Abstract
We study a functional version of nonstationary fractionally integrated time series, covering the functional unit root as a special case. The time series taking values in an infinite-dimensional separable Hilbert space are projected onto a finite number of sub-spaces, the level of nonstationarity allowed to vary over them. Under regularity conditions, we derive a weak convergence result for the projection of the fractionally integrated functional process onto the asymptotically dominant sub-space, which retains most of the sample information carried by the original functional time series. Through the classic functional principal component analysis of the sample variance operator, we obtain the eigenvalues and eigenfunctions which span a sample version of the dominant sub-space. Furthermore, we introduce a simple ratio criterion to consistently estimate the dimension of the dominant sub-space, and use a semiparametric local Whittle method to estimate the memory parameter. Monte-Carlo simulation studies are given to examine the finite-sample performance of the developed techniques.
Funding Statement
The first author is partially supported by the National Natural Science Foundation of China (No. 72033002).
Acknowledgements
The authors would like to thank two reviewers for the comments, which helped to improve the paper.
Citation
Degui Li. Peter M. Robinson. Han Lin Shang. "Nonstationary fractionally integrated functional time series." Bernoulli 29 (2) 1505 - 1526, May 2023. https://doi.org/10.3150/22-BEJ1508
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