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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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