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.