A COMPANION TO Theoretical Econometrics

Common trends representation

As mentioned above, there is a dual relationship between the number of cointegrating vectors (r) and the number of common trends (n - r) in an n - dimensional system. Hence, testing for the dimension of the set of "common trends" provides an alternative approach to testing for the cointegration order in a VAR//VECM representation. Stock and Watson (1988) provide a detailed study of this type of methodology based on the use of the so-called Beveridge-Nelson (1981) decomposition. This works from the Wold representation of an I(1) system, which we can write as in expression (30.11) with C(L) = Xj= 0 Cp, C0 = In. As shown in expression (30.12), C(L) can be expanded as C(L) = C(1) + C(L)(1 - L), so that, by integrating (30.11), we get

yt = C(1)Yt + +, (30.21)

where +t = C(L)et can be shown to be covariance stationary, and Yt = Xti=1ei is a latent or unobservable set of random walks which capture the I(1) nature of the data. However, as above mentioned, if the cointegration order is r, there must be an (r x n) Г matrix such that T'C(1) = 0 since, otherwise, r'yt would be I(1) instead of I(0). This means that the (n x n) C(1) matrix cannot have full rank. Indeed, from standard linear algebra arguments, it is easy to prove that the rank of C(1) is (n - r), implying that there are only (n - r) independent common trends in the system. Hence, there exists the so-called common trends representation of a cointegrated system, such that

Vt = Ф Vc + +t, (3°.22)

where Ф is an n x (n - r) matrix of loading coefficients such that Г'Ф = 0 and yC is an (n - r) vector random walk. In other words, Vt can be written as the sum of (n - r) common trends and an I(0) component. Thus, testing for (n - r) common trends in the system is equivalent to testing for r cointegrating vectors. In this sense, Stock and Watson's (1988) testing approach relies upon the observation that, under the null hypothesis, the first-order autoregressive matrix of yC should have (n - r) eigenvalues equal to unity, whereas, under the alternative hypothesis of higher cointegration order, some of those eigenvalues will be less than unity. It is worth noticing that there are other alternative strategies to identify the set of common trends, yc, which do not impose a vector random walk structure. In
particular, Gonzalo and Granger (1995), using arguments embedded in the Johansen's approach, suggest identifying yC as linear combinations of yt which are not caused in the long-run by the cointegration relationships r'yt_1. These linear combinations are the orthogonal complement of matrix B in (30.16), yC = B±yt, where B± is an (n x (n _ r)) full ranked matrix, such that B'B± = 0, that can be estimated as the last (n _ r) eigenvectors of the second moments matrix S01S11S10 with respect to S00. For instance, when some of the rows of matrix B are zero, the common trends will be linear combinations of those I(1) variables in the system where the cointegrating vectors do not enter into their respective adjust­ment equations. Since common trends are expressed in terms of observable vari­ables, instead of a latent set of random walks, economic theory can again be quite useful in helping to provide useful interpretation of their role. For example, the rational expectations version of the permanent income hypothesis of consumption states that consumption follows a random walk whilst saving (disposable income minus consumption) is I(0). Thus, if the theory is a valid one, the cointegrating vector in the system formed by consumption and disposable income should be P' = (1, _1) and it would only appear in the second equation (i. e. a' = (0, a 22)), implying that consumption should be the common trend behind the nonstationary behavior of both variables.

To give a simple illustration of the conceptual issues discussed in the previous two sections, let us consider the following Wold (MA) representation of the bivariate I(1) process yt = (y1t, y2t)',

Evaluating C(L) at L = 1 yields

so that rank C(1) = 1. Hence, yt ~ CI(1, 1). Next, inverting C(L), yields the VAR representation

where

0.4 -0.8

0.2 0.4

so that rank A(1) = 1 and

Hence, having normalized on the first element, the cointegrating vector is P' = (1, -2), leading to the following VECM representation of the system

Next, given C(1) and normalizing again on the first element, it is clear that the common factor is yct = ХЦ e1; + 2XU e2i, whereas the loading vector Ф and the common trend representation would be as follows

Notice that P'yt eliminates yct from the linear combination which achieves cointegration. In other words, Ф is the orthogonal complement of P once the normalization criteria has been chosen.

Finally, to examine the effects of drift terms, let us add a vector p = (pv p2)' of drift coefficients to the VAR representation. Then, it is easy to prove that y1t and y2t will have a linear trends with slopes equal to p1/2 + p 2 and p1/4 + p 2/2, respectively. When 2p1 + p2 Ф 0 the data will have linear trends, whereas the cointegrating relationship will not have them, since the linear combination in p annihilates the individual trends for any p1 and p2.

The interesting case arises when the restriction 2p1 + p2 = 0 holds, since now the linear trend is purged from the system, leading to the restricted ECM representation

where p?1 = p1/0.4.