Dynamic linear regression models

The dynamic linear regression model (28.35) discussed above, constitutes a reduc­tion of the VAR(1) model (28.31), given that we can decompose D(Z t|Z t-1; ф) further, based on the separation Z t = (yT, XT )T, yt : m1 x 1, to yield:

D(Z0, Z1,. . ., Zr; ¥) M= D(Zо; фо) П D(Zf | Zf_1; ф)

t=1 T

= D(Z0; Ф0) ПD(yt | Xt, ZM; ¥)D(Xt | Zt_1; ф).


Under the normality reduction assumption this gives rise to the multivariate dynamic linear regression model (MDLR):

yt = BTx t + BTx t-1 + ATyt-1 + ut, t Є T,

image671 image672

As we can see, the statistical parameters of the MDLR(1) (28-37) differ from those of the VAR(1) model (28-31) in so far as the former goes one step further pur­porting to model (in terms of the extra conditioning) the "contemporaneous dependence" captured in О in the context of the former model – This model is particularly interesting in econometrics because it provides a direct link to the simultaneous equations model; see Spanos (1986), p. 645.

4 Conclusion

Statistical models, such as AR(p), MA(q), ARMA(p, q) and the linear regression model with error autocorrelation, often used for modeling time series data, have been considered from a particular viewing angle we called the probabilistic reduction (PR) approach. The emphasis of this approach is placed on specifying statistical models in terms of a consistent set of probabilistic assumptions regard­ing the observable stochastic process underlying the data, as opposed to the error term – Although the discussion did not cover the more recent developments in time series econometrics, it is important to conclude with certain remarks regard­ing these developments – The recent literature on unit roots and cointegration, when viewed from the PR viewpoint, can be criticized on two grounds. The first criticism is that the literature has largely ignored the statistical adequacy issue – When the estimated AR( p) models are misspecified, however, the unit root test inference results will often be misleading. The second criticism concerns the in­adequate attention paid by the recent literature on the implicit parameterizations (discussed above) and what they entail (see Spanos and McGuirk, 1999).


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