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An Extension of Cointegration to Fractional Autoregressive Processes

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  • Søren Johansen

    (Department of Economics, University of Copenhagen)

Abstract

This paper contains an overview of some recent results on the statistical analysis of cofractional processes, see Johansen and Nielsen (2010b). We first give an brief summary of the analysis of cointegration in the vector autoregressive model and then show how this can be extended to fractional processes. The model allows the process X_{t} to be fractional of order d and cofractional of order d-b≥0; that is, there exist vectors β for which β′X_{t} is fractional of order d-b. We analyse the Gaussian likelihood function to derive estimators and test statistics. The asymptotic properties are derived without the Gaussian assumption, under suitable moment conditions. We assume that the initial values are bounded and show that they do not influence the asymptotic analysis. The estimator of β is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. The asymptotic distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion.

Suggested Citation

  • Søren Johansen, 2010. "An Extension of Cointegration to Fractional Autoregressive Processes," Discussion Papers 10-28, University of Copenhagen. Department of Economics.
    • Handle: RePEc:kud:kuiedp:1028
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    Cited by:

    1. Claudio Morana, 2014. "Factor Vector Autoregressive Estimation of Heteroskedastic Persistent and Non Persistent Processes Subject to Structural Breaks," Working Papers 273, University of Milano-Bicocca, Department of Economics, revised May 2014.
    2. Luis A. Gil-Alana & Antonio Moreno & Seonghoon Cho, 2012. "The Deaton paradox in a long memory context with structural breaks," Applied Economics, Taylor & Francis Journals, vol. 44(25), pages 3309-3322, September.
    3. Guglielmo Caporale & Luis Gil-Alana, 2014. "Fractional integration and cointegration in US financial time series data," Empirical Economics, Springer, vol. 47(4), pages 1389-1410, December.
    4. Daiki Maki, 2013. "Detecting cointegration relationships under nonlinear models: Monte Carlo analysis and some applications," Empirical Economics, Springer, vol. 45(1), pages 605-625, August.
    5. Stoyan V. Stoyanov & Yong Shin Kim & Svetlozar T. Rachev & Frank J. Fabozzi, 2017. "Option pricing for Informed Traders," Papers 1711.09445, arXiv.org.
    6. Guglielmo Maria Caporale & Luis Alberiko Gil-Alana & Robert Mudida, 2015. "Testing the Marshall–Lerner Condition in Kenya," South African Journal of Economics, Economic Society of South Africa, vol. 83(2), pages 253-268, June.

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    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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