Long Memory, Fractional Integration, and Cross-Sectional Aggregation

It is commonly argued that observed long memory in time series variables can result from cross-sectional aggregation of dynamic heterogeneous micro units. For instance, Granger (1980) demonstrated that aggregation of AR(1) processes with a Beta distributed AR coefficient can exhibit long memory under certain conditions and that the aggregated series will have an autocorrelation function that exhibits hyperbolic decay. In this paper, we further analyze this phenomenon. We demonstrate that the aggregation argument leading to long memory is consistent with a wide range of definitions of long memory. In a simulation study we seek to quantify Granger's result and find that indeed both the time series and cross-sectional dimensions have to be rather significant to reflect the theoretical asymptotic results. Long memory can result even for moderate T,N dimensions but can vary considerably from the theoretical degree of memory. Also, Granger's result is most precise in samples with a relatively high degree of memory. Finally, we show that even though the aggregated process will behave as generalized fractional process and thus converge to a fractional Brownian motion asymptotically, the fractionally differenced series does not behave according to an ARMA process. In particular, despite the autocorrelation function is summable and hence the fractionally differenced process satisfy the conditions for being I(0), it still exhibits hyperbolic decay. This may have consequences for the validity of ARFIMA time series modeling of long memory processes when the source of memory is due to aggregation.