Standard Long Memory and State Space Modeling of ARFIMA Process with White Noise

This chapter develops the state space approach for estimating and forecasting ARFIMA models, building on the theoretical framework established by Chan and Palma (1998). The ARFIMA(p,d,q) process is formally defined with conditions for stationarity (d < 0.5), invertibility (d > -0.5), and long memory (d > 0). The chapter demonstrates how the infinite MA and AR representations of ARFIMA processes can be truncated and cast into finite-dimensional state space form, enabling practical implementation of the Kalman filter. Two approximation approaches are developed: the MA(m) approximation based on the Wold representation and the AR(m) approximation based on the inverse polynomial expansion. The system matrices (Z, T, H) for each representation are explicitly constructed. The Kalman filter recursions are presented for computing the log-likelihood function, and quasi-maximum likelihood estimation (QMLE) methodology is developed. Monte Carlo simulation results demonstrate the consistency and efficiency of the state space estimators, with tables showing parameter estimates across different sample sizes. Empirical applications to Nile River flow data (important for African agricultural economics) and Australian Consumer Price Index data illustrate the practical utility of the methodology, confirming the presence of standard long memory in these benchmark time series.