Testing for Trends in High-Dimensional Time Series

ABSTRACT

The article considers statistical inference for trends of high-dimensional time series. Based on a modified ${\mathcal{L}}^2$ distance between parametric and nonparametric trend estimators, we propose a de-diagonalized quadratic form test statistic for testing patterns on trends, such as linear, quadratic, or parallel forms. We develop an asymptotic theory for the test statistic. A Gaussian multiplier testing procedure is proposed and it has an improved finite sample performance. Our testing procedure is applied to a spatial temporal temperature data gathered from various locations across America. A simulation study is also presented to illustrate the performance of our testing method. Supplementary materials for this article are available online.

Keywords:

• Asymptotic normality
• Gaussian multiplier testing procedure
• Long-run covariance matrix
• Parametric testing
• Temporal and spatial dependence

Supplementary Materials

The detailed proofs is provided in the online supplementary materials.

Acknowledgements

The authors thank the reviewers and the editor for very helpful suggestions, which substantially improve the article.

Additional information

Funding

The research was partially supported by NSF grant DMS-1405410.