Johansen and Juselius (1990) recommend using the standard trace test for the number r of non-zero eigenvalues in the matrix .5 The test statistic for

H0: r R against Hp r > R

Tr=-Tj2 ln(l-i,), (12.4)

i=R+l

where T is the sample size, n is the number of variables in the system and the eigenvalues of are real numbers X such thatO X < 1. In (12.4) the estimates of these eigenvalues are ordered so that X\ > X2 > ... > Xn. So the Tr statistic decreases as R increases. The Johansen method first computes the eigenvalues and then calculates the trace statistic for every R = 0 to n - 1. Critical values of the trace statistic (12.4) are given in Johansen and Juselius (1990). They depend on the specification of the underlying model, whether or not it includes a constant or trend, and the number of lags in the VAR.

The presence of the constant in (12.3) is necessary for variables that exhibit a drift in the stochastic trend and so that the cointegration test will be on the detrended data (§11.1.4). Likewise, if one or more variables is thought to contain a deterministic trend - that is they are 1(1) + trend - then a time trend may be included also. However, it is very unlikely that a time trend would be necessary for most financial markets. In fact the examples given in this section contain neither a constant nor a time trend, since there is no obvious trend in the data, but in many cases, when the prices do appear to be trending, a constant should be included in the Johansen test.

Table 12.5 summarizes the results of using the Johansen procedure to test for cointegration in the crude oil term structure data. The in-built Johansen cointegration procedure in the PcGive package on the CD gives the optimal lag length for the maintained VAR() model, and in this example it turns out to be 3. Actually the results are so robust that their qualitative nature is more or less independent of different lag specifications, and similar conclusions may be drawn whether the actual futures prices or their logarithms are used (both are reported).

The Johansen trace statistics reject the null hypothesis that there are R cointegration vectors in favour of the alternative that there are greater than R cointegration vectors at the 1% level, for all R up to and including 10. The probability value of the trace statistic for the null hypothesis r 9 against the alternative r > 9 is 0.004 for the price data and 0.002 for the log price data.

Another test, the maximal eigenvalue test, is described in their paper, and some packages offer this as well as the trace test as standard output from cointegration procedure. However, the maximal eigenvalue test does not have nested h\potheses and in some cases the maximal eigenvalue and trace tests imply different conclusions. In that case the results of the trace tests should be preferred.

Table 12.5: Johansen trace tests for cointegration in the crude oil term structure

For R - 10 the hypothesis is only rejected at the 5% level, since the probability value of the trace statistic is 0.02 (price data) or 0.014 (log price data). Naturally we cannot accept the hypothesis r > 11 in the last row of the table since in that case the log futures prices themselves would have to be stationary. It may be concluded that there are 10 or even 11 (the maximum number) cointegration vectors in the system.

Returning to the US yield curve data of Figure 6.1 that were discussed in §12.1.2, an Engle-Granger regression using the 1-month yield as dependent variable gives an ADF(2) statistic on the residuals of -6.49, so the system is definitely cointegrated. But the Engle-Granger method gives only one long-run equilibrium, viz.

ml = 0.73m2 + 0.127m3 + 0.273m6 - 0.016ml2 - 0.42y2 + 0.41y3 + 0.067y5 - 0.122y7 - 0.09yl0.

A different estimate of a long-run equilibrium would be obtained if the 3-month yield or any other maturity were used as dependent variable. Using the Johansen procedure, Table 12.6 indicates that there are not one but seven cointegration vectors in this system (the tests reject the null for all r 6, but the null hypothesis that r 7 cannot be rejected in favour of r > 7). The conclusion is that there are seven cointegration vectors in this US yield curve with ten maturities - not the maximum number of nine, but there is still a very high level of co-dependency in the yield data.

The Johansen procedure is more informative than the Engle-Granger procedure because it finds all possible cointegrating relationships. It is commonly employed for economic problems because there are usually many variables in the system and often there is no clear indication of which should be the

Table 12.6: Johansen tests for cointegration in US yields

dependent variable in an Engle-Granger regression. However, there are good reasons for choosing Engle-Granger as the preferred methodology for many financial applications of cointegration:

> It is very straightforward to implement and to interpret (in fact it can be

done in a simple spreadsheet). > From a risk management point of view the Engle-Granger criterion of

minimum variance is usually more important than the Johansen criterion of

maximum stationarity. > There is often a natural choice of dependent variable in the cointegrating

regressions, for example, in equity index tracking (§12.5). > The Engle-Granger small-sample bias may not be a problem since sample

sizes are generally quite large in financial analysis and the cointegration

vector is super-consistent.6

The Johansen procedure is more informative than the Engle-Granger procedure because it finds all possible cointegrating relationships. However, there are good reasons for choosing Engle-Granger as the preferred methodology for many financial applications of cointegration

Both the Johansen and the Engle-Granger tests have been applied in extensive empirical work on cointegration in financial markets. There are many other cointegration tests: for example, Phillips and Ouliaris (1990) propose a two-step cointegration test based on the residuals from a cointegrating regression, and a test described by Engle and Yoo (1987) is based on the significance of the disequilibrium terms in the ECM.

12.3 Error Correction and Causality

The mechanism which ties cointegrated series together is a causality, not in the sense that if we make a structural change to one series the other will change

6In §A.1.3 a consistent estimator is defined as one whose distribution converges to the true value of the parameter as the sample size increases to infinity. A super-consistent estimator is a consistent estimator with a very fast convergence.