Characterization of the Partial Autocorrelation Function

The conditions |φₖ| 1 for all k = 1,2, ⋯ and |φₖ| = 1 implies φₖ₊₁ = φₖ are both necessary and sufficient for a sequence of real numbers \φₖ; k = 1,2, ⋯\ to be the partial autocorrelation function for a real, discrete parameter, stationary time series. If all partial autocorrelations beyond the pth are zero, the series is an autoregression. If all beyond the pth have magnitude unity, the series satisfies a homogeneous stochastic difference equation. A stationary series is singular if and only if ∑N₁ φₖ² diverges with N. The likelihood function for the partial autocorrelation function is produced, assuming normality.