doi:10.1016/j.csda.2006.11.004 
Copyright © 2006 Elsevier B.V. All rights reserved.
Generalised long-memory GARCH models for intra-daily volatility
Silvano Bordignona, Massimiliano Caporinb and Francesco Lisia, 
, 
aDepartment of Statistical Sciences, University of Padova, via Cesare Battisti, 241, 35122 Padova, Italy
bDepartment of Economics, University of Padova, Italy
Received 7 February 2006;
revised 27 October 2006;
accepted 6 November 2006.
Available online 28 November 2006.
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Abstract
The class of fractionally integrated generalised autoregressive conditional heteroskedastic (FIGARCH) models is extended for modelling the periodic long-range dependence typically shown by volatility of most intra-daily financial returns. The proposed class of models introduces generalised periodic long-memory filters, based on Gegenbauer polynomials, into the equation describing the time-varying volatility of standard GARCH models. A fitting procedure is illustrated and its performance is evaluated by means of Monte Carlo simulations. The effectiveness of these models in describing periodic long-memory volatility patterns is shown through an empirical application to the Euro–Dollar intra-daily exchange rate.
Keywords: Long-memory; Intra-daily volatility; G-GARCH; Gegenbauer processes
Fig. 1. Autocorrelation function and periodogram of 
for hourly data.
Fig. 2. Periodogram of 
for hourly data.
Fig. 3. Periodogram of deseasonalised 
. The Y-axis is truncated at 30: actually the peak at zero frequency is about 100.
Fig. 4. Periodogram of standardised squared residuals, 
, of the FIGARCH model with seasonal dummies.
Fig. 5. Autocorrelation function and periodogram of log-squared residuals of Log-G-GARCH model for hourly data.
Fig. 6. Periodograms of log-squared returns for 4-hourly (left) and half-hourly (right) data.
Fig. 7. Periodograms of deseasonalised 
for 4-hourly (left) and half-hourly (right) data. Y-axes are truncated at 30: actually, the peaks at zero frequency are about 40 for 4-hourly and about 200 for half-hourly data.
Table 1.
Finite sample properties of QML estimators for Log-G-GARCH: n=1500, M=1000, S1=5
Estimated bias, Monte Carlo root mean squared error (RMSE) and average of standard errors based on finite sample approximation 10 for 
over 1000 Monte Carlo simulations.
Table 2.
Finite sample properties of QML estimators for Log-G-GARCH
True parameters are γ=-0.5, d0=0.1, d1=0.2, d2=0.3. Lengths of series are n=1500,1000,500,300,150, M=1000, S1=5. Bias, RMSE and Std.Err. as in Table 1.
Table 3.
Finite sample properties of QML estimators for Log-G-GARCH
n=1500, M=1000 and variable S, ratios n/S are 500, 300, 414 and 166 corresponding to S=3,5,7,9. Bias, RMSE and Std.Err. as in Table 1.
Table 4.
Simulation results for Log-G-GARCH
n=1500, M=1000. Model parameters: first (from top) simulation: γ=-0.05, d0=0.2, d1=0.2, d2=0.3, d3=0.3, d4=0.3; second simulation: γ=-0.05, d0=0.4, d1=0.1, d2=0.3, d3=0.1, d4=0.1; third simulation: γ=-0.05, d0=0.2, d1=0.2, d2=0.2, d3=0.4, d4=0.2; fourth simulation: γ=-0.05, d0=0.4, d1=0.3, d2=0.1, d3=0.2, d4=0.1. Bias, RMSE and Std.Err. as in Table 1.
Table 5.
Log-G-GARCH model: estimated parameters for hourly data
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