LIang, Hua & Gao, Jiti, 2000. "Partially linear models," MPRA Paper 39562, University Library of Munich, Germany, revised 01 Sep 2000.
  10. Ahn, Hyungtaik & Powell, James L., 1993. "Semiparametric estimation of censored selection models with a nonparametric selection mechanism," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 3-29, July.
  11. 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.
  12. 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.
  13. Robinson, Peter M, 1988. "Root- N-Consistent Semiparametric Regression," Econometrica, Econometric Society, vol. 56(4), pages 931-54, July.
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