Seasonal Nonstationarity and Near-Nonstationarity*
Eric Ghysels, Denise R. Osborn,
and Paulo M. M. Rodrigues*
Over the last three decades there has been an increasing interest in modeling seasonality. Progressing from the traditional view that the seasonal pattern is a nuisance which needed to be removed, it is now considered to be an informative feature of economic time series which should be modeled explicitly (see for instance Ghysels (1994) for a review).
Since the seminal work by Box and Jenkins (1970), the stochastic properties of seasonality have been a major focus of research. In particular, the recognition that the seasonal behavior of economic time series may be varying and changing over time due to the presence of seasonal unit roots (see for example Canova and Ghysels (1994), Hylleberg (1994), Hylleberg, J0rgensen, and S0rensen (1993) and Osborn (1990)), has led to the development of a considerable number of testing procedures (inter alia, Canova and Hansen (1995), Dickey, Hasza, and Fuller (1984), Franses (1994), Hylleberg, Engle, Granger, and Yoo (1990) and Osborn, Chui, Smith, and Birchenhall (1988)).
In this chapter, we review the properties of stochastic seasonal nonstationary processes, as well as the properties of several seasonal unit root tests. More specifically, in Section 2 we analyze the characteristics of the seasonal random walk and generalize our discussion for seasonally integrated ARMA (autoregressive moving average) processes. Furthermore, we also illustrate the implications that can emerge when nonstationary stochastic seasonality is posited as deterministic.
In Section 3 we consider the asymptotic properties of the seasonal unit root test procedures proposed by Dickey et al. (1984) and Hylleberg et al. (1990). Section 4 generalizes most of the results of Section 3 by considering the behavior of the test procedures in a near seasonally integrated framework. Finally, Section 5 concludes the chapter.
To devote a chapter on the narrow subject of seasonal nonstationarity deserves some explanation. Seasonal time series appear nonstationary, a feature shared by many economic data recorded at fixed time intervals. Whether we study so – called seasonal adjusted data or raw series, the question of seasonal unit roots looms behind the univariate models being used. The standard seasonal adjustment programs like X-11 and X-12/ARIMA involve removal of seasonal unit roots (for details of seasonal adjustment programs see for instance Findley et al.
(1998) or Ghysels and Osborn (2000)). Removing such roots may be unwarranted if they are not present and may cause statistical problems such as non-invertible MA roots (see Maravall, 1993, for further discussion). When unadjusted series are considered, then the question of seasonal unit roots is a basic issue of univariate time series model specification. Since we are particularly interested in how the asymptotic results for conventional unit root processes generalize to the seasonal context we will use the large sample distribution theory involving Brownian motion representations.