Estimation and Testing for High-dimensional Near Unit Root Time Series

Abstract

This paper considers a p-dimensional time series model of the form x(t)=Π x(t-1)+Σ^(1/2)y(t), 1≤t≤T, where y(t)=(y(t1),...,y(tp))^T and Σ is the square root of a symmetric positive definite matrix. Here Π is a symmetric matrix which satisfies that ∥Π ∥_2≤ 1 and T(1-∥Π ∥_min) is bounded. The linear processes Y(tj) is of the form ∑_{k=0}^∞b(k)Z(t-k,j) where ∑_{i=0}^∞|b(i)| < ∞ and {Z(ij) } are 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< ∞. We first investigate the asymptotic behavior of the first k largest eigenvalues of the sample covariance matrices of the time series model. Then we propose a new estimator for the high-dimensional near unit root setting through using the largest eigenvalues of the sample covariance matrices and use it to test for near unit roots. Such an approach is theoretically novel and addresses some important estimation and testing issues in the high-dimensional near unit root setting. Simulations are also conducted to demonstrate the finite-sample performance of the proposed test statistic.