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Nonlinearity and Temporal Dependence

Nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models. We study this link using three measures of temporal dependence: rho-mixing, beta-mixing and alpha-mixing. Stationary diffusions that are rho-mixing have mixing coefficients that decay exponentially to zero. When they fail to be rho-mixing, they are still beta-mixing and alpha-mixing; but coefficient decay is slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state-dependent, Poisson sampling alters the temporal dependence.

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File URL: http://cowles.econ.yale.edu/P/cd/d16b/d1652-r.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1652R.

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Length: 40 pages
Date of creation: Oct 2009
Date of revision:
Handle: RePEc:cwl:cwldpp:1652r
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Web page: http://cowles.econ.yale.edu/

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

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  1. Conley, Timothy G, et al, 1997. "Short-Term Interest Rates as Subordinated Diffusions," Review of Financial Studies, Society for Financial Studies, vol. 10(3), pages 525-77.
  2. Hansen, Lars Peter & Alexandre Scheinkman, Jose & Touzi, Nizar, 1998. "Spectral methods for identifying scalar diffusions," Journal of Econometrics, Elsevier, vol. 86(1), pages 1-32, June.
  3. Xiaohong Chen & Lars Peter Hansen & Jose A. Scheinkman, 2009. "Principal components and the long run," CeMMAP working papers CWP07/09, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  4. Hidalgo, Javier & Robinson, Peter M., 1996. "Testing for structural change in a long-memory environment," Journal of Econometrics, Elsevier, vol. 70(1), pages 159-174, January.
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  6. Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," Econometrica, Econometric Society, vol. 63(4), pages 767-804, July.
  7. Granger, Clive W. J. & Terasvirta, Timo, 1999. "A simple nonlinear time series model with misleading linear properties," Economics Letters, Elsevier, vol. 62(2), pages 161-165, February.
  8. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
  9. Veretennikov, A. Yu., 1997. "On polynomial mixing bounds for stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 115-127, October.
  10. Xiaohong Chen & Lars Peter Hansen & Jose Scheinkman, 2009. "Principal Components and Long Run Implications of Multivariate Diffusions," Cowles Foundation Discussion Papers 1694, Cowles Foundation for Research in Economics, Yale University.
  11. Harrison, J Michael & Pitbladdo, Richard & Schaefer, Stephen M, 1984. "Continuous Price Processes in Frictionless Markets Have Infinite Variation," The Journal of Business, University of Chicago Press, vol. 57(3), pages 353-65, July.
  12. Andersen T. G & Bollerslev T. & Diebold F. X & Labys P., 2001. "The Distribution of Realized Exchange Rate Volatility," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 42-55, March.
  13. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous-time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323.
  14. Comte, F. & Renault, E., 1996. "Long memory continuous time models," Journal of Econometrics, Elsevier, vol. 73(1), pages 101-149, July.
  15. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
  16. Diebold, Francis X. & Inoue, Atsushi, 2001. "Long memory and regime switching," Journal of Econometrics, Elsevier, vol. 105(1), pages 131-159, November.
  17. Darrell Duffie & Peter Glynn, 2004. "Estimation of Continuous-Time Markov Processes Sampled at Random Time Intervals," Econometrica, Econometric Society, vol. 72(6), pages 1773-1808, November.
  18. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-55, January.
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