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Time Series Analysis - Bayesian Multiple Time Series Modeling

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Time Series Analysis - Bayesian Multiple Time Series Modeling

(V.IV-1)

The likelihood function is

(V.IV-2)

On combining (V.IV-1), (V.IV-2), and Bayes' theorem, the posterior pdf is obtained as

(V.IV-3)

which can be drastically simplified on using

(V.IV-4)

such that the following is obtained

(V.IV-5)

where the Bayesian estimator is

(V.IV-6)

From (V.IV-5) it follows that

(V.IV-7)

with parameter covariance

(V.IV-8)

such that it can be concluded that the posterior distribution for the Bayesian estimator (V.IV-6) is normal

(V.IV-9)

Since in practice the error covariance matrix is not known, it is replaced by its estimated value (on using an LS estimator). Also, it may be practical to estimate the parameters in the context of SUR (since the exogenous variables are the same in each equation). The Bayesian k-th equation parameter estimator is

(V.IV-10)

Note that this LS procedure also applies in case of time series with cointegration (even with included constant).

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