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Quantile inference for near-integrated autoregressive time series under infinite variance and strong dependence

  • Chan, Ngai Hang
  • Zhang, Rong-Mao
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    Consider a near-integrated time series driven by a heavy-tailed and long-memory noise , where {[eta]j} is a sequence of i.i.d random variables belonging to the domain of attraction of a stable law with index [alpha]. The limit distribution of the quantile estimate and the semi-parametric estimate of the autoregressive parameters with long- and short-range dependent innovations are established in this paper. Under certain regularity conditions, it is shown that when the noise is short-memory, the quantile estimate converges weakly to a mixture of a Gaussian process and a stable Ornstein-Uhlenbeck (O-U) process while the semi-parametric estimate converges weakly to a normal distribution. But when the noise is long-memory, the limit distribution of the quantile estimate becomes substantially different. Depending on the range of the stable index [alpha], the limit distribution is shown to be either a functional of a fractional stable O-U process or a mixture of a stable process and a stable O-U process. These results indicate that although the quantile estimate tends to be more efficient for infinite variance time series, extreme caution should be exercised in the long-memory situation.

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    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 119 (2009)
    Issue (Month): 12 (December)
    Pages: 4124-4148

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    Handle: RePEc:eee:spapps:v:119:y:2009:i:12:p:4124-4148
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    1. repec:cup:cbooks:9780521608275 is not listed on IDEAS
    2. Davis, Richard A. & Knight, Keith & Liu, Jian, 1992. "M-estimation for autoregressions with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 145-180, February.
    3. Peter C.B. Phillips, 1989. "Time Series Regression with a Unit Root and Infinite Variance Errors," Cowles Foundation Discussion Papers 897R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1989.
    4. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
    5. Benoit Mandelbrot, 1967. "The Variation of Some Other Speculative Prices," The Journal of Business, University of Chicago Press, vol. 40, pages 393.
    6. Knight, Keith, 1991. "Limit Theory for M-Estimates in an Integrated Infinite Variance," Econometric Theory, Cambridge University Press, vol. 7(02), pages 200-212, June.
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    8. Resnick, Sidney & Greenwood, Priscilla, 1979. "A bivariate stable characterization and domains of attraction," Journal of Multivariate Analysis, Elsevier, vol. 9(2), pages 206-221, June.
    9. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
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