Long-Range Dependent Curve Time Series

Abstract

We introduce methods and theory for functional or curve time series with long-range dependence. The temporal sum of the curve process is shown to be asymptotically normally distributed, the conditions for this covering a functional version of fractionally integrated autoregressive moving averages. We also construct an estimate of the long-run covariance function, which we use, via functional principal component analysis, in estimating the orthonormal functions spanning the dominant subspace of the curves. In a semiparametric context, we propose an estimate of the memory parameter and establish its consistency. A Monte Carlo study of finite-sample performance is included, along with two empirical applications. The first of these finds a degree of stability and persistence in intraday stock returns. The second finds similarity in the extent of long memory in incremental age-specific fertility rates across some developed nations. Supplementary materials for this article are available online.

Keywords:

• Curve process
• Functional FARIMA
• Functional principal component analysis
• Limit theorems
• Long-range dependence

Acknowledgments

The authors would like to thank an associate editor and a reviewer for their insightful comments and suggestions, which substantially improved the article. They also thank Professor Hong Miao for providing the financial dataset.