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

ABSTRACT

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 L₂-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.

KEYWORDS:

• B-spline
• Locally stationary process
• SCAD penalty function
• Varying-coefficient additive model

Supplementary Material

The supplementary material cover all the proofs and additional numerical studies.

Acknowledgments

The authors are grateful to an associate editor and four referees for their constructive comments that have substantially improved the quality of this article.

Notes

1 We define over-fitting for the true model as ${\widehat{\mathcal{S}}}_{\mathrm{A}}\cup {\widehat{\mathcal{S}}}_{\mathrm{C}}\neg \supset \mathcal{S}$, and under-fitting as ${\widehat{\mathcal{S}}}_{\mathrm{A}}\cup {\widehat{\mathcal{S}}}_{\mathrm{C}}⫌\mathcal{S}$, where $\mathcal{S}={\mathcal{S}}_{\mathrm{A}}\cup {\mathcal{S}}_{\mathrm{C}}$, ${\widehat{\mathcal{S}}}_{\mathrm{A}}$ and ${\widehat{\mathcal{S}}}_{\mathrm{C}}$ are the identified index sets of AT and VCT, respectively.

Additional information

Funding

Tao Huang’s research was supported in part by the State Key Program in the Major Research Plan of NSFC (No. 91546202) and Program for Innovative Research Team of SHUFE. Jinhong You’s research was supported in part by the National Natural Science Foundation of China (NSFC) (No. 11471203).