System-Based Approaches to Cointegration
Whereas in the previous section we confined the analysis to the case where there is at most a single cointegrating vector in a bivariate system, this setup is usually quite restrictive when analyzing the cointegrating properties of an n-dimensional vector of I(1) variables where several cointegration relationships may arise. For example, when dealing with a trivariate system formed by the logarithms of nominal wages, prices, and labor productivity, there may exist two relationships, one determining an employment equation and another determining a wage equation. In this section we survey some of the popular estimation and testing procedures for cointegration in this more general multivariate context, which will be denoted as system-based approaches.
In general, if yt now represents a vector of n I(1) variables its Wold representation (assuming again no deterministic terms) is given by
Ayt = C(L)e f, (30.11)
where now et ~ nid(0, X), X being the covariance matrix of e t and C(L) an (n x n) invertible matrix of polynomial lags, where the term "invertible" means that | C(L) = 0| has all its roots strictly larger than unity in absolute value. If there is a cointegrating (n x 1) vector, в’ = (ви,…, вм), then, premultiplying (30.11) by в’ yields
в^у = в'[С(1) + C(L)A]ef, (30.12)
where C(L) has been expanded around L = 1 using a first-order Taylor expansion and C(L) can be shown to be an invertible lag matrix. Since the cointegration property implies that в’уt is I(0), then it must be that в С(1) = 0 and hence A(= 1 – L) will cancel out on both sides of (30.12). Moreover, given that C(L) is invertible, then y has a vector autoregressive representation such that
A(L)y t = et, (30.13)
where A(L)C(L) = (1 – L)In, In being the (n x n) identity matrix. Hence, we must have that A(1)C(1) = 0, implying that A(1) can be written as a linear combination of the elements в, namely, A(1) = ав’, with a being another (n x 1) vector. In the same manner, if there were r cointegrating vectors (0 < r < n), then A(1) = БГ’, where B and Г are this time (n x r) matrices which collect the r different a and в vectors. Matrix B is known as the loading matrix since its rows determine how many cointegrating relationships enter each of the individual dynamic equations in (30.13). Testing the rank of A(1) or C(1), which happen to be r and n – r, respectively, constitutes the basis of the following two procedures.