Seasonal Time Series and Autocorrelation Function Estimation

Abstract

Lorsqu'on calcule une fonction d'autocorrélation, il est normal d'enlever d'une série la moyenne non conditionnelle. Cette pratique s'applique également dans le cas des séries saisonnières. Pourtant, il serait plus logique d'utiliser des moyennes saisonnières. Hasza (1980) et Bierens (1993) ont étudié l'effet de la moyenne sur l'estimation d'une fonction d'autocorrélation pour un processus avec racine unitaire. Nous examinons le cas de processus avec racines unitaires saisonnières. Nos résultats théoriques de distribution asymptotique, de même que nos simulations de petits échantillons, démontrent l'importance d'enlever les moyennes saisonnières quand on veut identifier proprement les processus saisonniers.

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.