## The Annals of Statistics

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

#### Abstract

Let $\{Z_{ij}\}$ be independent and identically distributed (i.i.d.) random variables with $EZ_{ij}=0$, $E\vert Z_{ij}\vert^{2}=1$ and $E\vert Z_{ij}\vert^{4}<\infty$. Define linear processes $Y_{tj}=\sum_{k=0}^{\infty}b_{k}Z_{t-k,j}$ with $\sum_{i=0}^{\infty}\vert b_{i}\vert <\infty$. Consider a $p$-dimensional time series model of the form $\mathbf{x}_{t}=\boldsymbol{\Pi} \mathbf{x}_{t-1}+\Sigma^{1/2}\mathbf{y}_{t},\ 1\leq t\leq T$ with $\mathbf{y}_{t}=(Y_{t1},\ldots,Y_{tp})'$ and $\Sigma^{1/2}$ be the square root of a symmetric positive definite matrix. Let $\mathbf{B}=(1/p)\mathbf{XX}^{*}$ with $\mathbf{X}=(\mathbf{x_{1}},\ldots,\mathbf{x_{T}})'$ 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 $\mathbf{B}$ when $\mathbf{x}_{t}$ 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.

#### Article information

Source
Ann. Statist., Volume 46, Number 5 (2018), 2186-2215.

Dates
Revised: July 2017
First available in Project Euclid: 17 August 2018

https://projecteuclid.org/euclid.aos/1534492833

Digital Object Identifier
doi:10.1214/17-AOS1616

Mathematical Reviews number (MathSciNet)
MR3845015

Zentralblatt MATH identifier
06964330

#### Citation

Zhang, Bo; Pan, Guangming; Gao, Jiti. CLT for largest eigenvalues and unit root testing for high-dimensional nonstationary time series. Ann. Statist. 46 (2018), no. 5, 2186--2215. doi:10.1214/17-AOS1616. https://projecteuclid.org/euclid.aos/1534492833

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#### Supplemental materials

• Supplement to “CLT for largest eigenvalues and unit root testing for high-dimensional nonstationary time series”. The supplement [35] provides the proofs of the results in Appedix A and some more discussions about other models.