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On multiple periodic autoregression. (English) Zbl 0634.62086
The basic model considered in this work is that of a vector autoregression for $$X_ t$$, which has r components, with the parameters in the autoregression varying periodically with period p. Thus, for $$k=1,...,p$$, $$j=1,2,...,$$ $$X_{n+(j- 1)p+k}=\sum^{n}_{i=1}U_{ki}X_{n+(j-1)p+k-i}+Y_{n+(j-1)p+k}.$$It is assumed that the $$Y_{n+(j-1)p+k}$$ are independent random vectors with covariance matrices $$G_ k^{-1}$$ and Gaussian (though this latter is not made apparent in the introduction).
The estimation procedure, which is too complicated to describe here in detail, is essentially one of regressing, for each $$k=1,...,p$$, $$x_{n+k+(j-1)p}$$ (after mean correction) on $$x_{n+k+(j-1)p-i}$$, $$i=1,...,n$$. If $$X_ t$$, $$t=1,...,N$$, is observed the data in the regression are for $$j=1,2,..$$. [(N-k-n)/p]. This estimation procedure is justified as being obtained from modal values for a posterior distribution commencing from a uniform density for the parameters, other than the $$G_ k$$, and proportional to $$| G_ k|^{-}$$ for the $$G_ k$$ (with zero mass for indefinite $$G_ k).$$
An asymptotic posterior chi-square distribution is obtained for an expression quadratic in the estimates $$\hat U_ k$$ from which confidence regions for the $$U_ k$$ can be derived, and similarly for the $$\mu_ k$$, and test procedures to test for the need for a periodic model are also provided.
Reviewer: E.J.Hannan
##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F15 Bayesian inference 62M07 Non-Markovian processes: hypothesis testing 62M09 Non-Markovian processes: estimation
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##### References:
  J. Anděl: The Statistical Analysis of Time Series. SNTL Prague 1976  J. Anděl: Mathematical Statistics. SNTL Prague 1978  J. Anděl: Statistical analysis of periodic autoregression. Apl. mat. 28 (1983), 164-185.  J. Anděl A. Rubio A. Insua: On periodic autoregression with unknown mean. Apl. mat. 30(1985), 126-139. · Zbl 0585.62152  T. W. Anderson: An Introduction to Multivariate Statistical Analysis. Wiley, New York 1958. · Zbl 0083.14601  W. P. Cleveland G. C. Tiao: Modeling seasonal time series. Rev. Economic Appliquée 32 (1979), 107-129.  H. Cramér: Mathematical Methods of Statistics. Princeton Univ. Press, Princeton 1946. · Zbl 0063.01014  E. G. Gladyshev: Periodically correlated random sequences. Soviet Math. 2 (1961), 385-388. · Zbl 0212.21401  R. H. Jones W. M. Brelsford: Time series with periodic structure. Biometrika 54 (1967), 403-408. · Zbl 0153.47706  H. Neudecker: Some theorems on matrix differentiation with special reference to Kronecker matrix products. J. Amer. Statist. Assoc. 64 (1969), 953-963. · Zbl 0179.33102  M. Pagano: On periodic and multiple autoregression. Ann. Statist. 6 (1978), 1310-1317. · Zbl 0392.62073  C. R. Rao: Linear Statistical Inference and Its Application. Wiley, New York 1965. · Zbl 0137.36203  C. G. Tiao M. R. Grupe: Hidden periodic autoregressive - moving average models in time series data. Biometrika 67 (1980), 365-373. · Zbl 0436.62076
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