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Asymptotic Expansions in Nonstationary Vector Autoregressions

Published online by Cambridge University Press:  11 February 2009

P. C. B. Phillips
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Abstract

This paper studies the statistical properties of vector autoregressions (VAR's) for quite general multiple time series which are integrated processes of order one. Functional central limit theorems are given for multivariate partial sums of weakly dependent innovations and these are applied to yield first-order asymptotics in nonstationary VAR's. Characteristic and cumulant functionals for generalized random processes are introduced as a means of developing a refinement of central limit theory on function spaces. The theory is used to find asymptotic expansions of the regression coefficients in nonstationary VAR's under very general conditions. The results are specialized to the scalar case and are related to other recent work by the author [21].

Type
Research Article
Information
Econometric Theory , Volume 3 , Issue 1 , February 1987 , pp. 45 - 68
Copyright
Copyright © Cambridge University Press 1987

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References

REFERENCES

1. Billingsley, P. Convergence of Probability Measures. John Wiley: New York, 1968.Google Scholar
2. Bochner, S. Harmonic Analysis and the Theory of Probability. Berkeley: University of California Press, 1960.Google Scholar
3. Cooley, T. F. and LeRoy, S. F.. Atheoretical macroeconometrics: A critique. Journal of Monetary Economics 16 (1985): 283–308.CrossRefGoogle Scholar
4. Doan, T., Litterman, R. B., and Sims, C.. Forecasting and conditional projection using real istic prior distributions. Econometric Reviews 3 (1984): 1–100.CrossRefGoogle Scholar
5. Evans, G.B.A. and Savin, N. E.. Testing for unit roots: 1. Econometrica 49 (1981): 753–779.CrossRefGoogle Scholar
6. Gelfand, I. M. and Vilenkin, N. Y.. Generalized Functions, Vol. 4. New York: Academic Press, 1964.Google Scholar
7. Herrndorf, N. The invariance principle for φ-mixing sequences. Z. Wahrscheinlichkeitstheorie verw. Gebiete 63 (1983): 97–108.CrossRefGoogle Scholar
8. Herrndorf, N. A functional central limit theorem for ρ-mixing sequences. Journal of Multivariate Analysis 15 (1984): 141–146.CrossRefGoogle Scholar
9. Herrndorf, N. A functional central limit theorem for weakly dependent sequences of random variables. Annals of Probability 12 (1984): 141–153.CrossRefGoogle Scholar
10. Hida, T. Brownian Motion. Springer Verlag: New York, 1980.CrossRefGoogle Scholar
11. Ibragimov, I. A. and Linnik, Y. V.. Independent and Stationary Sequences of Random Variables. Groningen: Woller-Nordhoff, 1971.Google Scholar
12. Kolmogorov, A. N. and Rozanov, Y. A.. On strong mixing conditions for stationary Gaussian processes. Theory of Probability and Applications 5 (1960): 204–208.CrossRefGoogle Scholar
13. Koopmans, T. C. Statistical Inference in Dynamic Economic Models. New York: Wiley, 1950.Google Scholar
14. Litterman, R. B. Forecasting with Bayesian vector autoregressions: Four years of experience. Mimeographed (Federal Reserve Bank of Minneapolis), 1984.CrossRefGoogle Scholar
15. Magnus, J. R. and Neudecker, H.. The elimination matrix: Some lemmas and applications. SIAM Journal on Algebraic and Discrete Methods 1 (1980): 442–449.CrossRefGoogle Scholar
16. Mann, H. B. and Wald, A.. On the statistical treatment of linear stochastic difference equations. Econometrica 11 (1943): 173–220.CrossRefGoogle Scholar
17. Pagan, A. Three econometric methodologies: A critical appraisal. Australian National University (mimeo), 1986.Google Scholar
18. Phillips, P.C.B. Approximations to some finite sample distributions associated with a first-order stochastic difference equation. Econometrica 45 (1977): 463–485.CrossRefGoogle Scholar
19. Phillips, P.C.B. Edgeworth and saddlepoint approximations in a first-order noncircular autoregression. Biometrika 65 (1978): 91–98.CrossRefGoogle Scholar
20. Phillips, P.C.B. Exact small sample theory in the simultaneous equations model. Chapter 8 and pp. 449–516 in Intriligator, M. D. and Griliches, Z. (Eds.), Handbook of Econometrics. Amsterdam: North-Holland, 1983.CrossRefGoogle Scholar
21. Phillips, P.C.B. Time series regression with a unit root. Cowles Foundation Discussion Paper No. 740R, Yale University (April 1985), forthcoming in Econometrica, 1986.Google Scholar
22. Phillips, P.C.B. Weak convergence to the matrix stochastic integral BdB'. Cowles Foundation Discussion Paper No. 796, Yale University, July 1986.Google Scholar
23. Phillips, P.C.B. Asymptotic expansions in stationary vector autoregressions (in preparation).Google Scholar
24. Phillips, P.C.B. and Durlauf, S. N.. Multiple time series regression with integrated processes. Review of Economic Studies (07 1986), Vol. 53, pp. 473–496.CrossRefGoogle Scholar
25. Prohorov, Y. V. The method of characteristic functionals, in Neyman, J. (Ed.), Proceedings of 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, pp. 403–419. Berkeley: University of California Press, 1961.Google Scholar
26. Rozanov, Y. A. Stationary Random Processes. San Francisco: Holden Day, 1967.Google Scholar
27. Sims, C. A. Macroeconomics and reality. Econometrica 48 (1980): 1–48.CrossRefGoogle Scholar
28. White, H. Asymptotic Theory for Econometricians. New York: Academic Press, 1984.Google Scholar
29. White, H. and Domowitz, I.. Nonlinear regression with dependent observations. Econometrica 52 (1984): 143–162.CrossRefGoogle Scholar
30. Yaglom, A. M. An Introduction to the Theory of Stationary Random Functions. New York: Dover, 1973.Google Scholar