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Consistent Testing of Cointegrating Relationships

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  • Francesc Marmol
  • Carlos Velasco

Abstract

In this paper we investigate methods for testing the existence of a cointegration relationship among the components of a nonstationary fractionally integrated (NFI) vector time series. Our framework generalizes previous studies restricted to unit root integrated processes and permits simultaneous analysis of spurious and cointegrated NFI vectors. We propose a modified F-statistic, based on a particular studentization, which converges weakly under both hypotheses, despite the fact that OLS estimates are only consistent under cointegration. This statistic leads to a Wald-type test of cointegration when combined with a narrow band GLS-type estimate. Our semiparametric methodology allows consistent testing of the spurious regression hypothesis against the alternative of fractional cointegration without prior knowledge on the memory of the original series, their short run properties, the cointegrating vector, or the degree of cointegration. This semiparametric aspect of the modelization does not lead to an asymptotic loss of power, permitting the Wald statistic to diverge faster under the alternative of cointegration than when testing for a hypothesized cointegration vector. In our simulations we show that the method has comparable power to customary procedures under the unit root cointegration setup, and maintains good properties in a general framework where other methods may fail. We illustrate our method testing the cointegration hypothesis of nominal GNP and simple-sum (M1, M2, M3) monetary aggregates. Copyright The Econometric Society 2004.

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File URL: http://hdl.handle.net/10.1111/j.1468-0262.2004.00554.x
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Bibliographic Info

Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 72 (2004)
Issue (Month): 6 (November)
Pages: 1809-1844

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Handle: RePEc:ecm:emetrp:v:72:y:2004:i:6:p:1809-1844

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  1. Gonzalo, Jesús & Lee, Tae-Hwy, . "Pitfalls in testing for long run relationships," Open Access publications from Universidad Carlos III de Madrid info:hdl:10016/753, Universidad Carlos III de Madrid.
  2. Clive Granger & Namwon Hyung & Yongil Jeon, 2001. "Spurious regressions with stationary series," Applied Economics, Taylor and Francis Journals, vol. 33(7), pages 899-904.
  3. Engle, Robert F & Granger, Clive W J, 1987. "Co-integration and Error Correction: Representation, Estimation, and Testing," Econometrica, Econometric Society, vol. 55(2), pages 251-76, March.
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  5. Granger, E.J. & Swanson, N.R., 1996. "An introduction to stochastic Unit Root Processes," Papers 4-96-3, Pennsylvania State - Department of Economics.
  6. Marmol, Francesc & Velasco, Carlos, . "Trend stationarity versus long-range dependence in time series analysis," Open Access publications from Universidad Carlos III de Madrid info:hdl:10016/4349, Universidad Carlos III de Madrid.
  7. Hassler, U. & Marmol, Francesc & Velasco, Carlos, . "Residual log-periodogram inference for long-run relationships," Open Access publications from Universidad Carlos III de Madrid info:hdl:10016/4359, Universidad Carlos III de Madrid.
  8. Robinson, Peter M. & Yajima, Yoshihiro, 2002. "Determination of cointegrating rank in fractional systems," Journal of Econometrics, Elsevier, vol. 106(2), pages 217-241, February.
  9. Tsay, Wen-Jen & Chung, Ching-Fan, 2000. "The spurious regression of fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 96(1), pages 155-182, May.
  10. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-80, November.
  11. Cheung, Yin-Wong & Lai, Kon S, 1993. "A Fractional Cointegration Analysis of Purchasing Power Parity," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(1), pages 103-12, January.
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  13. Peter C.B. Phillips & Sam Ouliaris, 1987. "Asymptotic Properties of Residual Based Tests for Cointegration," Cowles Foundation Discussion Papers 847R, Cowles Foundation for Research in Economics, Yale University, revised Jul 1988.
  14. Lobato, I. & Robinson, P. M., 1996. "Averaged periodogram estimation of long memory," Journal of Econometrics, Elsevier, vol. 73(1), pages 303-324, July.
  15. Robinson, P.M. & Velasco, Carlos, . "Autocorrelation-Robust Inference," Open Access publications from Universidad Carlos III de Madrid info:hdl:10016/4529, Universidad Carlos III de Madrid.
  16. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-58, May.
  17. Kramer, Walter & Marmol, Francesc, 2004. "The power of residual-based tests for cointegration when residuals are fractionally integrated," Economics Letters, Elsevier, vol. 82(1), pages 63-69, January.
  18. Gil-Alana, L. A. & Robinson, P. M., 1997. "Testing of unit root and other nonstationary hypotheses in macroeconomic time series," Journal of Econometrics, Elsevier, vol. 80(2), pages 241-268, October.
  19. Abdol Soofi, 1998. "A fractional cointegration test of purchasing power parity: the case of selected members of OPEC," Applied Financial Economics, Taylor and Francis Journals, vol. 8(6), pages 559-566.
  20. Baillie, R.T. & Bollerslev, T., 1993. "Cointegration, Fractional Cointegration, and Exchange RAte Dynamics," Papers 9103, Michigan State - Econometrics and Economic Theory.
  21. Jansson, Michael & Haldrup, Niels, 2002. "Regression Theory For Nearly Cointegrated Time Series," Econometric Theory, Cambridge University Press, vol. 18(06), pages 1309-1335, December.
  22. Marinucci, D. & Robinson, P. M., 2000. "Weak convergence of multivariate fractional processes," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 103-120, March.
  23. Marmol, Francesc, 1998. "Spurious regression theory with nonstationary fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 84(2), pages 233-250, June.
  24. Haldrup, Niels, 1994. "The asymptotics of single-equation cointegration regressions with I(1) and I(2) variables," Journal of Econometrics, Elsevier, vol. 63(1), pages 153-181, July.
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Citations

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Cited by:
  1. Esther Stroe-Kunold & Joachim Werner, 2009. "A drunk and her dog: a spurious relation? Cointegration tests as instruments to detect spurious correlations between integrated time series," Quality & Quantity: International Journal of Methodology, Springer, vol. 43(6), pages 913-940, November.
  2. Nielsen, Morten Orregaard & Shimotsu, Katsumi, 2007. "Determining the cointegrating rank in nonstationary fractional systems by the exact local Whittle approach," Journal of Econometrics, Elsevier, vol. 141(2), pages 574-596, December.
  3. Katarzyna Lasak, 2008. "Likelihood based testing for no fractional cointegration," CREATES Research Papers 2008-52, School of Economics and Management, University of Aarhus.
  4. Javier Hualde, 2005. "Unbalanced Cointegration," Faculty Working Papers 06/05, School of Economics and Business Administration, University of Navarra.
  5. Javier Hualde & Peter M Robinson, 2006. "Root-N-Consistent Estimation Of Weakfractional Cointegration," STICERD - Econometrics Paper Series /2006/499, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  6. Hurvich, Cliiford & Wang, Yi, 2006. "A Pure-Jump Transaction-Level Price Model Yielding Cointegration, Leverage, and Nonsynchronous Trading Effects," MPRA Paper 1413, University Library of Munich, Germany.
  7. Katarzyna Lasak, 2008. "Maximum likelihood estimation of fractionally cointegrated systems," CREATES Research Papers 2008-53, School of Economics and Management, University of Aarhus.
  8. Mauro Costantini & Roy Cerqueti, 2007. "Non parametric Fractional Cointegration Analysis," ISAE Working Papers 78, ISTAT - Italian National Institute of Statistics - (Rome, ITALY).
  9. Avarucci, Marco & Velasco, Carlos, 2008. "A Wald Test for the Cointegration Rank in Nonstationary Fractional Systems," Research Memoranda 049, Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization.

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