Temporal Aggregation of Stationary And Nonstationary Discrete‐Time Processes

We study the autocorrelation structure and the spectral density function of aggregates from a discrete‐time process. The underlying discrete‐time process is assumed to be a stationary AutoRegressive Fractionally Integrated Moving‐Average (ARFIMA) process, after suitable number of differencing if necessary. We derive closed‐form expressions for the limiting autocorrelation function and the normalized spectral density of the aggregates, as the extent of aggregation increases to infinity. These results are then used to assess the loss of forecasting efficiency due to aggregation.