Estimation and Identification of a Varying-Coefficient Additive Model for Locally Stationary Processes

The additive model and the varying-coefficient model are both powerful regression tools, with wide practical applications. However, our empirical study on a financial data has shown that both of these models have drawbacks when applied to locally stationary time series. For the analysis of functional data, Zhang and Wang have proposed a flexible regression method, called the varying-coefficient additive model (VCAM), and presented a two-step spline estimation method. Motivated by their approach, we adopt the VCAM to characterize the time-varying regression function in a locally stationary context. We propose a three-step spline estimation method and show its consistency and asymptotic normality. For the purpose of model diagnosis, we suggest an L 2 -distance test statistic to check multiplicative assumption, and raise a two-stage penalty procedure to identify the additive terms and the varying-coefficient terms provided that the VCAM is applicable. We also present the asymptotic distribution of the proposed test statistics and demonstrate the consistency of the two-stage model identification procedure. Simulation studies investigating the finite-sample performance of the estimation and model diagnosis methods confirm the validity of our asymptotic theory. The financial data are also considered. Supplementary materials for this article are available online.