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Canonical partial autocorrelation function of a multivariate time series. (English) Zbl 0703.62095

Summary: 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 \(\beta (-n)=\beta (n)'\), \(n=0,1,...\), where \(\beta\) (0) is nonnegative definite, \(\{\beta\) (n), \(n=1,2,...\}\) 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 shown 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.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62H20 Measures of association (correlation, canonical correlation, etc.)
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