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Effect of temporal aggregation on multiple time series in the frequency domain


Correspondence to: Uwe Hassler, Goethe University Frankfurt, RuW Building, Grueneburgplatz 1, 60323 Frankfurt, Germany.



The effect of temporal aggregation on bivariate spectral measures is investigated. First, the low-frequency regression coefficient turns out to be invariant under aggregation irrespective of differencing, with the exception of when the aggregation of flow and stock variables is combined. Second, the long-run squared coherency is invariant with respect to aggregation irrespective of differencing. Third, for frequencies different from zero, limiting results for a growing aggregation level m are obtained equal to those at frequency 0 of the underlying basic series. Hence, all frequency domain information is distorted by aggregation apart from the long-run one. This also holds true for the phase angle that always approaches zero with growing aggregation level m. The sole exception to these findings is the case of the skip sampling stationary series. Moreover, for finite aggregation level, one may exactly quantify the aggregational effect on each cycle of interest. Numerical examples illustrate our results.