Canonical Partial Autocorrelation Function of a Multivariate Time Series

Abstract

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.