CLT FOR LARGEST EIGENVALUES AND UNIT ROOT TESTING FOR HIGH-DIMENSIONAL NONSTATIONARY TIME SERIES

• 成果类型: Article
• 署名作者: Zhang, Bo; Pan, Guangming; Gao, Jiti
• 署名单位: Nanyang Technological University; Monash University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1616
• 发表日期: 2018
• 页码: 2186-2215
• 关键词: sample covariance matrices; panel-data; distributions; UNIVERSALITY; statistics; limit

摘要

Let {Z(ij)} be independent and identically distributed (i.i.d.) random variables with EZ(ij) = 0, E vertical bar Z(ij)vertical bar(2) = 1 and E vertical bar Z(ij)vertical bar(4) < infinity. Define linear processes Y-tj = Sigma(infinity)(k=0) b(k) Z(t -k,j) with Sigma(infinity)(i=0) vertical bar b(i)vertical bar < infinity. Consider a p-dimensional time series model of the form x(t) = Pi x(t-1) + Sigma(1/2)y(t), 1 <= t <= T with y(t) = (Y-t(1), ..., Y-tp)' and Sigma(1/2) be the square root of a symmetric positive definite matrix. Let B = (1/p)XX* with X = (x(1), ..., 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.