On periodic autoregression with unknown mean. (English) Zbl 0585.62152

Periodic autoregressive models of the form \(X_ t- \sum^{p}_{1}b_{ti}X_{t-i}=\epsilon_ t,\) where the time-varying coefficients \(b_{ti}\) are periodical functions of time, and \(\{\epsilon_ t\}\) is a Gaussian white noise (mean zero, variance \(\sigma^ 2)\), have been studied in detail in a previous paper by the first author, ibid. 28, 364-385 (1983; Zbl 0537.62073).
The previous results are extended here to the more general case where \(X_ t\) has an unknown mean \(\mu_ t\), itself a periodical function of time: \[ (X_ t-\mu_ t)-\sum^{p}_{1}b_{ti}(X_{t-i}-\mu_{t- i})=\epsilon_ t; \] the innovation variance \(E(\epsilon^ 2_ t)\) can also be considered periodical. Periodical autoregressions should provide interesting models for seasonal series. A Bayesian approach is adopted, which however leads to results that are asymptotically equivalent to those of Gaussian maximum likelihood methods.
Reviewer: M.Hallin


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F15 Bayesian inference


Zbl 0537.62073
Full Text: EuDML


[1] J. Anděl: Statistical analysis of periodic autoregression. Apl. mat. 28 (1983), 364-365. · Zbl 0537.62073
[2] G. E. P. Box G. C. Tiao: Intervention analysis with applications to economic and environmental problems. J. Amer. Statist. Assoc. 70 (1975), 70-79. · Zbl 0316.62045
[3] E. G. Gladyshev: Periodically correlated random sequences. Soviet Math. 2 (1961), 385-388. · Zbl 0212.21401
[4] E. G. Gladyshev: Periodically and almost periodically correlated random process with continuous time parameter. Theory Prob. Appl. 8 (1963), 173-177. · Zbl 0138.11003
[5] M. Pagano: On periodic and multiple autoregression. Ann. Statist. 6 (1978), 1310-1317. · Zbl 0392.62073
[6] A. Zellner: An introduction to Bayesian Inference in Econometrics. Wiley, New York, 1971. · Zbl 0246.62098
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