Estimating a VAR(l) on the S&P 500 and FT 100 daily return data gives

/ 0.018534 0.30033 \ V0.22758 -0.18147 J

Now the solutions to 1 - tr(A)z + A z2 = 0 are z = -2.76719 and z = 5.03926. Both lie outside the unit circle and we may conclude that the S&P 500 and the FTSE 100 returns are jointly stationary. Similar results are obtained when testing the joint stationarity of these returns with the CAC returns.

The correlations of individual stocks with an index can also be very unstable over time. Figure 11.10 shows EWMA correlations with X = 0.94 of three of the main Brazilian stocks with the Ibovespa equity index. While the correlations with the main stock Telebras are quite stable, those with two other major stocks Eletrobras and Petrobras are very unstable indeed. It difficult to find However joint stationarity tests are not rejected on these data. Indeed, it is examples where the difficult to find examples where the assumption of joint stationarity does not assumption of joint hold. For example, all of the US stocks in Table 11.3 (§11.3.2) pass bivariate stationarity does not hold joint stationarity tests, performed over each year and over the whole data period. The conclusion is that the observed instability of correlations between individual equities, between equities and their index, and between different equity indices is not due to lack of joint stationarity; more likely it is a result of non-linear relationships.

11.4.3 Granger Causality

Granger causality means that a lead-lag relationship is evident between variables in a multivariate time series

After the seminal work of Granger (1988), the term Granger causality means that a lead-lag relationship is evident between variables in a multivariate time series. In a bivariate system of jointly stationary time series {xt\ and {y,}, the variable x is said to Granger cause if lagged x improves the predictions of y, even after lagged variables have been included as explanatory variables.

A lead-lag is to be expected in the co-dependent relationships that are observed between many financial markets and dynamic models of multivariate time series need to be applied if one is to gain direct insights into the nature of lead-lag relationships. This subsection shows how vector autoregressive models may be used to investigate any lead-lag behaviour between financial markets.

Consider the bivariate VAR(/;) model (11.29) written out in full as:

x, = cl+Y + Y Kyt-t + ei/>

/=i i=i

Oct-94 Mar-95 Jul-95 Dec-95 Apr-96 Aug-96 Jan-97 May-97 Sep-97

-Ibovespa -Telebras - Ibovespa-Electrobras Ibovespa-Petrobras Figure 11.10 EWMA correlations in Brazilian equities.

The test for Granger causality from x to is an F-test for the joint significance of a2b . . ., a2p in an OLS regression. Similarly, the test for Granger causality from to x is an F-test for the joint significance of 6n, . . ., $lp.

To illustrate the method, consider testing for Granger causal flows between WTI crude oil spot and futures using the VAR(2) model of Table 11.4. Following §A.2.3, the -statistic is calculated from residual sums of squares in the restricted and unrestricted models, and for this example .F(spot -> futures) = 118.195 and /"(futures -> spot) = 7.197. The 1% critical value of the F2666 distribution is 4.64, and so there was significant causality from futures to spot markets but a very much more significant causality from spot to futures. This model provides some very convincing empirical evidence that spot crude oil prices are good predictors of the near futures prices.

Every empirical finding from market data modelling depends on both the length of data period used for the test and the time at which it is performed. If the finding is not robust to changes in data then it will not have much practical use, so it is important to backtest the model very thoroughly, checking the recorded Granger causal flows for stability. In fact it is always the case that the strength of Granger causality can change over time, the direction of causality can change depending on the time that it is measured, or there can be bidirectional causality.

An example of this sort of change in causal flows will be given in Chapter 12. There it is shown that although the VAR(2) model of Table 11.4 is certainly an

The strength of Granger causality can change over time, the direction of causality can change depending on the time that it is measured, or there can be bidirectional causality

improvement on the earlier AR(2) specification for WTI futures alone, it is still not well specified. An important variable has been omitted: these spot and futures prices are cointegrated, so they have an equilibrium relationship. Therefore a VAR model of their returns requires a variable that is defined by the equilibrium between these price series.