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Fractional Cointegration

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Abstract

We describe a variety of seimparametric models and estimators for fractional cointegration. All of the estimators we consider are based on the discrete Fourier transform of the data. This includes the ordinary least squares estimator as a special case.We make a distinction between Type I and Type II models, which differ from each other in terms of assumptions about initialization, and which lead to different functional limit laws for the partial sum processes. We compare the estimators in terms of rate of convergence. We briefly discuss the problems of testing for cointegration and determining the cointegrating rank. We also discuss relevant modeling issues, such as the local parametrization of the phase function.

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References

  • Adenstedt, R. K. (1974): On large-sample estimation for the mean of a stationary random sequence. Annals of Statistics 2, 1095–1107.

    Article  MATH  MathSciNet  Google Scholar 

  • Akonom, J. and Gourieroux, C. (1987): A functional central limit theorem for fractional processes. Preprint, CEREMAP, Paris.

    Google Scholar 

  • Breitung, J. and Hassler, U. (2002): Inference on the cointegration rank in fractionally integrated processes. Journal of Econometrics 110, 167–185.

    Article  MATH  MathSciNet  Google Scholar 

  • Brockwell, P. J. and R. A. Davis (1991): Time series: theory and methods: Second Edition. Springer, New York.

    Google Scholar 

  • Chan, N. H. and Terrin, N. (1995): Inference for unstable long-memory processes with applications to fractional unit root autoregressions. Annals of Statistics 23, 1662–1683.

    Article  MATH  MathSciNet  Google Scholar 

  • Chen, W. (2006): Efficiency in estimation of memory Preprint, SSRN e-Library.

    Google Scholar 

  • Chen, W. and Hurvich, C. (2003a): Estimating fractional cointegration in the presence of polynomial trends. Journal of Econometrics 117, 95–121.

    Article  MATH  MathSciNet  Google Scholar 

  • Chen, W. and Hurvich, C. (2003b): Semiparametric estimation of multivariate fractional cointegration. Journal of the American Statistical Association 98, 629–642.

    Article  MATH  MathSciNet  Google Scholar 

  • Chen, W. and Hurvich, C. (2006): Semiparametric estimation of fractional cointegrating subspaces. Annals of Statistics 34.

    Google Scholar 

  • Christensen, B. J. and Nielsen, M. O. (2006): Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting. Journal of Econometrics 133, 343–371.

    Article  MathSciNet  Google Scholar 

  • Davidson, J. (2002): A model of fractional cointegration, and tests for cointegration using the bootstrap. Journal of Econometrics 110, 187–212.

    Article  MATH  MathSciNet  Google Scholar 

  • Dolado, J.J., Gonzalo, J. and Mayoral, L. (2003): Long range dependence in Spanish political opinion poll data. Journal of Applied Econometrics 18, 137–155.

    Article  Google Scholar 

  • Dueker, M. and Startz, R. (1998): Maximum-likelihood estimation of fractional cointegration with an application to U.S. and Canadian bond rates. The Review of Economics and Statistics 80, 420–426.

    Article  Google Scholar 

  • Engle, R. and Granger C. W. J. (1987): Co-integration and error correction: representation, estimation and testing. Econometrica 55, 251–276.

    Article  MATH  MathSciNet  Google Scholar 

  • Granger, C. W. J. and Joyeux, R. (1980): An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis 1, 15–29.

    Article  MATH  MathSciNet  Google Scholar 

  • Granger, C. W. J. (1986): Developments in the study of cointegrated economic variables. Oxford Bulletin of Economics and Statistics 48, 213–228.

    Google Scholar 

  • Hosking, J. R. M. (1981): Fractional differencing. Biometrika 68, 165–176.

    Article  MATH  MathSciNet  Google Scholar 

  • Hurvich, C. M. and Chen, W. (2000): An efficient taper for potentially overdifferenced long-memory time series. Journal of Time Series Analysis 21, 155–180.

    Article  MATH  MathSciNet  Google Scholar 

  • Hurvich, C. M. and Ray, B. K. (1995): Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes. Journal of Time Series Analysis 16, 17–41.

    Article  MATH  MathSciNet  Google Scholar 

  • Jeganathan, P. (1999): On asymptotic inference in cointegrated time series with fractionally integrated errors. Econometric Theory 15, 583–621.

    Article  MATH  MathSciNet  Google Scholar 

  • Johansen, S. (1988): Statistical analysis of cointegration vectors Journal of Economic Dynamics and Control 12, 231–254.

    Article  MATH  MathSciNet  Google Scholar 

  • Johansen, S. (1991): Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551–1580.

    Article  MATH  MathSciNet  Google Scholar 

  • Lobato, I. N. (1999): A semiparametric two-step estimator in a multivariate long memory model. Journal of Econometrics 90, 129–153.

    Article  MATH  MathSciNet  Google Scholar 

  • Mandelbrot, B. and Van Ness, J. (1968): Fractional Brownian motions, fractional noises and applications. SIAM Review 10, 422–437.

    Article  MATH  MathSciNet  Google Scholar 

  • Marinucci, D. and Robinson, P.M. (1999): Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 111–122.

    Article  MATH  MathSciNet  Google Scholar 

  • Marinucci, D. and Robinson, P.M. (2000): Weak convergence of multivariate fractional processes. Stochastic Processes and their Applications 86, 103–120.

    Article  MATH  MathSciNet  Google Scholar 

  • Marinucci, D. and Robinson, P.M. (2001): Journal of Econometrics Journal of Econometrics 105, 225–247.

    Article  MATH  MathSciNet  Google Scholar 

  • Marmol, F. and Velasco, C. (2004): Consistent testing of cointegrating relationships. Econometrica 72, 1809–1844.

    Article  MATH  MathSciNet  Google Scholar 

  • Nielsen, M. O. and Shimotsu, K. (2007): Determining the cointegrating rank in nonstationary fractional systems by the exact local Whittle approach. Journal of Econometrics to appear.

    Google Scholar 

  • Pham, T. D., and Tran, L. T. (1985): Some mixing properties of time series models. Stochastic Processes and their Applications 19, 297–303.

    Article  MATH  MathSciNet  Google Scholar 

  • Robinson, P. M. (1994): Semiparametric analysis of long-memory time series. Annals of Statistics 22, 515–539.

    Article  MATH  MathSciNet  Google Scholar 

  • Robinson, P. M. (1995b): Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630–1661.

    Article  MATH  Google Scholar 

  • Robinson, P. M. (2006): Multiple local Whittle estimation in nonstationary systems. Preprint.

    Google Scholar 

  • Robinson, P. M. and Hualde, J. (2003): Cointegration in fractional systems with unknown integration orders. Econometrica 71, 1727–1766.

    Article  MATH  MathSciNet  Google Scholar 

  • Robinson, P. M. and Marinucci, D. (2001): Narrow-band analysis of nonstationary processes. Annals of Statistics 29, 947–986.

    Article  MATH  MathSciNet  Google Scholar 

  • Robinson, P. M. and Marinucci, D. (2003): Semiparametric frequency domain analysis of fractional cointegration. In: Robinson, P. (Ed.): Time series with long memory, Advanced Texts in Econometrics. Oxford University Press, Oxford.

    Google Scholar 

  • Robinson, P. M. and Yajima, Y. (2002): Determination of cointegrating rank in fractional systems. Econometrics 106, 217–241.

    Article  MATH  MathSciNet  Google Scholar 

  • Shimotsu, K. (2006): Gaussian semiparametric estimation of multivariate fractionally integrated processes. Journal of Econometrics, forthcoming.

    Google Scholar 

  • Shimotsu, K. and Phillips, P. C. B. (2005): Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890–1933.

    Article  MATH  MathSciNet  Google Scholar 

  • Silveira, G. (1991): Contributions to Strong Approximations in time series with applications in nonparametric statistics and functional central limit theorems. Ph.D. Thesis, University of London.

    Google Scholar 

  • Sowell, F. (1990): The fractional unit root distribution. Econometrica 58, 495–505.

    Article  MATH  MathSciNet  Google Scholar 

  • Taqqu, M. S. (2003): Fractional Brownian motion and long-range dependence. In: Doukhan, P., Oppenheim, G. and Taqqu, M. (Eds.): Theory and Applications of Long-Range Dependence. Birkhäuser, Boston.

    Google Scholar 

  • Velasco, C. (1999a): Non-stationary log-periodogram regression. J. Econometrics 91, 325–371.

    Article  MATH  MathSciNet  Google Scholar 

  • Velasco, C. (1999b): Gaussian semiparametric estimation of non-stationary time series Journal of Time Series Analysis 20, 87–127.

    Article  MathSciNet  Google Scholar 

  • Velasco, C. (2003): Gaussian semi-parametric estimation of fractional cointegration. Journal of Time Series Analysis 24, 345–378.

    Article  MATH  MathSciNet  Google Scholar 

  • Velasco, C. and Marmol, F. (2004): Consistent testing of cointegration relationships. Econometrica 72, 1809–1844.

    Article  MATH  MathSciNet  Google Scholar 

  • Velasco, C. (2007): The periodogram of fractional processes. Journal of Time Series Analysis, forthcoming.

    Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, Texas A&M University, 77843, College Station, TX, USA

    Willa W. Chen

  2. New York University, 44 W. 4’th, 10012, New York, NY, USA

    Clifford M. Hurvich

Authors
  1. Willa W. Chen
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  2. Clifford M. Hurvich
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Correspondence to Willa W. Chen .

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    Chen, W.W., Hurvich, C.M. (2009). Fractional Cointegration. In: Mikosch, T., Kreiß, JP., Davis, R., Andersen, T. (eds) Handbook of Financial Time Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71297-8_31

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