Advancement of Fractionally Differenced Gegenbauer Processes with Long Memory

Abstract

The class of long memory time series models involving Gegenbauer processes is investigated in detail in terms of formulation, parameter estimation, prediction and testing. Corresponding truncated AR (autoregressive) and MA (moving average) approximations driven by Gaussian white noise are analysed through state space modelling and Kalman filtering to assess the viability of estimating techniques . The optimal approximation option is employed to proceed with the estimation of model parameters. The resulting mean square errors are validated by the predictive accuracy to establish an optimal lag order through a large scale simulation study. It is shown that the use of this newly established lag order for a real data application provides benchmarks which are comparable and mostly better than a number of existing results in the literature. It is followed by an execution of this technique to extract and assess seasonal models through a Monte Carlo experiment. Thereafter empirical applications were provided. The above approach has been extended to model fractionally differenced Gegenbauer processes with conditional heteroskedastic errors and models with seasonality. Potential applications are provided. In addition, quasi-likelihood type ratio tests have been developed for testing unit roots, stationarity versus non-stationarity and Gegenbauer long memory versus standard long memory.