# 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).

(28.36)

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,  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).

References

Anderson, T. W. (1971). The Statistical Analysis of Time Series. Wiley, New York.

Box, G. E.P., and G. M. Jenkins (1970). Time Series Analysis: Forecasting and Control, revised edn 1976. Holden-Day, San Francisco.

Cochrane, D., and G. H. Orcutt (1949). Application of least-squares regression to relation­ships containing autocorrelated error terms. Journal of the American Statistical Association 44, 32-61.

Cramer, H. (1937). Random Variables and Probability Distributions. Cambridge: Cambridge University Press.

Davis, T. H. (1941). The Analysis of Economic Time Series. Cowles Commission Monograph No. 6, The Principia Press, Indiana.

Dhrymes, P. J. (1971). Distributed Lags: Problems of Estimation and Formulation. Edinburgh: Oliver and Boyd.

Dickey, D. A., and W. A. Fuller (1979). Distribution of the estimators for autore­gressive time series with a unit root. Journal of the American Statistical Association 74, 427-31.

Dickey, D. A., and W. A. Fuller (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057-72.

Durbin, J., and G. S. Watson (1950). Testing for serial correlation in least squares regression

I. Biometrika 37, 409-28.

Durbin, J., and G. S. Watson (1951). Testing for serial correlation in least squares regression

II. Biometrika 38, 159-78.

Engle, R. F., and C. W.J. Granger (1987). Cointegration and error-correction representation: estimation and testing. Econometrica 55, 251-76.

Frisch, R. (1933). Propagation problems and impulse problems in dynamic economics. In Economic Essays in Honor of Gustav Cassel. London: Macmillan.

Granger, C. W.J. (1980). Long memory relationships and the aggregation of dynamic models. Journal of Econometrics 14, 227-38.

Granger, C. W.J. (1983). Cointegrated variables and error-correcting models. UCSD dis­cussion paper 83-13.

Granger, C. W.J. (ed.) (1990). Modelling Economic Series: Readings on the Methodology of Econometric Modeling. Oxford: Oxford University Press.

Granger, C. W.J., and P. Newbold (1974). Spurious regressions in econometrics. Journal of Econometrics 2, 111-20.

Granger, C. W.J., and P. Newbold (1977). Forecasting Economic Time Series. London: Aca­demic Press.

Granger, C. W.J., and T. Terasvirta (1993). Modelling Nonlinear Economic relationships. Oxford: Oxford University Press.

Hamilton, J. D. (1994). Time Series Analysis. New Jersey: Princeton University Press. Hendry, D. F. (1993). Econometrics: Alchemy or Science?. Oxford: Blackwell.

Hendry, D. F., and G. E. Mizon (1978). Serial correlation as a convenient simplification not a nuisance: a comment on a study of the demand for money by the Bank of England. Economic Journal 88, 549-63.

Hendry, D. F., and M. S. Morgan (1995). The Foundations of Economic Analysis: An Introduc­tion. New York: Cambridge University Press.

Heyde, C. C., and E. Seneta (1977). I. J. Bieyname: Statistical Theory Anticipated. New York: Springer-Verlag.

Hooker, R. (1901). Correlation of the marriage rate with trade. Journal of the Royal Statistical Society 64, 485-603.

Hooker, R. (1905). On the correlation of successive observations: illustrated by corn prices.

Journal of the Royal Statistical Society 68, 696 -703.

Johansen, S. (1991). Estimation and hypothesis testing of cointegrating vectors in Gaussian vector autoregressive models. Econometrica 59, 1551-81.

Khinchine, A. Y. (1932). Selle successioni stazioarie di eventi. Giorn. Ist. Ital. Attuari 3, 267-74. Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitrechnung, Berlin. Foundations of the Theory of Probability, 2nd English edn. New York: Chelsea Publishing Co. Kolmogorov, A. N. (1941). Stationary sequences in Hilbert space. Byull. Moskov. Gos. Univ.

Mat. 2, 1-40. English translation reprinted in Shiryayev (1992), pp. 228-71.

Mann, H. B. and A. Wald (1943). On the statistical treatment of linear stochastic difference equations. Econometrica 11, 173 -220.

Moore, H. L. (1914). Economic Cycles: Their Law and Cause. New York: Macmillan.

Nelson, C. R., and C. I. Plosser (1982). Trends and random walks in macro-economic time series: some evidence and implications. Journal of Monetary Economics 10, 139-62. Norton, J. (1902). Statistical Studies in the New York Money Market. New York: Macmillan. Phillips, P. C.B. (1986). Understanding spurious regression in econometrics. Journal of Econometrics 33, 311-40.

Phillips, P. C.B. (1987). Time series regressions with a unit root. Econometrica 55, 227-301. Phillips, P. C.B. (1991). Optimal inference in cointegrating systems. Econometrica 59, 283-306.

Schuster, A. (1906). On the periodicities of sunspots. Philosophical Transactions of Royal Society of London A, 206, 69-100.

Shiryayev, A. N. (ed.) (1992). Selected Works of A. N. Kolmogorov, vol. II: Probability Theory and Mathematical Statistics. Dordrecht: Kluwer.

Sims, C. A. (1980). Macroeconomics and reality. Econometrica 48, 1-48.

Slutsky, E. (1927). The summation of random causes as the source of cyclic processes (in Russian); English translation in Econometrica 5, (1937).

Spanos, A. (1986). Statistical Foundations of Econometric Modelling. Cambridge: Cambridge University Press.

Spanos, A. (1987). Error autocorrelation revisited: the AR(1) case. Econometric Reviews 6, 285-94.

Spanos, A. (1990). Unit roots and their dependence of the conditioning information set. Advances in Econometrics 8, 271-92.

Spanos, A. (1995). On normality and the linear regression model. Econometric Reviews 14, 195-203.

Spanos, A. (1999). Probability Theory and Statistical Inference: Econometric Modeling with Observational Data. Cambridge: Cambridge University Press.

Spanos, A., and A. McGuirk (1999). The power of unit root tests revisited. Mimeo, Virginia Polytechnic Institute and State University.

Stigler, S. M. (1986). The History of Statistics: the Measurement of Uncertainty before 1900, Cambridge, MA: Harvard University Press.

Wold, H. O. (1938). A Study in the Analysis of Stationary Time Series (revised 1954) Uppsala: Almquist and Wicksell.

Yule, G. U. (1921). On the time-correlation problem. Journal of the Royal Statistical Society 84, 497-526.

Yule, G. U. (1926). Why do we sometimes get nonsense correlations between time series – a study in sampling and the nature of time series. Journal of the Royal Statistical Society 89, 1-64.

Yule, G. U. (1927). On a method of investigating periodicities in disturbed series, with special reference to Wolfer’s sunspot numbers. Philosophical Transactions of the Royal Society series A, 226, 267-98.