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Fully Modified Least Squares and Vector Autoregression

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Fully modified least squares (FM-OLS) regression was originally designed in work by Phillips and Hansen (1990) to provide optimal estimates of cointegrating regressions. The method modifies least squares to account for serial correlation effects and for the endogeneity in the regressors that results from the existence of a cointegrating relationship. This paper provides a general framework which makes it possible to study the asymptotic behavior of FM-OLS in models with full rank I(1) regressors, models with I(1) and I(0) regressors, models with unit roots, and models with only stationary regressors. This framework enables us to consider the use of FM regression in the context of vector autoregressions (VAR's) with some unit roots and some cointegrating relations. The resulting FM-VAR regressions are shown to have some interesting properties. For example, when there is some cointegration in the system, FM-VAR estimation has a limit theory that is normal for all of the stationary coefficients and mixed normal for all of the nonstationary coefficients. Thus, there are no unit root limit distributions even in the case of the unit root coefficient submatrix (i.e., I_{n-r}, for an n-dimensional VAR with r cointegrating vectors). Moreover, optimal estimation of the cointegration space is attained in FM-VAR regression without prior knowledge of the number of unit roots in the system, without pretesting to determine the dimension of the cointegration space and without the use of restricted regression techniques like reduced rank regression. The paper also develops an asymptotic theory for inference based on FM-OLS and FM-VAR regression. The limit theory for Wald tests that rely on the FM estimator is shown to involve a linear combination of independent chi-squared variates. This limit distribution is bounded above by the conventional chi-squared distribution with degrees of freedom equal to the number of restrictions. Thus, conventional critical values can be used to construct valid (but conservative) asymptotic tests in quite general FM time series regressions. This theory applies to causality testing in VAR's and is therefore potentially useful in empirical applications.

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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 1047.

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Length: 79 pages
Date of creation: May 1993
Date of revision:
Publication status: Published in Econometrica (September 1995), 63(5): 1023-1078
Handle: RePEc:cwl:cwldpp:1047

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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Causality testing; cointegration; fully modified regression; fully modified vector autoregression; hyperconsistency; long-run covariance matrix; one-sided long-run covariance matrix; some unit roots;

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  1. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, vol. 58(1), pages 113-44, January.
  2. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  3. Hendry, David F, 1980. "Econometrics-Alchemy or Science?," Economica, London School of Economics and Political Science, vol. 47(188), pages 387-406, November.
  4. Toda, Hiro Y & Phillips, Peter C B, 1993. "Vector Autoregressions and Causality," Econometrica, Econometric Society, vol. 61(6), pages 1367-93, November.
  5. Peter C.B. Phillips & Joon Y. Park, 1986. "Statistical Inference in Regressions with Integrated Processes: Part 1," Cowles Foundation Discussion Papers 811R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1987.
  6. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-58, May.
  7. Hiro Y. Toda & Peter C.B. Phillips, 1991. "Vector Autoregression and Causality: A Theoretical Overview and Simulation Study," Cowles Foundation Discussion Papers 1001, Cowles Foundation for Research in Economics, Yale University.
  8. Peter C.B. Phillips, 1988. "Spectral Regression for Cointegrated Time Series," Cowles Foundation Discussion Papers 872, Cowles Foundation for Research in Economics, Yale University.
  9. Peter C.B. Phillips & Mico Loretan, 1989. "Estimating Long Run Economic Equilibria," Cowles Foundation Discussion Papers 928, Cowles Foundation for Research in Economics, Yale University.
  10. Bowden,Roger J. & Turkington,Darrell A., 1990. "Instrumental Variables," Cambridge Books, Cambridge University Press, number 9780521385824, April.
  11. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
  12. Peter C.B. Phillips & Bruce E. Hansen, 1988. "Statistical Inference in Instrumental Variables," Cowles Foundation Discussion Papers 869R, Cowles Foundation for Research in Economics, Yale University, revised Apr 1989.
  13. Peter C.B. Phillips, 1992. "Hyper-Consistent Estimation of a Unit Root in Time Series Regression," Cowles Foundation Discussion Papers 1040, Cowles Foundation for Research in Economics, Yale University.
  14. Peter C.B. Phillips, 1991. "Unidentified Components in Reduced Rank Regression Estimation of ECM's," Cowles Foundation Discussion Papers 1003, Cowles Foundation for Research in Economics, Yale University.
  15. Park, Joon Y, 1992. "Canonical Cointegrating Regressions," Econometrica, Econometric Society, vol. 60(1), pages 119-43, January.
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