Aggregation and long-memory: An analysis based on the discrete Fourier transform

- We derive dft of aggregated series using aliasing effect.
- We analyze spectrum of stationary series and periodogram of non-stationary series.
- Same bandwidths yield same long-memory estimate for aggregated and original series.
- The theoretical findings are illustrated via the analysis of S&P 500 volatility.

Datasets constructed via temporal aggregation or skip sampling are widely used by empirical studies in economics and finance, which leads to substantive discussion and debates on the effects of temporal aggregation and choice of sampling frequency. This paper studies a key feature of data aggregation by deriving the representation of the discrete Fourier transform (dft) of the aggregated series considering the aliasing effect. Analyses are not limited to the spectrum of the stationary series under aggregation, but extended to the periodogram of the non-stationary series. We further apply our results of the dft to a particular example of fractional processes under aggregation. We show that the estimates of the long-memory parameter are the same for the temporally aggregated series and the original one if the same bandwidths are used, regardless of the stationarity of the series. The theoretical findings are empirically verified by the analysis of S&P 500 volatility from 1928 to 2011.