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  • James Davidson (a1) and Nigar Hashimzade (a1)

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.

Corresponding author
Address correspondence to James Davidson, School of Business and Economics, University of Exeter, Exeter EX4 4PU, U.K.; e-mail:
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Davidson, J. & R.M. de Jong (2000) The functional central limit theorem and convergence to stochastic integrals, part II: Fractionally integrated processes. Econometric Theory 16, 643666.

Duncan, T.E., Y. Hu, & B. Pasik-Duncan (2000) Stochastic calculus for fractional Brownian motion, part I: Theory. SIAM Journal of Control Optimization 38, 582612.

Flandrin, P. (1989) On the spectrum of fractional Brownian motions. IEEE Transactions on Information Theory 35, 197199.

Granger, C.W.J. & R. Joyeux (1980) An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis 1, 1529.

Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68, 165176.

Mandelbrot, B.B. & J.W. van Ness (1968) Fractional Brownian motions, fractional noises and applications. SIAM Review 10, 422437.

Marinucci, D. & P.M. Robinson (1999) Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 111122.

Marinucci, D. & P.M. Robinson (2000) Weak convergence of multivariate fractional processes. Stochastic Processes and Their Applications 86, 103120.

Marinucci, D. & P.M. Robinson (2001) Semiparametric fractional cointegration analysis. Journal of Econometrics 105, 225247.

Reed, I.S., P.C. Lee, & T.K. Truong (1995) Spectral representation of fractional Brownian motion in n dimensions and its properties. IEEE Transactions on Information Theory 41, 14391451.

Robinson, P.M. (1994b) Semiparametric analysis of long memory time series. Annals of Statistics 22, 515539.

Taqqu, M.S. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte Gebiete 31, 287302.

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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
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