Abstract
This paper is concerned with the estimating problem of seemingly unrelated (SU) nonparametric regression models. The authors propose a new method to estimate the unknown functions, which is an extension of the two-stage procedure in the longitudinal data framework. The authors show the resulted estimators are asymptotically normal and more efficient than those based on only the individual regression equation. Some simulation studies are given in support of the asymptotic results. A real data from an ongoing environmental epidemiologic study are used to illustrate the proposed procedure.
Similar content being viewed by others
References
A. Zellner, An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias, J. Amer. Statist. Assoc., 1962, 57(298): 348–368.
A. Zellner, Estimators for seemingly unrelated equations: Some exact finite sample results, J. Amer. Statist. Assoc., 1963, 58(304): 977–992.
N. C. Kakwani, A note on the efficiency of the Zellner’s seemingly unrelated regressions estimator, Ann. Inst. Statist. Math., 1974, 26(3): 361–362.
R. Gallant, Seemingly unrelated nonlinear regressions, J. Econometrics, 1975, 3(1): 35–50.
M. J. Silvapulle, Asymptotic behavior of robust estimators of regression and scale parameters with fixed carriers, Ann. Statist., 1985, 13(4): 1490–1497.
G. A. Ghazal, The error of forecast for seemingly unrelated regression (SUR) equations, Egyptian Statist. J., 1986, 30(2): 35–64.
V. K. Srivastava and E. A. Giles, Seemingly Unrelated Regression Equations Models: Estimation and Inference, Statistics: Textbooks and Monographs, Vol. 80, Marcel Dekker Inc., New York, 1987.
A. Gould and J. F. Lawless, Consistency and efficiency of regression coefficient estimates in location-scale models, Biometrika, 1988, 75(3): 535–540.
D. M. Rocke, Bootstrap Bartlett adjustment in seemingly unrelated regression, J. Amer. Statist. Assoc., 1989, 84(406): 598–601.
H. Neudecker and F. A. G. Windmeijer, R 2 in seemingly unrelated regression equations, Statist. Neerlandica, 1991, 45(4): 405–411.
D. M. Mandy and C. Martins-Filho, Seemingly unrelated regressions under additive heteroscedasticity: Theory and share equation applications, J. Econometrics, 1993, 58(3): 315–346.
H. Kurata, On the efficiencies of several generalized least squares estimators in a seemingly unrelated regression model and a heteroscedastic model, J. Multivariate Anal., 1999, 70(1): 86–94.
P. Hougarrd, Analysis of Multivariate Survival Data, Springer, New York, 2000.
A. Liu, Efficient estimation of two seemingly unrelated regression equations, J. Multivariate Anal., 2002, 82(2): 445–456.
V. M. Ng, Robust Bayesian inference for seemingly unrelated regressions with elliptical errors, J. Multivariate Anal., 2002, 83(3): 409–414.
J. D. Kalbfleisch and R. L. Prentice, The Statistical Analysis of Failure Time Data (2nd Edition), Wiely, New York, 2002.
R. J. Carroll, M. Doug, F. Larry, and K. Victor, Seemingly unrelated measurement error models, with application to nutritional epidemiology, Biometrics, 2006, 62(1): 75–84.
W. He and J. F. Lawless, Bivariate location-scale models for regression analysis, with applications to lifetime data, J. R. Stat. Soc. Ser. B Stat. Methodel, 67(1): 63–78.
M. Smith, and R. Kohn, Nonparametric seemingly unrelated regression, J. Econom., 2000, 98(2): 257–281.
Y. D. Wang, W. S. Guo, and B. Brown, Spline smoothing for bivariate data with application between hormones, Statistica Sinica, 2000, 10(2): 377–397.
S. Lang, S. Adebayo, and L. Fahremir, Bayesian semiparametric seemingly unrelated regression, Processdings in Computational Statistics (ed. by W. Härdle and B. Rönz), Physika-Verlag, Heidelberg, 2002, 195–200.
S. Lang, S. Adebayo, L. Fahremir, and W. Steiner, Bayesian geoadditive seemingly unrealted regression, Compu. Statist., 2003, 18(2): 263–292.
G. Koop, D. Poirer, and J. Tobias, Semiparametric Bayesian inference in multiple equation models, J. Applied Econometrics, 2005, 20(6): 723–747.
A. H. Welsh and T. W. Yee, Local regression for vector responses, J. Stat. Plann. Infere., 2005, to appear.
N. Wang, Marginal nonparametric kernel regression accounting for within-subject correlation, Biometrika, 2003, 90(1): 43–52.
O. B. Linton, E. Mammen, X. Lin, and R. J. Carroll, Correlation and marginal longitudinal kernel nonparametric regression, Proceedings of the Second Seattle Symposium in Biostatistics (Lecture Notes in Statist., Vol. 179), Springer, New York, 2004, 23–33.
J. Fan and I. Gijbels, Local Polynomial Modeling and its Applications, Chapman and Hall, London, 1996.
J. L. Horowitz and E. Mammen, Nonparametric estimation of an additive model with a link function, Ann. Statist., 2004, 32(6): 2412–2443.
K. A. Gary, M. P. Longnecker, M. A. Klebanoff, J. W. Brock, H. Zhou, and L. Needham, In Utero exposure to background levels of Polychlorinated Biphenls and cognitive functioning among school-aged children, Am. J. Epidemiology, 2005, 162(1): 17–26.
C. J. Stone, Optimal rates of convergence for nonparametric estimators, Ann. Statist, 8(6): 1348–1360
Y. P. Mack and B. W. Silverman, Weak and strong uniform consistency of kernel regression estimates, Z. Wahrsch. Verw. Gebiete, 1982, 61(3): 405–415.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was supported in part by National Natural Science Foundation of China (NSFC) under Grants No. 10471140 and No. 10731010, the National Basic Research Program of China (973 Program) under Grant No. 2007CB814902, and Science Fund for Creative Research Groups.
Rights and permissions
About this article
Cite this article
You, J., Xie, S. & Zhou, Y. Two-Stage Estimation for Seemingly Unrelated Nonparametric Regression Models. Jrl Syst Sci & Complex 20, 509–520 (2007). https://doi.org/10.1007/s11424-007-9048-8
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11424-007-9048-8