Table 12.3: ADF(l) tests for stationarity of crude oil term structure spreads

cointegration vector is estimated, but all of the independent cointegration vectors should be employed in the ECM (§12.3). So there are some disadvantages in using the Engle-Granger methodology for more than two variable systems.

Consider an example of the application of the Engle-Granger method to a multivariate system when it is not appropriate to do so. The term structure of crude oil futures from 1 month to 12 months, using prices quoted on NYMEX from 4 February 1993 to 24 March 1999, is illustrated in Figure 6.4. From a purely visual inspection a high degree of cointegration seems likely. Indeed, looking at the ADF(l) statistics on the 11 independent spreads with the 1-month future log price shown in Table 12.3, many of these are stationary at the 1 % level. The longer spreads have less stationarity than the shorter spreads, in fact since the 5% critical value is -2.88 only the spreads up to 8 months are stationary at the 5% level.

Only one cointegration vector can be identified from an Engle-Granger regression

There will be more than one independent cointegration vector, because so many of the spreads are stationary.4 However, only one cointegration vector can be identified from an Engle-Granger regression. So when there are more than two variables the first step in the Engle-Granger is to select a dependent variable - and in this example the choice must be arbitrary. Using the log price of the 1-month future as the dependent variable, and the other 11 log futures prices as explanatory variables, gives the OLS regression reported in Table 12.4.

The Engle-Granger test for cointegration examines whether the residuals from this model are stationary, and with an ADF(l) statistic of -18.41 it is certainly

In fact from the results in the next section based on the Johansen methodology, it appears that there might be as many as 10 cointegration vectors.

Table 12.4: Engle-Granger regression for crude oil futures term structure

Variable Coefficient /-ratio

the case that they are. Similar regressions using other log futures prices as the dependent variable confirm the finding of cointegration, but each of these regressions gives a different result for the cointegration vector. So which ones should be used to model the equilibrium relationships?

The Engle-Granger procedure is only applicable to systems with more than two variables in very special circumstances, when there are clear answers to the following questions:

> Are there several cointegration vectors, that is, several long-run equilibria? If so, which equilibrium is being identified by the Engle-Granger regression, and is it the most appropriate?

> Which variable should be chosen as the dependent variable in the Engle-Granger regression? How different would the results be if another variable were used as dependent variable?

The Engle-Granger procedure is only applicable to systems with more than two variables in very special circumstances

12.2.2 The Johansen Methodology

Johansens methodology for investigating cointegration in a multivariate system has been preferred by economists. It employs a power function with better properties than the Engle-Granger method (Kremers et al., 1992), and has less bias when the number of variables is greater than two (Johansen, 1988, 1991; Johansen and Juselius, 1990). The Johansen tests are based on the eigenvalues of a stochastic matrix and in fact reduce to a canonical correlation problem similar to that of principal components. The Johansen tests seek the linear combination which is most stationary whereas the Engle-Granger tests, being based on OLS, seek the linear combination having minimum variance.

The Johansen tests are a multivariate generalization of the unit root tests that were described in §11.1.5. There it was shown that an AR(1) process may be

The Johansen tests seek the linear combination which is most stationary whereas the Engle-Granger tests seek the linear combination having minimum variance

rewritten in the form (11.10), where the first difference Ay, is regressed on the lagged level y, \. The test for a stochastic trend is based on the fact that the coefficient on the lagged level should be zero if the process has a unit root. Generalizing this argument for a VAR(l) process motivates the Johansen tests for a common stochastic trend, that is, for cointegration. The VAR(l) model (11.28) may be rewritten with Ay, as the dependent variable in a regression on

Now if each variable in is 1(1) then each equation in (12.2) has a stationary variable on the left-hand side. The errors are stationary and therefore each term in (A - I)yr , must be stationary for the equation to be balanced. If A - I has rank zero, so it is equivalent to the zero matrix, this condition implies nothing about relationships between the variables. But if A - I has rank r > 0, then there are r independent linear relations between the variables that must be stationary. Therefore the 1(1) variables in will have a common stochastic trend - that is, they will be cointegrated - if the rank of A - I is non-zero; the number of cointegration vectors is the rank of A - I. The rank of a matrix is given by the number of non-zero eigenvalues, so the Johansen procedure based on (12.2) tests for the number of non-zero eigenvalues in

The model (12.2) is not the only possible maintained model for Johansen tests: a VAR(l) model with a constant may not be the most appropriate representation of the data. Returning to the univariate analogy, recall that the Dickey-Fuller regression can contain more than just a lagged levels term. It may contain a constant if there is a drift in the stochastic trend, a time trend if the process also contains a deterministic trend, and it can be augmented with sufficient lagged dependent variables to remove autocorrelation in residuals. The same applies to the Johansen test: the maintained model may or may not contain a constant or a trend term, and the number of lagged first differences is chosen so that residuals are not autocorrelated.

If a higher-order VAR(/?) model is used to motivate the Johansen tests, the first difference formulation becomes

and the Johansen method is a test for the number of non-zero eigenvalues of the matrix

Ay, = «o + (A - I)y, ! + e,.

(12.2)

A-I.

Ay, = a0 + (A, - T)Ayt x + (A, + A2 - I)Ay, 2 + . . . + (A, + A2 + ... + A, , - I)Ay + (A, + A2 + ... + A, - I)y, „ + s,

(12.3)

= A, + A2 + ... +Ap-l.