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Alternative Frequency And Time Domain Versions Of Fractional Brownian Motion

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  • Davidson, James
  • Hashimzade, Nigar

Abstract

This paper compares models of fractional processes and associated weak convergence results based on moving average representations in the time domain with spectral representations. Both approaches have been applied in the literature on fractional processes. We point out that the conventional forms of these models are not equivalent, as is commonly assumed, even under a Gaussianity assumption. We show that it is necessary to distinguish between two-sided processes depending on both leads and lags from one-sided or causal processes, because in the case of fractional processes these models yield different limiting properties. We derive new representations of fractional Brownian motion and show how different results are obtained for, in particular, the distribution of stochastic integrals in the multivariate context. Our results have implications for valid statistical inference in fractional integration and cointegration models.We thank F. Hashimzade and two anonymous referees for their valuable comments.

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Bibliographic Info

Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 24 (2008)
Issue (Month): 01 (February)
Pages: 256-293

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Handle: RePEc:cup:etheor:v:24:y:2008:i:01:p:256-293_08

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Cited by:
  1. C.S. Bos & S.J. Koopman & M. Ooms, 2007. "Long Memory Modelling of Inflation with Stochastic Variance and Structural Breaks," Tinbergen Institute Discussion Papers 07-099/4, Tinbergen Institute.
  2. Davidson, James & Hashimzade, Nigar, 2009. "Representation And Weak Convergence Of Stochastic Integrals With Fractional Integrator Processes," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 25(06), pages 1589-1624, December.
  3. Guillaume Chevillon & Sophocles Mavroeidis, 2013. "Learning generates Long Memory," Post-Print hal-00661012, HAL.
  4. Peter M Robinson, 2007. "Multiple Local Whittle Estimation in StationarySystems," STICERD - Econometrics Paper Series, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE /2007/525, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  5. Bos, Charles S. & Koopman, Siem Jan & Ooms, Marius, 2014. "Long memory with stochastic variance model: A recursive analysis for US inflation," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 76(C), pages 144-157.
  6. Lavancier, Frédéric & Philippe, Anne & Surgailis, Donatas, 2009. "Covariance function of vector self-similar processes," Statistics & Probability Letters, Elsevier, Elsevier, vol. 79(23), pages 2415-2421, December.
  7. James Davidson & Nigar Hashimzade, 2008. "Type I and Type II Fractional Brownian Motions: a Reconsideration," Discussion Papers, Exeter University, Department of Economics 0816, Exeter University, Department of Economics.

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