Long memory continuous time models. (English) Zbl 0856.62104

Summary: This paper presents a new family of long memory models: the continuous-time moving average fractional process. The continuous-time framework allows to reconcile two competitive types of modelling: fractional integration of ARMA processes and fractional Brownian Motion. A comparison with usual discrete-time ARFIMA models is made. Some well-known empirical evidence on macroeconomic and financial time series, such as variability of forward rates, aggregation of responses across heterogeneous agents, are well-captured by this continuous-time modelling. Moreover, the usual statistical tools for long memory series and for stochastic differential equations can be jointly applied in this setting.


62P20 Applications of statistics to economics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
91B84 Economic time series analysis
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[1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions. (1972) · Zbl 0543.33001
[2] Backus, D. K.; Zin, S. E.: Long-memory inflation uncertainty: evidence from the term structure of interest rates. Journal of money, credit and banking 25, 681-700 (1995)
[3] Baillie, R. T.; Bollerslev, T.; Mikkelsen, H. O. A.: Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of econometrics (1993) · Zbl 0865.62085
[4] Bergstrom, A. R.: Continuous time econometric modelling. Advanced text in econometrics (1990)
[5] Cheung, Y. W.; Lai, K. S.: A fractional cointegration analysis. Journal of business and economic statistics 11, 103-112 (1993)
[6] Comte, F.: Simulation and estimation of long memory continuous time models. Journal of time series analysis (1993) · Zbl 0836.62060
[7] Comte, F.; Renault, E.: Noncausality in continuous time VARMA models. Econometric theory (1992)
[8] Cox, J.; Ingersoll, J.; Ross, S.: A re-examination of traditional hypotheses about the term structure of interest rates. Journal of finance 36, 769-799 (1981)
[9] Dahlhaus, R.: Efficient parameter estimation for self-similar processes. Annals of statistics 17, 1749-1766 (1989) · Zbl 0703.62091
[10] Ding, Z.; Granger, C. W. J.; Engle, R. F.: A long memory property of stock market returns and a new model. Journal of empirical finance 1, 83-108 (1993)
[11] Fox, R.; Taqqu, M. S.: Large sample properties of parameter estimates for strongly dependent stationary time series. Annals of statistics 14, 517-532 (1986) · Zbl 0606.62096
[12] Geweke, J.; Porter-Hudak, S.: The estimation and application of long memory time series models. Journal of time series analysis 4, 221-238 (1983) · Zbl 0534.62062
[13] Goncalves, E.; Gourieroux, C.: Agrégation de processus autorégressifs d’ordre I. Annales d’economie et de statistiques 12, 127-149 (1988)
[14] Gourieroux, C.; Monfort, A.: Séries temporelles et modèles dynamiques. (1990)
[15] Granger, C. W. J.: Long memory relationships and the aggregation of dynamic models. Journal of econometrics 14, 227-238 (1980) · Zbl 0466.62108
[16] Granger, C. W. J.; Joyeux, R.: An introduction to long memory time series models and fractional differencing. Journal of time series analysis 1, 15-29 (1980) · Zbl 0503.62079
[17] Harrison, J.; Kreps, D.: Martingale and arbitrage in multiperiods securities markets. Journal of financial economic theory 20, 380-408 (1979) · Zbl 0431.90019
[18] Haubrich, J.; Lo, A.: The sources and nature of long-term memory in the business cycle. (1991)
[19] Hosking, J. M. R.: Fractional differencing. Biometrica 68, 165-176 (1981) · Zbl 0464.62088
[20] Kunsch, H.: Discrimination between monotonic trends and long range dependence. Journal of applied probabilities 23, 1025-1030 (1987) · Zbl 0623.62085
[21] Lawrance, A. J.; Kottegoda, N. T.: Stochastic modelling of riverflow times series. Journal of the royal statistical society A 140, 1-47 (1977)
[22] Lo, A. W.: Long term memory in stock market prices. Econometrica 59, 1279-1313 (1991) · Zbl 0781.90023
[23] Mandelbrot, B. B.; Ness, Van: Fractional Brownian motions, fractional noises and applications. SIAM review 10, 422-437 (1968) · Zbl 0179.47801
[24] Mandelbrot, B. B.: When can price be arbitraged efficiently? A limit to the validity of the random walk and martingale models. Review of economics and statistics 53, 225-236 (1971)
[25] Merton, R. C.: Continuous-time finance. (1990) · Zbl 1019.91502
[26] Protter, P.: Stochastic integration and differential equations. (1990) · Zbl 0694.60047
[27] Robinson, P. M.: Log-periodogram regression of time series with long range dependence. London school of economics invited session of the European congress of the econometric society (1992) · Zbl 0838.62085
[28] Rozanov, Yu.A.: Stationary random processes. (1968)
[29] Schwartz, L.: Théorie des distributions. (1966)
[30] Sims, C. A.: Martingale-like behavior of asset prices and interest rates. Discussion paper 205 (1984)
[31] Sowell, F.: Modeling long-run behavior with the fractional ARMA model. Journal of monetary economics 29, 277-302 (1992)
[32] Vasicek, O.: An equilibrium characterization of the term structure. Journal of financial economics 5, 177-188 (1977) · Zbl 1372.91113
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