ARMA, ARIMA, ARFIMA, and GARMA Models with GARCH Errors

This chapter extends long memory time series models to incorporate conditional heteroskedasticity through GARCH error structures, addressing the volatility clustering commonly observed in financial time series. The GARMA(0,d,0)-GARCH(r,s) process is formally defined, combining the Gegenbauer polynomial filter for capturing long memory with GARCH dynamics for modeling time-varying volatility. Theoretical properties including stationarity conditions, causal representations, and autocorrelation structures are established through rigorous proofs. The state space approach is adapted to accommodate GARCH errors, with MA and AR approximations developed and their system matrices explicitly specified. The Kalman filter is integrated with GARCH estimation to obtain QMLE parameter estimates. Extensive Monte Carlo experiments compare the MA and AR approximations across stationary cases with different parameter configurations, establishing optimal lag order intervals ([6-9] for MA, [9-12] for AR) and demonstrating the superiority of the MA approach. The methodology is extended to nonstationary ARFIMA-GARCH models, with comparative analysis showing that the proposed QMLE approach outperforms traditional MLE methods. Empirical applications to the daily S&P 500 index (8,435 observations) and Chicago Board Options Exchange volatility index (2,744 observations) validate the model’s ability to capture both long memory persistence and conditional heteroskedasticity in real financial data.