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Cointegrated Linear Processes in Hilbert Space
Author(s) -
Beare Brendan K.,
Seo Juwon,
Seo WonKi
Publication year - 2017
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/jtsa.12251
Subject(s) - mathematics , hilbert space , autoregressive model , reproducing kernel hilbert space , covariance , series (stratigraphy) , cointegration , representation (politics) , kernel (algebra) , pure mathematics , statistics , paleontology , politics , political science , law , biology
We extend the notion of cointegration for multivariate time series to a potentially infinite‐dimensional setting in which our time series takes values in a complex separable Hilbert space. In this setting, standard linear processes with nonzero long‐run covariance operator play the role of I 0 processes. We show that the cointegrating space for an I 1 process may be sensibly defined as the kernel of the long‐run covariance operator of its difference. The inner product of an I 1 process with an element of its cointegrating space is a stationary complex‐valued process. Our main result is a version of the Granger–Johansen representation theorem: we obtain a geometric reformulation of the Johansen I(1) condition that extends naturally to a Hilbert space setting, and show that an autoregressive Hilbertian process satisfying this condition, and possibly also a compactness condition, admits an I 1 representation.