R. F. Engle and B. S. Yoo, "Forecasting and Testing in Cointegrating Systems," Journal of Econometrics, Vol. 35, 1987, pp. 143-159.

"S. Johansen and K. Juselius, "Maximum Likelihood Estimation and Inference on Cointegration with Applications to the Demand for Money," Oxford Bulletin of Economics and Statistics, Vol. 52, 1990, pp. 169-210.

not be of full rank, where = -/ + A, + 2 + • • • + A. Note that in the earlier discussion we had = I, and hence we considered (-/ + A,).

The preceding discussion also suggests that if some of the variables in a VAR model are cointegrated, this implies some restrictions on the parameters of the VAR model. In Section 14.3 we pointed out that predictions from the unrestricted VAR models were not good, and hence some restrictions on the parameters were imposed in the Bayesian VAR (BVAR) approach. The cointegration theory gives a theoretical basis for imposing some restrictions on the VAR model. It has been found that predictions from the VAR model improved with restrictions imposed by cointegration theory." However, many of the comparisons made have been with unrestricted VAR rather than the BVAR. What we need to do is compare the predictions from VAR models that use cointegration restrictions with those generated by BVAR. Note that as we described in Section 14.3, the Bayesian VAR approach, by assuming a prior coefficient of unity on the own lagged terms, implicitly assumes that all the variables in the VAR model are unit root processes.

Suppose we consider a set of 3 variables, all of which are unit root processes. Suppose also that there are 2 cointegrating relationships among them. Then since any linear combination of cointegrated relationships is also cointegrated, it becomes very difficult to give any economic interpretation to the cointegrated relations. Each of them is a long-run equilibrium relationship, and all linear combinations are equilibrium relationships. There is, thus, an identification problem, and unless we have some extraneous information, we cannot identify the long-run equilibrium relationship. This has been the experience with the estimation of some long-run demand for money functions. For instance, Johansen and Juselius" estimate the demand for money functions for Denmark and Finland using quarterly data. For the Danish data the sample was 1974-1 to 1987-3 (55 observations). For the Finnish data the sample was 1958-1 to 1984-3 (67 observations). For the Danish data, there was only one cointegrated relationship, and this simplified the interpretation of the cointegrating vector as a long-run demand for money function. But for the Finnish data there were 3 cointegrating vectors, and this caused problems of interpretation.

This is not surprising because "cointegration" is a purely statistical concept based on properties of the time-series considered. It is "A-theoretical Econometrics." Cointegrated relationships need not have any economic meaning. But even if they do not, they can be used to improve predictions from the VAR models.

One important contribution of cointegration tests is to the modeUing of VAR systems, whether they should be in levels or first differences or both with some

"See P. C. B. Phillips, "Optimal Inference in Cointegrated Systems," Econometrica. Vol. 59. 1991, pp. 283-306.

"G. W. J. Granger and Tae-Hwy Lee, "Multicointegration," in Fomby and Rhoades (eds.). Advances in Econometrics, pp. 71-84.

restrictions. For this purpose the cointegration relationships need not have any economic interpretation. The cointegrated relations are of value only in determining the restrictions of the VAR system. (It is all part of a-theoretical econometrics anyway.)

If a set of unit root variables satisfies a cointegration reiation, simple first differencing of all the variables can lead to econometric problems.** In the general VAR system with n variables, if all the variables are nonstationary, using an unrestricted VAR in levels is appropriate. If the variables are id 1(11 l»it no-cointegration relation exists, then appUcation of an unrestricted VAR in first differences is appropriate. If there are r cointegrating relationships, then we need to model the system as a VAR in the r stationary combinations and [n - r) differences of the original variables.

In any case, when our interest is in forecasting, the existence of some cointegrating relationships, even if they do not have any economic interpretation, should help us to improve the forecasts from the VAR models. The multipUcity of cointegrating vectors (and the resulting identification problems mentioned earlier) need not worry us. Cointegration relationships that do not make any economic sense need not be discarded. In fact, this is the most important use of cointegration tests and cointegrating relationships.

14.9 Cointegration and

Error Correction Models

If X, and y, are cointegrated, there is a long-run relationship between them. Furthermore, the short-run dynamics can be described by the error correction model (ECM). This is known as the Granger representation theorem.

If X, ~ 1(1), y, ~ 1(1), and z, = y, - fix, is 1(0), then x and are said to be cointegrated. The Granger representation theorem says that in this case x, and y, may be considered to be generated by ECMs of the form

Ax, = pZ, , + lagged(Ax„ Ay,) + e„ Ay, = 92Z,-i + lagged(Ax„ Ay,) + ej,

where at least one of p, and p2 is nonzero and e„ and ej, are white-noise errors.

Granger and Lee* suggest a further generalization of the concept of cointegration. Define w, = Xj-o , j that is, w, is an accumulated sum of z, or Aw, = z,. Since z, ~ 1(0), w, will be 1(1). Then x, and y, are said to be multi-cointegrated if x, and w, are cointegrated. In this case y, and w, will also be cointegrated. It follows that u, = w, - ax, ~ 1(0), where a is the cointegration

14.10 Tests for Cointegration

An important ingredient in the analysis of cointegrated systems is tests for cointegration. Consider, first, the case of two variables, x and y. We first apply unit root tests to check that x and are both 1(1). We next regress on x (or x on y) and consider = - . We then apply unit root tests on .

If X and are cointegrated, = - ( is 1(0). On the other hand, if they are not cointegrated, will be 1(1). Since unit root tests will be applied to u, the null hypothesis (as we discussed in Section 14.5) is that there is a unit root. Thus the null hypothesis and the alternative in cointegration tests are:

: has a unit root or x and are not cointegrated H{. X and are cointegrated

The additional problem here is that is not observed. Hence we use the estimated residual from the cointegrating regression. Engle and Granger (1987) suggest several cointegration tests but suggest that using the ADF test to test for unit roots in , is best.

An alternative procedure is to use the VAR model, compute the characteristic roots of the matrix A of the coefficients of the VAR model (or the roots of the matrix B in equation (14.13) in the case of a general AR model) and apply the tests described earUer; that is, consider n(X - 1) and use the tables in Fuller (1976, p. 371).

The case with more than two variables is more complicated. If there is only one unit root, the procedures we described earUer based on the VAR model can be used. If there is more than one unit root, the Johansen procedure (referred to earlier) has to be used.

Of particular importance since the significance levels used are the conventional 1% and 5% levels is the meaning of the null and alternative hypotheses in cointegration tests. In the unit root tests, the null hypothesis is that there is a unit root. That is, we maintain that the time series are difference stationary. This null hypothesis is maintained unless there is overwhelming evidence to reject it. In the case of cointegration tests, the null hypothesis is that there is no cointegration. That is, we maintain that there are no long-run relationships. This null hypothesis is maintained unless there is overwhelming evidence to reject it. The way the null hypothesis and the alternatives are formulated and

constant. If X, and y, are multicointegrated, Granger and Lee show that they have the following (generalized) ECM representation:

At, = PiZ,-i + + lagged(Ax„ ,) + e„

Ay, = p2Z,-i + 82 , , + lagged(Ax„ ,) + ,

Examples of this are: x, = sales, y, = production, z, = y, - x, = inventory change and w, = inventory. Sales, production, and inventory are all 1(1) and possibly cointegrated; z„ the inventory change, is 1(0).