Abstract
We consider the problem of modeling heteroscedasticity in semiparametric regression analysis of cross-sectional data. Existing work in this setting is rather limited and mostly adopts a fully nonparametric variance structure. This approach is hampered by curse of dimensionality in practical applications. Moreover, the corresponding asymptotic theory is largely restricted to estimators that minimize certain smooth objective functions. The asymptotic derivation thus excludes semiparametric quantile regression models. To overcome these drawbacks, we study a general class of location-dispersion regression models, in which both the location function and the dispersion function are semiparametrically modeled. We establish unified asymptotic theory which is valid for many commonly used semiparametric structures such as the partially linear structure and single-index structure. We provide easy to check sufficient conditions and illustrate them through examples. Our theory permits non-smooth location or dispersion functions, thus allows for semiparametric quantile heteroscedastic regression and robust estimation in semiparametric mean regression. Simulation studies indicate significant efficiency gain in estimating the parametric component of the location function. The results are applied to analyzing a data set on gasoline consumption.
Highlights
The problem of heteroscedasticity, which traditionally means nonconstant variance function, frequently arises in regression analysis of economic data
We study a general class of locationdispersion regression models, in which both the location function and the dispersion function are semiparametrically modeled
We assume that the unknown function g0 belongs to some space G of uniformly bounded functions that depend on X and β through a variable U = U (X, β), where β belongs to a compact set B in Rl, with l ≥ 1 depending on the model (e.g. U (X, β) = X(2) and U (X, β) = βT X for the above partial linear and single index structures respectively)
Summary
The problem of heteroscedasticity, which traditionally means nonconstant variance function, frequently arises in regression analysis of economic data. In the semiparametric regression setting, Schick (1996), Liang, Hardle and Carroll (1999), Hardle, Liang and Gao (2000, §2), Ma, Chiou and Wang (2006) have studied heteroscedastic partially linear mean regression models, where the variance function σ2(x) is assumed to be smooth but unknown They estimate the variance function nonparametrically, and use the estimator to construct weights to achieve more efficient estimation of the parametric component of the mean regression function. This approach avoids the rigid assumption imposed by a parametric dispersion function; at the same time it circumvents the curse of dimensionality introduced by a nonparametric dispersion function In this general framework, we establish an asymptotic normality theory for estimating the form of heteroscedasticity by building on the work of Chen, Linton and Van Keilegom (2003), who developed a general theory for semiparametric estimation with a non-smooth criterion function.
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