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June, 1990 Canonical Partial Autocorrelation Function of a Multivariate Time Series
Serge Degerine
Ann. Statist. 18(2): 961-971 (June, 1990). DOI: 10.1214/aos/1176347635


We propose a definition of the partial autocorrelation function $\beta(\cdot)$ for multivariate stationary time series suggested by the canonical analysis of the forward and backward innovations. Here $\beta(\cdot)$ satisfies $\beta(-n) = \beta(n)', n = 0, 1, \cdots,$ where $\beta(0)$ is nonnegative definite, $\{\beta(n), n = 1, 2, \cdots\}$ is a sequence of square matrices having singular values less than or equal to 1 and such that the order of $\beta(n + 1)$ is equal to the rank of $I - \beta(n)\beta(n)',$ the order of $\beta(1)$ being equal to the rank of $\beta(0)$. We show that there exists a one-to-one correspondence between the set of matrix autocovariance functions $\Lambda(\cdot)$, with the positive definiteness property, and the set of canonical partial autocorrelation functions $\beta(\cdot)$ as described above.


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Serge Degerine. "Canonical Partial Autocorrelation Function of a Multivariate Time Series." Ann. Statist. 18 (2) 961 - 971, June, 1990.


Published: June, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0703.62095
MathSciNet: MR1056346
Digital Object Identifier: 10.1214/aos/1176347635

Primary: 62M10
Secondary: 60G10, 62H20, 62M15

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 2 • June, 1990
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