Statistical inference for partially linear stochastic models with heteroscedastic errors

Partially linear models are extended linear models where one covariate is nonparametric, which is a good balance between flexibility and parsimony. The partially linear stochastic model with heteroscedastic errors is considered, where the nonparametric part can act as a trend. The estimators of the parametric component, the nonparametric component and the volatility function are proposed. Furthermore, simultaneous confidence bands about the nonparametric part and the volatility function are constructed based on their coverage probabilities, which are shown to be asymptotically correct. By the confidence bands, the problems of hypothesis testing in this model can be solved effectively from a global view. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies, and demonstrated by the analyses of non-stationary Australian annual temperature anomaly series and non-homoscedastic daily air quality measurements in New York, where the simultaneous confidence bands provide more comprehensive information about the nonparametric and volatility functions.