Temporal aggregation, systematic sampling, and the Hodrick-Prescott filter

The time aggregation properties of the Hodrick-Prescott (HP) filter, which decomposes a time series into trend and cycle, are analyzed for the case of annual, quarterly, and monthly data. Aggregation of the disaggregate components cannot be obtained as the exact result from direct application of an HP filter to the aggregate series. Employing several criteria, HP decompositions for different levels of aggregation that provide similar results can be found. The aggregation is guided by the principle that the period associated with the frequency for which the filter gain is should not be altered. This criterion is intuitive and easy to apply. It is shown that it is approximated, to the first order, by an already proposed empirical rule and that alternative, more complex criteria yield similar results. Moreover, the values of the smoothing parameter of the HP filter that provide results which are approximately consistent under aggregation are considerably robust with respect to the ARIMA model of the series. Aggregation is found to perform better for the case of temporal aggregation than for systematic sampling. The desirability of exact aggregation consistency is investigated. A clarification concerning the supposed spuriousness of the cycles obtained by the HP filter is discussed.