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

Let {Z_ij} be independent and identically distributed (i.i.d.) random variables with EZ_ij = 0, E|Z_ij|² = 1 and E|Z_ij|⁴ < ∞. Define linear processes Y_tj =^∞ _k=₀ b_kZ_t−_k,j with^∞ _i=₀ |b_i| < ∞. Consider a p-dimensional time series model of the form x_t = x_{t −1} +¹/²yt, 1 ≤ t ≤ T with yt = (Y_t1, . . ., Y_tp) and¹/² be the square root of a symmetric positive definite matrix. Let B = (1/p)XX^∗ with X = (x₁, . . ., 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 B when 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.