Abstract
Long memory in conditional variance is one of the empirical features exhibited by many financial time series. One class of models that was suggested to capture this behavior is the so-called Fractionally Integrated GARCH (Baillie, Bollerslev and Mikkelsen 1996) in which the ideas of fractional integration originally introduced by Granger (1980) and Hosking (1981) for processes for the mean are applied to a GARCH framework. In this paper we derive analytic expressions for the second-order derivatives of the log-likelihood function of FIGARCH processes with a view to the advantages that can be gained in computational speed and estimation accuracy. The comparison is computationally intensive given the typical sample size of the time series involved and the way the likelihood function is built. An illustration is provided on exchange rate and stock index data.
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References
Baillie R (1996) Long memory processes and fractional integration in econometrics. Journal of Econometrics73, 5–59
Baillie R, Bollerslev T, Mikkelsen H (1996) Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics74, 3–30
Beran J (1994) Statistics for long memory processes. Chapman and Hall, New York
Berndt E, Hall B, Hall, R, Hausman J (1974) Modelling the persistence of conditional variances. Annals of Economic and Social Measurement3, 653–665
Bollerslev T (1988) On the correlation structure for the GARCH process. Journal of Time Series Analysis9, 121–131
Caporin M (2001) Estimation and identification of FIGARCH, Università Ca' Foscari di Venezia, Venezia, I
Chung C-F (1999) Estimating the fractionally integrated GARCH model. National Taiwan University, Taipei, TW
Davidson J (2001) Moment and memory properties of linear conditional heteroscedasticity models. Cardiff University, Cardiff, UK
Davidson R, McKinnon J (1993) Estimation and inference in econometrics. Oxford University Press, Oxford
Ding Z-X, Granger C (1996) Modelling volatility persistence of speculative returns: a new approach. Journal of Econometrics73, 185–215
Ding Z-X, Granger C, Engle R (1993) A long memory property of stock market returns and a new approach. Journal of Empirical Finance1, 83–106
Efron B, and Hinkley D (1978) Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information. Biometrika65, 457–482
Engle R (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica50, 987–1008
Engle R, Bollerslev T (1986) Modelling the persistence of conditional variances. Econometric Reviews5, 1–50
Fiorentini G, Calzolari G, Panattoni L (1996) Analytic derivatives and the computation of GARCH estimates. Journal of Applied Econometrics11, 399–417
Granger C (1980) Long-memory relationships and the aggregation of dynamic models, Journal of Econometrics14, 228–238
Hamilton J (1994) Time series analysis. Princeton University Press, Princeton
Hosking J (1981) Fractional differencing. Biometrika68, 165–176
Li W, McLeod A (1986) Fractional time series modelling. Biometrika73, 217–221
Lombardi M (2001) Forecasting volatility of financial assets: Long memory processes and fractionally integrated GARCH. Master's thesis, Università di Firenze. In Italian
McCullough B, Renfro C (1998) Benchmarks and software standards: A case study of GARCH procedures. Journal of Economic and Social Measurement25, 59–71
Press W, Teukolsky S, Vetterling W, Flannery B (1992) Numerical recipes in C: The art of scientific computing (2nd edn). Cambridge University Press, Cambridge
Sowell F (1992) Maximum likelihood estimation of stationary univariate fractionally integrated time series models. Journal of Econometrics53, 165–188
Teyssière G (1996) Double long memory financial time series. QMW working paper 348 University of London
Thisted R (1988) Elements of statistical computing. Chapman Hall, New York
White H (1982) Maximum likelihood estimation of misspecified models. Econometrica50, 1–25
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Lombardi, M.J., Gallo, G.M. Analytic Hessian matrices and the computation of FIGARCH estimates. Statistical Methods & Applications 11, 247–264 (2002). https://doi.org/10.1007/BF02511490
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DOI: https://doi.org/10.1007/BF02511490