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. In this paper we demonstrate that the aggregation argument is consistent with a range of different long memory definitions. A simulation study shows that the cross-section dimension needs to be rather large to reflect the theoretical memory when using commonly used methods to estimate the memory parameter, especially when the theoretical memory is not too high. We show that the aggregated process will converge to a generalized fractional process in the limit. The coefficients of the moving average representation of the series decay hyperbolically but they differ from the coefficients arising from inversion of the fractional difference filter. It appears that the fractionally differenced series will have an autocorrelation function that still exhibits hyperbolic decay, but at a rate that ensures summability. The fractionally differenced series is thus I(0) but standard ARFIMA modeling is invalid when the long memory is caused by aggregation. It is shown that standard methods for estimating and selecting ARFIMA specifications fail in properly fitting the dynamics of the series.