Bootstrap Inference in Semiparametric Generalized Additive Models

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.

A simple bootstrap method for constructing nonparametric confidence bands for functions

“…Hall and Titterington (1988) also used explicit bias correction, in the sense that their bands required a known bound on an appropriate derivative of the target function. Bjerve, Doksum and Yandell (1985), Hall (1992b), Hall and Owen (1993), Neumann (1995), Chen (1996), Neumann and Polzehl (1998), Picard and Tribouley (2000), Chen, Härdle and Li (2003) (in the context of hypothesis testing), Claeskens and Van Keilegom (2003), Härdle et al (2004) and McMurry and Politis (2008) employed methods that involve undersmoothing. There is also a theoretical literature which addresses the bias issue through consideration of the technical function class from which a regression mean or density came; see, for example, Low (1997) and Genovese and Wasserman (2008).…”

Feasible estimation in generalized structured models

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Penalized partially linear models using sparse representations with an application to fMRI time series

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