Properties of Seasonal Unit Root Processes
The case of primary interest in the context of seasonal unit roots occurs when the process yt is nonstationary and annual differencing is required to induce stationarity. This is often referred to as seasonal integration. More formally:
Definition 1. The nonstationary stochastic process yt, observed at § equally spaced time intervals per year, is said to be seasonally integrated of order d, denoted yt ~ SI(d), if Ad§yt = (1 – L§)dyt is a stationary, invertible ARMA process.
Therefore, if first order annual differencing renders yt a stationary and invertible process, then yt ~ SI(1). The simplest case of such a process is the seasonal random walk, which will be the focus of analysis throughout most of this chapter. We refer to § as the number of seasons per year for yt.
The seasonal random walk is a seasonal autoregressive process of order 1, or SAR(1), such that
yt = yt-§ + £f, t = 1, 2,…, T (31.1)
with et ~ iid(0, о2). Denoting the season in which observation t falls as st, with st = 1 + (t – 1) mod §, backward substitution for lagged yt in this process implies that
Vt = VstS + X£t-S j, (31-2)
where nt = 1 + [(t – 1)/S] and [■] represents the greatest integer less or equal to (t – 1)/S. As noted by Dickey et al. (1984) and emphasized by Osborn (1993), the random walk in this case is defined in terms of the disturbances for the specific season st only, with the summation over the current disturbance et and the disturbance for this season in the nt – 1 previous years of the observation period. The term Vs, s = Vt-n, s, refers to the appropriate starting value for the process. Equation (31.1) is, of course, a generalization of the conventional nonseasonal random walk. Note that the unconditional mean of yt from (31.2) is
E( у) = E( Vs(-s). (31.3)
Thus, although the process (31.1) does not explicitly contain deterministic seasonal effects, these are implicitly included when E( ySt-s) is nonzero and varies over st = 1,…, S. In their analysis of seasonal unit roots, Dickey et al. (1984) separate the Vt corresponding to each of the S seasons into distinct series. Notationally, this is conveniently achieved using two subscripts, the first referring to the season and the second to the year. Then
Vt = Vs+S(n-1) = Vsn, (31.4)
where st and nt are here written as s and n for simplicity of notation. Correspondingly S disturbance series can be defined as
et + s(nt-1) Cn. (31.5)
Using these definitions, and assuming that observations are available for precisely N (N = T/S) complete years, then (31.1) can be written as
Vsn = Vs, o + Xеj s = ^..^ S and n = ^..^ N (31.6)
which simply defines a random walk for each season s = 1,…, S. Because the disturbances e t of (31.1) are uncorrelated, the random walks defined by (31.6) for the S seasons of the year are also uncorrelated. Thus, any linear combination of these processes can itself be represented as a random walk. The accumulation of disturbances allows the differences to wander far from the mean over time, giving rise to the phenomenon that "summer may become winter."