Skip to main content
SpringerLink
Log in

Efficient estimation of seemingly unrelated additive nonparametric regression models

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper is concerned with the estimating problem of seemingly unrelated (SU) nonparametric additive regression models. A polynomial spline based two-stage efficient approach is proposed to estimate the nonparametric components, which takes both of the additive structure and correlation between equations into account. The asymptotic normality of the derived estimators are establishedi. The authors also show they own some advantages, including they are asymptotically more efficient than those based on only the individual regression equation and have an oracle property, which is the asymptotic distribution of each additive component is the same as it would be if the other components were known with certainty. Some simulation studies are conducted to illustrate the finite sample performance of the proposed procedure. Applying the proposed procedure to a real data set is also made.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Zellner A, An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias, J. Amer. Statist. Assoc., 1962, 57: 348–368.

    Article  MathSciNet  MATH  Google Scholar 

  2. Zellner A, Estimators for seemingly unrelated equations: Some exact finite sample results, J. Amer. Statist. Assoc., 1963, 58: 977–992.

    Article  MathSciNet  MATH  Google Scholar 

  3. Kakwani N C, A note on the efficiency of the Zellner’s seemingly unrelated regressions estimator, Ann. Inst. Statist. Math., 1974, 26: 361–362.

    Article  MathSciNet  MATH  Google Scholar 

  4. Gallant R, Seemingly unrelated nonlinear regressions, J. Econometrics, 1975, 3: 35–50.

    Article  MathSciNet  MATH  Google Scholar 

  5. Silvapulle M J, Asymptotic behavior of robust estimators of regression and scale parameters with fixed carriers, Ann. Statist., 1985, 13: 1490–1497.

    Article  MathSciNet  MATH  Google Scholar 

  6. Ghazal G A, The error of forecast for seemingly unrelated regression (SUR) equations, Egyptian Statist. J., 1986, 30: 35–64.

    MathSciNet  Google Scholar 

  7. Srivastava V K and Giles E A, Seemingly Unrelated Regression Equations Models, Estimation and inference Statistics: Textbooks and Monographs, 80. Marcel Dekker, Inc., New York, 1987.

    MATH  Google Scholar 

  8. Gould A and Lawless J F, Consistency and efficiency of regression coefficient estimates in location-scale models, Biometrika, 1988, 75: 535–540.

    MathSciNet  MATH  Google Scholar 

  9. Rocke D M, Bootstrap Bartlett adjustment in seemingly unrelated regression, J. Amer. Statist. Assoc., 1989, 84, 598–601.

    Google Scholar 

  10. Neudecker H and Windmeijer F A G, R 2 in seemingly unrelated regression equations, Statist. Neerlandica, 1991, 45: 405–411.

    Article  MathSciNet  MATH  Google Scholar 

  11. Mandy D M and Martins-Filho C, Seemingly unrelated regressions under additive heteroscedasticity: theory and share equation applications, J. Econometrics, 1993, 58: 315–346.

    Article  MathSciNet  MATH  Google Scholar 

  12. Kurata H, On the efficiencies of several generalized least squares estimators in a seemingly unrelated regression model and a heteroscedastic model, J. Multivariate Anal., 1999, 70: 86–94.

    Article  MathSciNet  MATH  Google Scholar 

  13. Hougaard P, Analysis of Multivariate Survival Data, Springer, New York, 2000.

    Book  MATH  Google Scholar 

  14. Liu A, Efficient estimation of two seemingly unrelated regression equations, J. Multivariate Anal., 2002, 82: 445–456.

    Article  MathSciNet  MATH  Google Scholar 

  15. Ng V M, Robust Bayesian inference for seemingly unrelated regressions with elliptical errors, J. Multivariate Anal., 2002, 83: 409–414.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kalbfleisch J D and Prentice R L, The Statistical Analysis of Failure Time Data, 2nd ed, Wiely, New York, 2002.

    Book  MATH  Google Scholar 

  17. He W and Lawless J F, Bivariate location-scale models for regression analysis, with applications to lifetime data, J. R. Stat. Soc. Ser. B, 2005, 67: 63–78.

    Article  MathSciNet  MATH  Google Scholar 

  18. Carroll R J, Doug M, Larry F, and Victor K, Seemingly unrelated measurement error models, with application to nutritional epidemiology, Biometrics, 2006, 62: 75–84.

    Article  MathSciNet  MATH  Google Scholar 

  19. Smith M and Kohn R, Nonparametric seemingly unrelated regression, J. Econometrics, 2000, 98: 257–281.

    Article  MATH  Google Scholar 

  20. Wang Y D, Guo W S, and Brown B, Spline smoothing for bivariate data with application between hormones, Statistic Sinica, 2000, 10: 377–397.

    MathSciNet  MATH  Google Scholar 

  21. Welsh A H and Yee T W, Local regression for vector responses, J. Stat. Plann. Infere., 2006, 136: 3007–3031.

    Article  MathSciNet  MATH  Google Scholar 

  22. You J, Xie S, and Zhou Y, Two-stage estimation for seemingly unrelated nonparametric regression models, Journal of System Science & Complexity, 2007, 20(4): 509–520.

    Article  MathSciNet  MATH  Google Scholar 

  23. Stone C J, Optimal rates of convergence for nonparametric estimators, Ann. Statist., 1980, 8: 1348–1360.

    Article  MathSciNet  MATH  Google Scholar 

  24. Lang S, Adebayo S, and Fahremir L, Bayesian semiparametric seemingly unrelated regression. Ed. by W. Härdle and B. Rönz. Proceedings in Computational Statistics, Physika-Verlag, Heidelberg, 2002.

    Google Scholar 

  25. Lang S, Adebayo S, Fahremir L, and Steiner W, Bayesian geoadditive seemingly unrealted regression, Compu. Statist., 2003, 18: 263–292.

    MATH  Google Scholar 

  26. Koop G, Poirer D, and Tobias J, Semiparametric Bayesian inference in multiple equation models, J. Applied Econometrics, 2005, 20: 723–747.

    Article  Google Scholar 

  27. Stone C J, Additive regression and other nonparametric models, Ann. Statist., 1985, 13: 689–705.

    Article  MathSciNet  MATH  Google Scholar 

  28. Stone C J, The dimensionality reduction principle for generalized additive models, Ann. Statist., 1986, 14: 590–606.

    Article  MathSciNet  MATH  Google Scholar 

  29. Buja A, Hastie T, and Tibshirani R, Linear smoothers and additive models, Ann. Statist. 1989, 17: 453–555.

    Article  MathSciNet  MATH  Google Scholar 

  30. Hastie T J and Tibshirani R J, Generalized additive models. Monographs on Statistics and Applied Probability, 43. Chapman and Hall, Ltd., London, 1990.

    MATH  Google Scholar 

  31. Tjøstheim D and Auestad B H, Nonparametric identification of nonlinear time series: projections, J. Amer. Statist. Assoc., 1994, 89: 1398–1409.

    MathSciNet  Google Scholar 

  32. Horowitz J L and Mammen E, Oracle-Efficient Nonparametric Estimation of an Additive Model with an Unknown Link Function, Department of Economics, Northwestern University, U.S.A., 2005.

    Google Scholar 

  33. Gary K A, Longnecker M P, Klebanoff M A, Brock J W, Zhou H, and Needham L, In Utero exposure to background levels of Polychlorinated Biphenls and cognitive functioning among schoolaged children, Am. J. Epidemiology, 2005, 162: 17–26.

    Article  Google Scholar 

  34. de Boor C, A Practical Guide to Splines, Applied Mathematical Sciences, 27, Springer-Verlag, New York-Berlin, 1978.

    Book  MATH  Google Scholar 

  35. Schumaker L L, Spline Functions: Basic Theory, Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1981

    Google Scholar 

  36. Huang J and Stone C J, Extended linear modeling with splines. Nonlinear estimation and classification (Berkeley, CA, 2001), 213–233, Lecture Notes in Statist, 171, Springer, New York, 2003.

    Book  Google Scholar 

  37. Horowitz J L and Mammen E, Nonparametric estimation of an additive model with a link function, Ann. Statist., 2004, 32: 2412–2443.

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang N, Marginal nonparametric kernel regression accounting for within-subject correlation, Biometrika, 2003, 90: 43–52.

    Article  MathSciNet  MATH  Google Scholar 

  39. Schaalje G B and Butts R A, Some effects of ignoring correlated measurement errors in straight line regression, Biometrics, 1993, 49: 1262–1267.

    Article  Google Scholar 

  40. Tony M and Henderson R, Generalized least squares with ignored errors in variables, Technometrics, 2000, 42: 137–146.

    Google Scholar 

  41. Darrell T, Generalized vec operators and the seemingly unrelated regression equations model with vector correlated disturbances, J. Econometrics, 2000, 99: 225–253.

    Article  MathSciNet  MATH  Google Scholar 

  42. Mack Y P and Silverman BW, Weak and strong uniform consistency of kernel regression estimates, Z. Wahrsch. Verw. Gebiete, 1982, 61: 405–415.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Yuan Yuan, Jinhong You & Yong Zhou

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yong Zhou

Authors
  1. Yuan Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinhong You
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Zhou.

Additional information

ZHOU’s work was partially supported by National Natural Science Funds for Distinguished Young Scholar under Grant No. 70825004 and National Natural Science Foundation of China under Grant Nos. 10731010 and 10628104, the National Basic Research Program under Grant No. 2007CB814902, Creative Research Groups of China under Grant No. 10721101. Partially supported by leading Academic Discipline Program, 211 Project for Shanghai University of Finance and Economics (the 3rd phase) and project number: B803; YOU’s research was supported by grants from the National Natural Science Foundation of China under Grant No. 11071154.

This paper was recommended for publication by Editor ZOU Guohua.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yuan, Y., You, J. & Zhou, Y. Efficient estimation of seemingly unrelated additive nonparametric regression models. J Syst Sci Complex 26, 595–608 (2013). https://doi.org/10.1007/s11424-012-8351-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-012-8351-1

Key words

Search

Navigation