Estimation of Generalized Fractionally Differenced Processes with Conditionally Heteroskedastic Errors

An optimal estimation methodology based on state space modelling of long memory Gegenbauer processes driven by conditionally heteroskedastic errors is seemingly absent in the current literature. In lieu of it, this paper considers an approximation of long memory Gegenbauer processes driven by heteroskedastic errors using finite order moving average (MA) and autoregressive (AR) representations. A related state space form is used to estimate parameters by pseudo maximum likelihood and the Kalman filter. As a novel contribution, a comparative assessment of the suggested approximation techniques is performed using a large scale simulation study. It results in extensive Monte Carlo evidence that establish and validate the optimal order of the two approximations as interval estimates. The superiority of the created methodology is illustrated as an original contribution by extending it to non-stationary Gegenbauer processes to compare with similar existing estimation techniques and results in the literature. Finally, the approach is applied to the famous daily Standard and Poor (S and P) 500 series as a real application.