Characterization of the Partial Autocorrelation Function

Abstract

The conditions $|\phi_k| \leqq 1$ for all $k = 1,2, \cdots$ and $|\phi_k| = 1$ implies $\phi_{k+1} = \phi_k$ are both necessary and sufficient for a sequence of real numbers $\{\phi_k; k = 1,2, \cdots\}$ to be the partial autocorrelation function for a real, discrete parameter, stationary time series. If all partial autocorrelations beyond the $p$th are zero, the series is an autoregression. If all beyond the $p$th have magnitude unity, the series satisfies a homogeneous stochastic difference equation. A stationary series is singular if and only if $\sum^N_1 \phi_k^2$ diverges with $N$. The likelihood function for the partial autocorrelation function is produced, assuming normality.