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BOOTSTRAP INFERENCE IN SEMIPARAMETRIC GENERALIZED ADDITIVE MODELS

Published online by Cambridge University Press:  10 February 2004

Wolfgang Härdle
Affiliation:
Humboldt-Universität zu Berlin
Sylvie Huet
Affiliation:
Institut de Recherche Agronomique
Enno Mammen
Affiliation:
Universität Mannheim
Stefan Sperlich
Affiliation:
Universidad Carlos III de Madrid

Abstract

Semiparametric generalized additive models are a powerful tool in quantitative econometrics. With response Y, covariates X,T, the considered model is E(Y |X;T) = G{XTβ + α + m1(T1) + ··· + md(Td)}. Here, G is a known link, α and β are unknown parameters, and m1,…,md are unknown (smooth) functions of possibly higher dimensional covariates T1,…,Td. Estimates of m1,…,md, α, and β are presented, and asymptotic distributions are given for both the nonparametric and the parametric part. The main focus of the paper is application of bootstrap methods. It is shown how bootstrap can be used for bias correction, hypothesis testing (e.g., component-wise analysis), and the construction of uniform confidence bands. Further, bootstrap tests for model specification and parametrization are given, in particular for testing additivity and link function specification. The practical performance of the methods is illustrated in a simulation study.This research was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 373 “Quantifikation und Simulation ökonomischer Prozesse,” Humboldt-Universität zu Berlin, DFG project MA 1026/6-2, the Spanish “Dirección General de Enseñanza Superior,” no. BEC2001-1270, and the grant “Nonparametric methods in finance and insurance” from the Danish Social Science Research Council. We thank Marlene Müller, Oliver Linton, and two anonymous referees for helpful discussion.

Type
Research Article
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Ai, C. (1997) A semiparametric maximum likelihood estimator. Econometrica 65, 933963.Google Scholar
Beran, R. (1986) Comment on “Jackknife, bootstrap, and other resampling methods in regression analysis” by C.F.J. Wu. Annals of Statistics 14, 12951298.Google Scholar
Carroll, R.J., J. Fan, I. Gijbels, & M.P. Wand (1997) Generalized partially linear single-index models. Journal of the American Statistical Association 92, 477489.Google Scholar
Deaton, A. & J. Muellbauer (1980) Economics and Consumer Behavior. Cambridge University Press.
Eubank, R.L. & P.L. Speckman (1993) Confidence bands in nonparametric regression. Journal of the American Statistical Association 88, 12871301.Google Scholar
Fan, J., W. Härdle, & E. Mammen (1998) Direct estimation of low dimensional components in additive models. Annals of Statistics 26, 943971.Google Scholar
Goldberger, A.S. (1964) Econometric Theory. Wiley.
Gozalo, P.L. & O.B. Linton (2001) Testing additivity in generalized nonparametric regression models. Journal of Econometrics 104, 148.Google Scholar
Hansen, M.H., J.Z. Huang, C. Kooperberg, C.J. Stone, & Y.K. Truong (2002) Statistical Modeling with Spline Functions: Methodology and Theory. Springer-Verlag. In press.
Härdle, W., S. Huet, E. Mammen, & S. Sperlich (1998) Semiparametric additive indices for binary response and generalized additive models. Discussion Paper 95, Sanderforschungsbereich 373, Berlin.
Härdle, W. & E. Mammen (1993) Testing parametric versus nonparametric regression. Annals of Statistics 21, 19261947.Google Scholar
Härdle, W., E. Mammen, & M. Müller (1998) Testing parametric versus semiparametric modelling in generalized linear models. Journal of the American Statistical Association 93, 14611474.Google Scholar
Härdle, W., E. Mammen, & I. Proenca (2001) A bootstrap test for single index models. Statistics 35, 427452.Google Scholar
Hastie, T.J. & R.J. Tibshirani (1990) Generalized Additive Models. Chapman and Hall.
Horowitz, J.L. (1998) Semiparametric Methods in Econometrics. Lecture Notes in Statistics 131, Springer-Verlag.
Horowitz, J.L. (2001) Nonparametric estimation of a generalized additive model with an unknown link function. Econometrica 69, 499513.Google Scholar
Leontief, W. (1947) Introduction to a theory of the internal structure of functional relationships. Econometrica, 15 361373.Google Scholar
Linton, O.B. & W. Härdle (1996) Estimating additive regression models with known links. Biometrika 83, 529540.Google Scholar
Linton, O.B. & J.P. Nielsen (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82, 93101.Google Scholar
Mammen, E. (1992) When Does Bootstrap Work? Asymptotic Results and Simulations. Lecture Notes in Statistics 77, Springer-Verlag.
Mammen, E. & S. van de Geer (1997) Penalized quasi-likelihood estimation in partial linear models. Annals of Statistics 25, 10141035.Google Scholar
Mammen, E., O.B. Linton, & J.P. Nielsen (1999) The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Annals of Statistics 27, 14431490.Google Scholar
McCullagh, P. & J.A. Nelder (1989) Generalized Linear Models. Chapman and Hall.
Neumann, M. & J. Polzehl (1998) Simultaneous bootstrap confidence bands in nonparametric regression. Journal of Nonparametric Statistics 9, 307333.Google Scholar
Opsomer, J.D. & D. Ruppert (1999) A root-n consistent estimator for semiparametric additive modeling. Journal of Computational and Graphical Statistics 8, 715732.Google Scholar
Severini, T.A. & J.G. Staniswalis (1994) Quasi-likelihood estimation in semiparametric models. Journal of the American Statistical Association 89, 501511.Google Scholar
Sperlich, S., D. Tjøstheim, & L. Yang (2002) Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18, 197251.Google Scholar
Stone, C.J. (1985) Additive regression and other nonparametric models. Annals of Statistics 13, 685705.Google Scholar
Tjøstheim, D.J. & B.H. Auestadt (1994) Nonparametric identification of nonlinear time series: Projections. Journal of the American Statistical Association 89, 13981409.Google Scholar
Wu, C.F.G. (1986) Jackknife, bootstrap, and other resampling methods in regression analysis. (with discussion) Annals of Statistics 14, 12911380.Google Scholar
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