Estimation and Inference for Spatio-Temporal Single-Index Models

To better fit the actual data, this paper will consider both spatio-temporal correlation and heterogeneity to build the model. In order to overcome the “curse of dimensionality” problem in the nonparametric method, we improve the estimation method of the single-index model and combine it with the correlation and heterogeneity of the spatio-temporal model to obtain a good estimation method. In this paper, assuming that the spatio-temporal process obeys the $\alpha$ mixing condition, a nonparametric procedure is developed for estimating the variance function based on a fully nonparametric function or dimensional reduction structure, and the resulting estimator is consistent. Then, a reweighting estimation of the parametric component can be obtained via taking the estimated variance function into account. The rate of convergence and the asymptotic normality of the new estimators are established under mild conditions. Simulation studies are conducted to evaluate the efficacy of the proposed methodologies, and a case study about the estimation of the air quality evaluation index in Nanjing is provided for illustration.