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Autoregressive Errors in Singular Systems of Equations

Published online by Cambridge University Press:  11 February 2009

Phoebus J. Dhrymes
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Abstract

We consider a system of m general linear models, where the system error vector has a singular covariance matrix owing to various “adding up” requirements and, in addition, the error vector obeys an autoregressive scheme. The paper reformulates the problem considered earlier by Berndt and Savin [8] (BS), as well as others before them; the solution, thus obtained, is far simpler, being the natural extension of a restricted least-squares-like procedure to a system of equations. This reformulation enables us to treat all parameters symmetrically, and discloses a set of conditions which is different from, and much less stringent than, that exhibited in the framework provided by BS.

Finally, various extensions are discussed to (a) the case where the errors obey a stable autoregression scheme of order r; (b) the case where the errors obey a moving average scheme of order r; (c) the case of “dynamic” vector distributed lag models, that is, the case where the model is formulated as autoregressive (in the dependent variables), and moving average (in the explanatory variables), and the errors are specified to be i.i.d.

Type
Research Article
Information
Econometric Theory , Volume 10 , Issue 2 , June 1994 , pp. 254 - 285
Copyright
Copyright © Cambridge University Press 1994

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References

1. Aigner, D. J. On the behavior of an econometric model of short run bank behavior. Journal of Econometrics 14 (1980): 201–228.Google Scholar
2. Anderson, G.J. The structure of simultaneous equations estimators: A comment. Journal of Econometrics 14 (1980): 271–276.10.1016/0304-4076(80)90097-4CrossRefGoogle Scholar
3. Anderson, G.J. & Blundell, R.W., Estimation and Hypothesis Testing in Dynamic Singular Equation Systems. Econometrica 50 (1982): 1559–1571.10.2307/1913396CrossRefGoogle Scholar
4. Barten, A.P. Maximum likelihood estimation of a complete system of demand equations. European Economic Review 1 (1969): 7–80.10.1016/0014-2921(69)90017-8CrossRefGoogle Scholar
5. Barten, A.P. The systems of consumer demand approach: A review. Econometrica 45 (1977): 23–51.CrossRefGoogle Scholar
6. Berndt, E.R. & Christensen, L.R.. Testing for the existence of a consistent aggregate index of labor inputs. American Economic Review 64 (1974): 391–403.Google Scholar
7. Berndt, E.R. & Christensen, L.R.. The translog function and the substitution of equipment, structures and labor in US manufacturing, 1929–1969. Journal of Econometrics 1 (1973): 81–113.10.1016/0304-4076(73)90007-9CrossRefGoogle Scholar
8. Berndt, E.R. & Savin, N.E.. Estimation and hypothesis testing in singular equation systems with autoregressive disturbances, Econometrica 43 (1975): 937–957.CrossRefGoogle Scholar
9. Billingsley, P. Convergence of Probability Measures New York: Wiley, 1968.Google Scholar
10. Christensen, L.R., Jorgensen, D.W. & Lau, L.J.. Transcendental logarithmic production frontiers. Review of Economics and Statistics 55 (1973): 28–45.10.2307/1927992CrossRefGoogle Scholar
11. Christensen, L.R., Jorgensen, D.W. & Lau, L.J.. Transcendental logarithmic utility functions. American Economic Review 65 (1975): 367–383.Google Scholar
12. Cochrane, D. & Orcutt, G.H.. Application of least squares regressions to relationships containing autocorrelated error terms. Journal of the American Statistical Association 44 (1949): 32–61.Google Scholar
13. Deaton, A.S. Models and Projections of Demand in Postwar Britain. New York: Halstead Press, 1975.10.1007/978-1-4899-3113-9CrossRefGoogle Scholar
14. Deaton, A.S. & Muelbauer, J.. Economics and Consumer Behavior. Cambridge: Cambridge University Press, 1980.10.1017/CBO9780511805653CrossRefGoogle Scholar
15. Dacunha-Castelle, D. & Duflo, M.. Probability and Statistics, vol. II. New York: Springer-Verlag, 1986.Google Scholar
16. Dhrymes, P. J. On devising unbiased estimators for the parameters of a Cobb-Douglas production function. Econometrica 30 (1962): 297–304.10.2307/1910218CrossRefGoogle Scholar
17. Dhrymes, P.J. Introductory Econometrics. New York: Springer-Verlag, 1978.CrossRefGoogle Scholar
18. Dhrymes, P.J. Mathematics for Econometrics (2nd ed.). New York: Springer-Verlag, 1984.CrossRefGoogle Scholar
19. Dhrymes, P.J. Topics in Advanced Econometrics: Probability Foundations. New York: Springer-Verlag, 1989.10.1007/978-1-4612-4548-3CrossRefGoogle Scholar
20. Dhrymes, P.J. Topics in Advanced Econometrics: Linear and Nonlinear Simultaneous Equations. New York: Springer-Verlag, 1994.10.1007/978-1-4612-4302-1CrossRefGoogle Scholar
21. Dhrymes, P.J. & Schwarz, S.. On the existence of generalized inverse estimators in a singular system of equations. Journal of Forecasting 6 (1987): 181–193.CrossRefGoogle Scholar
22. Dhrymes, P.J. & Schwarz, S.. On the invariance of estimators for singular systems of demand equations. Greek Economic Review 9 (1987): 88–107.Google Scholar
23. Hall, P. & Heyde, C.C.. Martingale Limit Theory and its Applications. New York: Academic Press, 1980.Google Scholar
24. Powell, A.A. “Aitken estimators as a tool in allocating predetermined aggregates. Journal of the American Statistical Association 64 (1969): 913–922.10.1080/01621459.1969.10501023CrossRefGoogle Scholar
25. Rao, J.R. Alternative parametric models of sales advertising relationships. Journal of Marketing Research 9 (1972): 171–181.10.1177/002224377200900209CrossRefGoogle Scholar
26. Theil, H. Principles of Econometrics. New York: Wiley, 1971.Google Scholar