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124

When time series are cointegrated there must be some Granger causal flow in the system

too, but in the sense that turning points in one series precede turning points in the other. This is the concept of Granger causality that was introduced in §11.4.3. It was introduced without reference to cointegration because cointegration is not necessary for causality, though it is sufficient. When time series are cointegrated there must be some Granger causal flow in the system. Cointegration is not essential for a lead-lag relationship to exist: it may be that causal flows exist between time series because they have some other common feature (§12.6).

The Granger representation theorem states that a vector autoregressive model on differences of 1(1) variables will be misspecified if the variables are cointegrated (Engle and Granger, 1987). Engle and Granger showed that an equilibrium specification is missing from a VAR representation (11.29) but when lagged disequilibrium terms are included as explanatory variables the model becomes well specified. Such a model is called an error correction model because it has a self-regulating mechanism whereby deviations from the long-run equilibria are automatically corrected.

The ECM is a dynamic model of correlation in returns, and the t-statistics on its estimated coefficients provide much insight into the lead-lag behaviour between returns

The ECM is a dynamic model for first differences of the 1(1) variables that were used in the cointegrating regression. Thus if log prices are cointegrated and the cointegration vector is based on these, the ECM is a dynamic model of correlation in returns, and the -statistics on its estimated coefficients provide much insight into the lead-lag behaviour between returns. Note that the ECM is a short-run analysis of dynamic correlations, quite distinct from the first stage of finding cointegrating relationships in a long-run equilibrium analysis. The connection between the two stages is that the disequilibrium term z that is used in the ECM will be identified during the first stage.

The reason for the name error correction is that the model is structured so that short-run deviations from the long-run equilibrium will be corrected. This is simple to illustrate in the case of two cointegrated log price series x and y. The ECM takes the form

Ay, = a2 + B3;.Ax( ,- + B4iAy, ,. + , + e2„

(12.5)

where A denotes the first difference operator, z - x - ay is the disequilibrium term and the lag lengths and coefficients are determined by testing down OLS regressions (§11.3.3).

Suppose a > 0. The model (12.5) will only be an ECM if , < 0 and y2 > 0; only in that case will the last term in each equation constrain deviations from the long-run equilibrium so that errors will be corrected. To see this, suppose z is large and positive: then x will decrease because , < 0 and will increase because y2 > 0;



9.1

Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 Jul-98 Jan-99 Jul-99 Jan-00 Jul-00 AEX CAC - PAX]

Figure 12.3 Are European equity indices cointegrated?

both have the effect of reducing z, and in this way errors are corrected. Similarly if a < 0, for an ECM we must have , < 0 and y2 < 0; only then will equilibrium be maintained by the presence of the disequilibrium term.

The magnitude of the coefficients y, and y2 determines the speed of adjustment back to the long-run equilibrium following a market shock. When these coefficients are large, adjustment is quick so z will be highly stationary and reversion to the long-run equilibrium E(z) - E(x) - aE(y) will be rapid. In fact a test for cointegration proposed by Engle and Yoo (1987) is based on the significance of these speed-of-adjustment coefficients.

The magnitude of the coefficients yx and y2 determines the speed of adjustment back to the long-run equilibrium following a market shock

When x and are cointegrated log asset prices the ECM will capture dynamic correlations and causalities between their returns. If the coefficients on the lagged returns in the x equation are found to be significant then turning points in will lead turning points in x. That is, Granger causes x (§11.4.3). There must be causalities when a spread is mean-reverting and two asset prices are moving in line, but the direction of causality may change over time.

For an empirical example of cointegration and error correction modelling, consider the European equity indices shown in Figure 12.3. They are plotted as logarithms and transformed to take the same values at the beginning of 1996. The figure indicates that there is likely to be cointegration between these indices, and this is verified by the Johansen test results based on a VAR(l) model that are reported in Table 12.7.

There is clear indication of a cointegration vector (in fact it is significant at the 0.1% level) which is estimated by the Johansen procedure as

z = ln(AEX) - 0.193 ln(CAC) - 0.546 ln(DAX).



Table

12.7a: Johansen tests on

the AEX, CAC

and DAX

H0: i- R vs H,: r > R

Trace

Prob

35.88

0.0016

9.25

0.153

3.43

0.666

Table 12.7b: Error correction model for AEX, CAC and DAX

To AEX To CAC To DAX

Coefficient

t-statistic

Coefficient

t-statistic

Coefficient

t-statistic

-4.33E-04

-0.88695

9.86E-04

2.04443

7.39E-04

1.40834

RAEX(-

0.055848

1.97003

-8.61E-03

-0.30703

-6.42E-03

-0.21034

RCAC(-

-0.0674

-1.88246

-0.09058

-2.55783

0.139623

3.62226

RDAX(-

0.19267

5.75736

0.22842

6.90079

-0.04725

-1.31139

Z(-l)

-0.01525

-3.80111

1.48E-03

0.373361

-7.40E-04

-0.17139

The corresponding ECM is shown in Table 12.7b. There are three columns of coefficients and -statistics, corresponding to the dependent variable in the regression being the return to the AEX, CAC and DAX, respectively. The first column shows that the explanatory variables in each model are the lagged returns and the lagged cointegration vector z, given above.

The results indicate a strong positive causality from the DAX to the other two markets, and these effects have been highlighted by bold type in the table. The CAC and the AEX tend to follow the DAX, so that if today there is a large negative return on the DAX this will tend to depress the returns on the other two markets tomorrow. Similarly, if the DAX jumps up today, the other two markets are likely to see bigger returns tomorrow.

For another example of error correction modelling, return to the WTI spot and futures prices example. In §11.2.1 it was shown that they are very highly cointegrated over the period with cointegration vector approximately (1, -1). That is, the basis z = In F - In S defines the long-run stationary equilibrium and so, by the Granger representation theorem, the lagged basis needs to be added to VAR models on returns such as the VAR(2) model reported in Table 11.4.

Adding the basis has the effect of reducing the significance of the second-order lags in the VAR(2) model. So ECMs of the form (12.5) with only one lagged daily return to spot, and one lagged daily return to futures and one lag of the



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