Seasonal Time Series and Autocorrelation Function Estimation

Time series are demeaned when sample autocorrelation functions are computed. By the same logic it would seem appealing to remove seasonal means from seasonal time series before computing sample autocorrelation functions. Yet, standard practice is only to remove the overall mean and ignore the possibility of seasonal mean shifts in the data. Whether or not time series are seasonally demeaned has very important consequences on the asymptotic behavior of autocorrelation functions (henceforth ACF). Hasza (1980) and Bierens (1993) studied the asymptotic properties of the sample ACF of non-seasonal integrated processes and showed how they depend on the demeaning of the data. In this paper we study the large sample behavior of the ACF when the data generating processes are seasonal with or without seasonal unit roots. The effect on the asymptotic distribution of seasonal mean shifts and their removal is investigated and the practical consequences of these theoretical developments are also discussed. We also examine the small sample behavior of ACF estimates through Monte Carlo simulations.
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