Spatial autoregressive fractionally integrated moving average model

In this paper, we introduce the concept of fractional integration for spatial autoregressive models. We show that the range of the dependence can be spatially extended or diminished by introducing a further fractional integration parameter to spatial autoregressive moving average models (SARMA). This new model is called the spatial autoregressive fractionally integrated moving average model, briefly sp-ARFIMA. We show the relation to time-series ARFIMA models and also to (higher-order) spatial autoregressive models. Moreover, an estimation procedure based on the maximum-likelihood principle is introduced and analysed in a series of simulation studies. Eventually, the use of the model is illustrated by an empirical example of atmospheric fine particles, so-called aerosol optical thickness, which is important in weather, climate and environmental science.