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A Near Unit Root Test for High-Dimensional Nonstationary Time Series

This paper considers a p-dimensional time series model of the form x(t)-δ(t)=φ(x(t-1)-δ(t-1))+Σ^(1/2)y(t), 1≤t≤T, where y(t) = (y(t1),...,y(tp)) and and Σ^(1/2) is the square root of a symmetric positive definite matrix. Here φ≤1 and T(1-φ) is bounded and the linear processes y(tj) is of the form Σb(k)z(t-k,j) where Σ|bi| < infinity and {z(ij)} are independent and identically distributed (i.i.d.) random variables with E(z(ij))=0, E|z(ij)|^2=1 and E|z(ij)|^4< infinity. We first investigate the asymptotic behavior of the first k largest eigenvalues of the sample covariance matrices of the time series model. We then propose an estimator of φ and use it to test for near unit roots. Simulations and empirical applications are also conducted to demonstrate the performance of the statistic.