CLT for largest eigenvalues and unit root testing for high-dimensional nonstationary time series

Bo Zhang, Guangming Pan, Jiti Gao

Research output: Contribution to journalArticleResearchpeer-review

10 Citations (Scopus)


Let {Zij} be independent and identically distributed (i.i.d.) random variables with EZij = 0, E|Zij|2 = 1 and E|Zij|4 < ∞. Define linear processes Ytj = k=0 bkZtk,j with i=0 |bi| < ∞. Consider a p-dimensional time series model of the form xt = xt −1 +1/2yt, 1 ≤ t ≤ T with yt = (Yt1, . . ., Ytp) and1/2 be the square root of a symmetric positive definite matrix. Let B = (1/p)XX with X = (x1, . . ., xT) and X be the conjugate transpose. This paper establishes both the convergence in probability and the asymptotic joint distribution of the first k largest eigenvalues of B when xt is nonstationary. As an application, two new unit root tests for possible nonstationarity of high-dimensional time series are proposed and then studied both theoretically and numerically.

Original languageEnglish
Pages (from-to)2186-2215
Number of pages30
JournalAnnals of Statistics
Issue number5
Publication statusPublished - Oct 2018


  • Asymptotic normality
  • Largest eigenvalue
  • Linear process
  • Unit root test

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