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November, 1974 Characterization of the Partial Autocorrelation Function
Fred L. Ramsey
Ann. Statist. 2(6): 1296-1301 (November, 1974). DOI: 10.1214/aos/1176342881


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.


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Fred L. Ramsey. "Characterization of the Partial Autocorrelation Function." Ann. Statist. 2 (6) 1296 - 1301, November, 1974.


Published: November, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0301.62046
MathSciNet: MR359219
Digital Object Identifier: 10.1214/aos/1176342881

Primary: 62M10
Secondary: 60G10 , 62N15

Keywords: autoregressions , Partial autocorrelations , stationary random processes , stochastic difference equations , time series models

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 6 • November, 1974
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