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Fractionally integrated generalized autoregressive conditional heteroskedasticity. (English) Zbl 0865.62085
Summary: The new class of fractionally integrated generalized autoregressive conditionally heteroskedastic (FIGARCH) processes is introduced. The conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations. Unlike I(\(d\)) processes for the mean, maximum likelihood estimates (MLE) of the FIGARCH parameters are argued to be \(T^{1/2}\)-consistent. The small sample behavior of an approximate MLE procedure is assessed through a simulation study, which also documents how the estimation of a standard GARCH model tends to produce integrated, or IGARCH, like estimates. An empirical example with daily Deutschmark-U.S. dollar exchange rates illustrates the practical relevance of the new FIGARCH specification.

MSC:
62P20 Applications of statistics to economics
91B84 Economic time series analysis
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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[1] Adenstedt, R.: On large-sample estimation for the mean of a stationary random sequence. Annals of statistics 2, 1095-1107 (1974) · Zbl 0296.62081
[2] Andersen, T. G.: Stochastic autoregressive volatility: A framework for volatility modeling. Mathematical finance 4, 75-102 (1994) · Zbl 0884.90013
[3] Andersen, T. G.; Bollerslev, T.: Intraday seasonality and volatility persistence in financial markets. Journal of empirical finance (1996)
[4] Baillie, R. T.: Long memory processes and fractional integration in econometrics. Journal of econometrics (1996) · Zbl 0854.62099
[5] Baillie, R. T.; Bollerslev, T.: The message in daily exchange rates: A conditional variance tale. Journal of business and economic statistics 7, 297-305 (1989)
[6] Baillie, R. T.; Bollerslev, T.: Intra day and inter market volatility in foreign exchange rates. Review of economic studies 58, 565-585 (1991)
[7] Baillie, R. T.; Bollerslev, T.: The long-memory of the forward premium. Journal of international money and finance 13, 565-571 (1994)
[8] Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. Journal of econometrics 31, 307-327 (1986) · Zbl 0616.62119
[9] Bollerslev, T.: A conditional heteroskedastic time series model for speculative prices and rates of return. Review of economics and statistics 69, 542-547 (1987)
[10] Bollerslev, T.; Engle, R. F.: Common persistence in conditional variances. Econometrica 61, 167-186 (1993) · Zbl 0782.62102
[11] Bollerslev, T.; Mikkelsen, H. O.: Modeling and pricing long-memory in stock market volatility. Journal of econometrics (1996) · Zbl 0960.62560
[12] Bollerslev, T.; Wooldridge, J. M.: Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric reviews 11, 143-172 (1992) · Zbl 0850.62884
[13] Bollerslev, T.; Chou, R. Y.; Kroner, K. F.: ARCH modeling in finance: A review of the theory and empirical evidence. Journal of econometrics 52, 5-59 (1992) · Zbl 0825.90057
[14] Bollerslev, T.; Engle, R. F.; Nelson, D. B.: ARCH models. Handbook of econometrics 4 (1994)
[15] Bougerol, P.; Picard, N.: Stationarity of GARCH processes and of some nonnegative time series. Journal of econometrics 52, 115-128 (1992) · Zbl 0746.62087
[16] Cai, J.: A Markov model of unconditional variance in ARCH. Journal of business and economic statistics 12, 309-316 (1994)
[17] Cheung, Y. W.: Long memory in foreign exchange rates. Journal of business and economic statistics 11, 93-101 (1993)
[18] Cheung, Y. W.; Diebold, F. X.: On maximum likelihood estimation of the degree of fractional integration when the mean is unknown. Journal of econometrics 62, 301-316 (1994)
[19] Chung, C. F.; Baillie, R. T.: Small sample bias in conditional sum of squares estimators of fractionally integrated ARMA models. Empirical economics 18, 791-806 (1993)
[20] Dacorogna, M. M.; Mller, U. A.; Nagler, R. J.; Olsen, R. B.; Pichet, O. V.: A geographical model for the daily and weekly seasonal volatility in the foreign exchange market. Journal of international money and finance 12, 413-438 (1993)
[21] Dahlhaus, R.: Efficient parameter estimation for self-similar processes. Annals of statistics 17, 1749-1766 (1989) · Zbl 0703.62091
[22] Delima, P.; Breidt, F. J.; Crato, N.: Modeling long-memory stochastic volatility. (1994)
[23] Diebold, F. X.; Rudebusch, G. D.: On the power of Dickey-fuller tests against fractional alternatives. Economics letters 35, 155-160 (1991)
[24] Diebold, F. X.; Schuermann, T.: Exact maximum likelihood estimation of observation driven econometric models. Simulation based inference in econometrics: methods and applications (1996) · Zbl 1184.62205
[25] Diebold, F. X.; Husted, S.; Rush, M.: Real exchange rates under the gold standard. Journal of political economy 99, 1252-1271 (1991)
[26] 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, 83-106 (1993)
[27] Drost, F. C.; Nijman, T.: Temporal aggregation of GARCH processes. Econometrica 61, 909-927 (1993) · Zbl 0780.62099
[28] Engle, R. F.: Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica 50, 987-1008 (1982) · Zbl 0491.62099
[29] Engle, R. F.; Bollerslev, T.: Modelling the persistence of conditional variances. Econometric reviews 5, 1-50 (1986) · Zbl 0619.62105
[30] Engle, R. F.; Lee, G. G. J.: A permanent and transitory component model of stock return volatility. (1993)
[31] Engle, R. F.; Mustafa, C.: Implied ARCH models from options prices. Journal of econometrics 52, 289-311 (1992)
[32] Gallant, A. R.; Rossi, P. E.; Tauchen, G.: Nonlinear dynamic structures. Econometrica 61, 871-907 (1993) · Zbl 0780.62100
[33] Granger, C. W. J.: Long memory relationships and the aggregation of dynamic models. Journal of econometrics 14, 227-238 (1980) · Zbl 0466.62108
[34] Granger, C. W. J.: Some properties of time series data and their use in econometric model specification. Journal of econometrics 16, 121-130 (1981)
[35] Granger, C. W. J.; Joyeux, R.: An introduction to long memory time series models and fractional differencing. Journal of time series analysis 1, 15-39 (1980) · Zbl 0503.62079
[36] Hamilton, J. D.; Susmel, R.: Autoregressive conditinal heteroskedasticity and changes in regime. Journal of econometrics 64, 307-333 (1994) · Zbl 0825.62950
[37] Harvey, A. C.: Long memory in stochastic volatility. (1993)
[38] Harvey, A. C.; Shephard, N.: Estimation and testing of stochastic variance models. (1993)
[39] Harvey, A. C.; Ruiz, E.; Shephard, N.: Multivariate stochastic variance models. Review of economic studies 61, 247-264 (1994) · Zbl 0805.90026
[40] Haubrich, J. G.; Lo, A. W.: The sources and nature of long-term dependence in the business cycle. (1992)
[41] Hodrick, R. J.: The empirical evidence on the efficiency of forward and futures foreign exchange markets. (1987)
[42] Hosking, J. R. M.: Fractional differencing. Biometrika 68, 165-176 (1981) · Zbl 0464.62088
[43] Hsieh, D. A.: Modeling heteroskedasticity in daily foreign exchange rates. Journal of business and economic statistics 7, 307-317 (1989)
[44] Hull, J.; White, A.: The pricing of options on assets with stochastic volatilities. Journal of finance 42, 381-400 (1987)
[45] Hurst, H.: Long term storage capacity of reservoirs. Transactions of the American society of civil engineers 116, 770-799 (1951)
[46] Jacquier, E.; Polson, N. G.; Rossi, P. E.: A Bayesian analysis of stochastic volatility models. Journal of business and economic statistics 12, 371-389 (1994)
[47] Lamoureux, C. G.; Lastrapes, W. D.: Persistence in variance, structural change and the GARCH model. Journal of business and economic statistics 8, 225-234 (1990)
[48] Lee, S. W.; Hansen, B. E.: Asymptotic theory for the \(GARCH(1,1)\) quasi-maximum likelihood estimator. Econometric theory 10, 29-52 (1994)
[49] Ljung, G. M.; Box, G. E. P.: On a measure of lack of fit in time series models. Biometrika 65, 297-303 (1978) · Zbl 0386.62079
[50] Lo, A. W.: Long term memory in stock market prices. Econometrica 59, 1279-1313 (1991) · Zbl 0781.90023
[51] Lumsdaine, R. L.: Finite sample properties of the maximum likelihood estimator in \(GARCH(1,1)\) and \(IGARCH(1,1)\) models: A Monte Carlo investigation. Journal of business and economic statistics 13, 1-10 (1995)
[52] Lumsdaine, R. L.: Consistency and asymptotic normality of the quasi maximum likelihood estimator in IGARCH (1,1) and covariance stationary GARCH (1,1) models. Econometrica (1996) · Zbl 0844.62080
[53] Mandelbrot, B. B.: The variation of certain speculative prices. Journal of business 36, 394-419 (1963)
[54] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications. SIAM review 10, 422-437 (1968) · Zbl 0179.47801
[55] Mccurdy, T.; Morgan, I.: Testing the martingale hypothesis in deutschmark futures with models specifying the form of heteroskedasticity. Journal of applied econometrics 3, 187-202 (1988)
[56] Nelson, D. B.: Stationarity and persistence in the \(GARCH(1,1)\) model. Econometric theory 6, 318-334 (1990)
[57] Nelson, D. B.: ARCH models as diffusion approximations. Journal of econometrics 45, 7-38 (1990) · Zbl 0719.60089
[58] Nelson, D. B.: Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347-370 (1991) · Zbl 0722.62069
[59] Nelson, D. B.: Filtering and forecasting with misspecified ARCH models I: Getting the right variance with the wrong model. Journal of econometrics 52, 61-90 (1992) · Zbl 0761.62169
[60] Nelson, D. B.; Cao, C. Q.: Inequality constraints in the univariate GARCH models. Journal of business and economic statistics 10, 229-235 (1992)
[61] Nelson, D. B.; Foster, D. P.: Asymptotic filtering theory for univariate ARCH models. Econometrica 62, 1-41 (1994) · Zbl 0804.62085
[62] Nijman, T.; Sentana, E.: Marginalization and contemporaneous aggregation in multivariate GARCH processes. Journal of econometrics (1996) · Zbl 0843.62104
[63] Perron, P.: The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361-1401 (1989) · Zbl 0683.62066
[64] Robinson, P. M.: Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of econometrics 47, 67-84 (1991) · Zbl 0734.62070
[65] Robinson, P. M.: Semiparametric analysis of long-memory time series. Annals of statistics 22, 515-539 (1994) · Zbl 0795.62082
[66] Robinson, P. M.: Time series with strong dependence. Advances in econometrics – sixth world congress of the econometric society (1994)
[67] Schwert, G. W.: Stock volatility and the crash of 87. Review of financial studies 3, 77-102 (1990)
[68] Shiller, R. J.; Perron, P.: Testing the random walk hypothesis: power versus frequency of observations. Economic letters 18, 381-386 (1985) · Zbl 1273.91383
[69] Silverman, B. W.: Density estimation for statistics and data analysis. (1986) · Zbl 0617.62042
[70] Sowell, F. B.: Fractional unit root distribution. Econometrica 58, 495-506 (1990) · Zbl 0727.62025
[71] Sowell, F. B.: Maximum likelihood estimation of stationary univariate fractionally integrated time series models. Journal of econometrics 53, 165-188 (1992)
[72] Sowell, F. B.: Modeling long-run behavior with the fractional ARIMA model. Journal of monetary economics 29, 277-302 (1992)
[73] Taylor, S.: Modelling financial time series. (1986) · Zbl 1130.91345
[74] Taylor, S.: Modelling stochastic volatility. Mathematical finance 4, 183-204 (1994) · Zbl 0884.90054
[75] Tschernig, R.: Long memory in foreign exchange rates revisited. Journal of international markets, institutions, and money 5, 53-78 (1995)
[76] Weiss, A. A.: Asymptotic theory for ARCH models: estimation and testing. Econometric theory 2, 107-131 (1986)
[77] Yajima, Y.: On estimation of a regression model with long-memory stationary errors. Annals of statistics 16, 791-807 (1988) · Zbl 0661.62090
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