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Regression Theory for Near-Integrated Time Series

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Abstract

The concept of a near-integrated vector random process is introduced. Such processes help us to work towards a general asymptotic theory of regression for multiple time series in which some series may be integrated processes of the ARIMA type, others may be stable ARMA processes with near unit roots, and yet others may be mildly explosive. A limit theory for the sample moments of such time series is developed using weak convergence and is shown to involve a simple functionals of a vector diffusion. The results suggest finite sample approximations which in the stationary case correspond to conventional central limit theory. The theory is applied to the study of vector autoregressions and cointegrating regressions of the type recently advanced by Granger and Engle (1987). A noncentral limiting distribution theory is derived for some recently proposed multivariate unit root tests. This yields some interesting insights into the asymptotic power properties of the various tests. Models with drift and near integration are also studied. The asymptotic theory in this case helps to bridge the gap between the nonnormal asymptotics obtained by Phillips and Durlauf (1986) for regressions with integrated regressors and the normal asymptotics that usually apply in regressions with deterministic regressors.

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File URL: http://cowles.econ.yale.edu/P/cd/d07b/d0781-r.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 781R.

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Length: 42 pages
Date of creation: Jan 1986
Date of revision: Jan 1987
Publication status: Published in Econometrica (September 1988), 56(5): 1021-1043
Handle: RePEc:cwl:cwldpp:781r

Note: CFP 711.
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Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
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Web page: http://cowles.econ.yale.edu/
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

Related research

Keywords: Brownian motion; cointegration; diffusion; near-integration; unit root tests;

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References

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  1. Evans, G B A & Savin, N E, 1981. "Testing for Unit Roots: 1," Econometrica, Econometric Society, vol. 49(3), pages 753-79, May.
  2. Phillips, P.C.B., 1986. "Testing for a Unit Root in Time Series Regression," Cahiers de recherche 8633, Universite de Montreal, Departement de sciences economiques.
  3. Park, Joon Y. & Phillips, Peter C.B., 1988. "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, Cambridge University Press, vol. 4(03), pages 468-497, December.
  4. West, Kenneth D, 1988. "Asymptotic Normality, When Regressors Have a Unit Root," Econometrica, Econometric Society, vol. 56(6), pages 1397-1417, November.
  5. Phillips, P C B, 1974. "The Estimation of Some Continuous Time Models," Econometrica, Econometric Society, vol. 42(5), pages 803-23, September.
  6. Kramer, Walter, 1984. "On the consequences of trend for simultaneous equation estimation," Economics Letters, Elsevier, vol. 14(1), pages 23-30.
  7. Granger, C. W. J. & Newbold, P., 1974. "Spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 2(2), pages 111-120, July.
  8. Peter C.B. Phillips, 1986. "Weak Convergence to the Matrix Stochastic Integral BdB," Cowles Foundation Discussion Papers 796, Cowles Foundation for Research in Economics, Yale University.
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