Nonstationary fractionally integrated functional time series

Degui Li, Peter M. Robinson, Han Lin Shang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1505-1526
Number of pages22
Issue number2
Publication statusPublished - May 2023


  • fractional integration
  • functional principal component analysis
  • functional time series
  • local Whittle estimation
  • nonstationary process


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