More general processes
To generalize the above discussion, weakly stationary autocorrelations can be permitted in the SI(1) process. That is, (31.1) can be generalized to the seasonally integrated ARMA process:
^(L)Asyt = 0(L)et, t = 1, 2,, T, (31.7)
where, as before, et ~ iid(0, о2), while the polynomials ф(L) and 0(L) in the lag operator L have all roots outside the unit circle. It is, of course, permissible that these polynomials take the multiplicative form of the seasonal ARMA model of Box and Jenkins (1970). Inverting the stationary autoregressive polynomial and defining zt = ф^)-10^)et, we can write (31.7) as:
AsVt = Zt, t = 1,…, T. (31.8)
The process superficially looks like the seasonal random walk, namely (31.1). There is, however, a crucial difference in that zt here is a stationary, invertible ARMA process. Nevertheless, performing the same substitution for lagged yt as above leads to the corresponding result, which can be written as
ySn = Vs, o + X Zsj s = 1,…, s and n = 1,…, N. (31.9)
As in (31.6), (31.9) implies that there are S distinct unit root processes, one corresponding to each of the seasons. The important distinction is that these processes in (31.9) may be autocorrelated and cross-correlated. Nevertheless, it is only the stationary components which are correlated.
Defining the observation and (weakly stationary) disturbance vectors for year n as Yn = (y1n,…, ySn)’ and Zn = (z1n,…, zSn)’ respectively, the vector representation of (31.9) is:
AYn = Zn, n = 1,…, N. (31.10)
The disturbances here follow a stationary vector ARMA process
Ф^ = 0(L)En. (31.11)
It is sufficient to note that Ф^) and 0(L) are appropriately defined S x S polynomial matrices in L with all roots outside the unit circle and En = (e1n,…, eSn)’. The seasonal difference of (31.7) is converted to a first difference in (31.10) because AYn = Yn – Yn-1 defines an annual (that is, seasonal) difference of the vector Yt. Now, in (31.10) we have a vector ARMA process in AYn, which is a vector ARIMA process in Yn. In the terminology of Engle and Granger (1987), the S processes in the vector Yt cannot be cointegrated if this is the data generating process (DGP). Expressed in a slightly different way, if the process is written in terms of the level Yn, the vector process will contain S unit roots due to the presence of the factor A = 1 – L in each of the equations. Therefore, the implication drawn from the seasonal random walk of (31.1) that any linear combination of the separate seasonal series is itself an I(1) process carries over to this case too.
For the purpose of this chapter, only the simple seasonal random walk case will be considered in the subsequent analysis. It should, however, be recognized that the key results extend to more general seasonally integrated processes.