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.
"Semiparametric modeling and estimation of heteroscedasticity in regression analysis of cross-sectional data." Electron. J. Statist. 4 133 - 160, 2010. https://doi.org/10.1214/09-EJS547