# Higher order cointegrated systems

The statistical theory of I(d) systems with d = 2, 3,…, is much less developed than the theory for the I(1) model, partly because it is uncommon to find time series, at least in economics, whose degree of integration higher than two, partly because the theory is quite involved as it must deal with possibly multicointegrated cases where, for instance, linear combinations of levels and first differences can achieve stationarity. We refer the reader to Haldrup (1999) for a survey of the statistical treatment of I(2) models, restricting the discussion in this chapter to the basics of the CI(2, 2) case.

Assuming, thus, that yt ~ CI(2, 2), with Wold representation given by

(1 – L)2 y = C(L)e f, (30.23)

then, by means of a Taylor expansion, we can write C(L) as C(L) = C(1) – C *(1)(1 – L) + C(L)(1 – L)2,

with С*(1) being the first derivative of C(L) with respect to L, evaluated at L = 1. Following the arguments in the previous section, yt ~ CI(2, 2) implies that there exists a set of cointegrating vectors such that Г’С(1) = Г’С *(1) = 0, from which the following VECM representation can be derived

A*(L)(1 – L)2yt = – ВіГу-1 – В2ГА/М + et (30.24)

with A*(0) = In. Johansen (1992b) has developed the maximum likelihood estima­tion of this class of models, which, albeit more complicated than in the CI(1, 1) case, proceeds along similar lines to those discussed in Section 3.

Likewise, there are systems where the variables have unit roots at the seasonal frequencies. For example, if a seasonally integrated variable is measured every half-a-year, then it will have the following Wold representation

(1 – L2)y = C(L)e t. (30.25)

Since (1 – L2) = (1 – L)(1 + L), the {уt} process could be cointegrated by obtaining linear combinations which eliminate the unit root at the zero frequency, (1 – L), and/or at the seasonal frequency, (1 + L). Assuming that Г1 and Г2 are sets of cointegrating relationships at each of the two above mentioned frequencies, Hylleberg et al. (1990) have shown that the VECM representation of the system this time will be

A*(L)(1 – L2)yt = – B^’Ay^ – В2Г2( y-1 + yt-2) + Et, (30.26)

with A*(0) = In. Notice that if there is no cointegration in (1 + L), Г2 = 0 and the second term in the right-hand side of (30.26) will vanish, whereas lack of cointeg­ration in (1 – L) implies Г1 = 0 and the first term will disappear. Similar arguments can be used to obtain VECM representations for quarterly or monthly data with seasonal difference operators of the form (1 – L4) and (1 – L12), respectively.