Table 11.4: VAR(2) model for crude oil spot and futures returns

between the variables in the system, to look for lead-lag behaviour as described in §11.4.3 and perhaps also for predictive purposes.

As an example of a VAR model, Table 11.4 shows the estimated VAR(2) model for daily returns to WTI crude oil spot and near futures from 1 June 1998 to 26 February 1999. This model is the bivariate generalization of the AR(2) model in §11.3.1 to include spot returns in the system. Each equation is estimated separately using OLS and the -statistics (in parentheses) indicate that the model fits the futures equation very well. In fact the F4666 statistic for goodness of fit (§A.2.4) is 66.7 for the futures equation, and this is significant at the highest level (1% F4666 = 3.35). The F-statistic for the future to spot causality is only 7.2; while this is still highly significant, it is weaker than the causality from spot to future prices (§11.4.3). The results indicate that yesterdays closing crude oil spot price is a good predictor of the closing futures price today.13 This possibility will be investigated again in the context of a much better model in §12.3.

11.4.2 Testing for Joint Covariance Stationarity

Two covariance-stationary time series {x,} and {y,} are jointly covariance-stationary if cov(x„ y, s) depends only on the lag s. In particular, the contemporaneous covariance cov(x„ yt) is a constant, irrespective of the time at which it is measured, and this means that multivariate time series that are generated by jointly covariance-stationary processes have correlation measures that will be stable over time.

The instability over time of estimates of correlation between the returns to two financial markets has been encountered throughout this book. It has been attributed to various causes, including the following:

l3Spot is the first month Cushing until the futures expiry, then the second month Cushing until the 25th, then back to the first month Cushing. The Cushing prices are quoted by Platts and represent market prices paid for crude to be delivered over the next (or second nearest) calendar month. Thus the underlying asset is exactly the same as that represented by the futures contract. This cash market trades for only about 30 minutes each day following the close of trading in the futures contract.

> Co-dependencies between asset returns may be highly non-linear in nature,

but correlation is a only a linear measure. > Unconditional correlation is essentially a static measure, but dynamic

relationships between markets may exist with a lead-lag nature. > Unconditional correlation only exists when the returns are jointly

covariance-stationary. It may be that correlation estimates jump around

because they are being measured on non-jointly stationary series.

This subsection addresses the third of these causes of unstable correlation. It describes how a VAR representation may be used to check whether the system is jointly covariance-stationary. In fact the conditions for joint stationarity can be viewed as a simple generalization of the conditions for univariate stationarity.

Recall from §11.2.1 that a univariate AR(1) model represents a stationary time series only if it is stable. That is, the coefficient a must be less than 1 in absolute value, because if a > 1 the AR(1) model explodes, each successive observation increasing until y, -> ±oo as t -> oo, and if a = 1 the model represents a non-stationary random walk. More generally the AR(p) model represents a stationary process only if the solutions to the characteristic equation 1 - axx - a2x2 - ... - apxp - 0 lie outside the unit circle.

Analogously, a multivariate VAR(l) model represents a jointly stationary multivariate time series only if it is stable and this will be the case if the solutions to the determinant equation

11 - Az j = 0 (11.30)

lie outside the unit circle. Since the solutions to (11.30) are the inverse of the eigenvalues of A,14 an alternative characterization of the joint stationarity condition is that all eigenvalues of A lie inside the unit circle. A useful check for joint stationarity is therefore to fit a VAR(l) and use this condition to test for stability.15

In a bivariate VAR(l) the coefficient matrix A is a 2 x 2 matrix, and (11.30) reduces to the simple quadratic equation

14The eigenvalues of a square matrix A are those scalars X such that Ax = Xx for some non-zero vector x (x is called the eigenvector of X). Thus (A - Xl)x = 0 for a non-zero x, which implies that (A - XI) is a singular matrix and the determinant A - Xl \ = 0. This equation is called the characteristic equation and the eigenvalues are the solutions.

!5The condition for joint stationarity based on a VAR(/>) representation (11.29) is that the solutions to the equation

I-A,z-A2z2- . . . -A,z=0

lie outside the unit circle. This equation is not very nice to solve when p > 2. But if the VAR(l) representation is not stable there is little practical advantage in checking for stability in higher-order VAR models. Even if a higher-order VAR is found to be stable when a VAR(l) is not, the lack of robustness of the basic properties of the data to different VAR formulations should lead one to question this type of modelling in the first place. Also if joint stationarity is tested using a VAR(/>) specification one can use the fact that a VARQ>) model has an equivalent representation as a VAR(l) model with p times as many variables, so that, in fact, joint stationarity only ever needs checking using the VAR(l) condition.

Unconditional correlation only exists when the returns are jointly covariance-stationary

If the VAR(l) representation is not stable there is little practical advantage in checking for stability in higher-order VAR models

Sep-97 Nov-97 Jan-98 Mar-98 May-98 Jul-98

I-UK-US - UK-FR -FR-USl

Figure 11.9 EWMA correlations of equity indices.

1 -tr(A)z+Az2 =0,

where tr(A) is the sum of the diagonal elements and A is the determinant of A. So the two series are jointly covariance-stationary if the solutions of this equation lie outside the unit circle.

Some correlations between the crude oil spot and futures returns have been shown in Figure 3.4. The correlations are relatively stable over time, but could this be just an artefact of the historic correlation method being applied to series with stress events? Or are these correlations stable over time because crude oil spot and near futures returns are indeed jointly covariance-stationary? To answer this question we estimate a VAR(l) representation of the crude oil spot and futures returns and obtain

/0.196 -0.254 \ v0.715 -0.536/

So tr(A) = -0.34 and A = 0.07655. The solutions of 1 + 0.34z + 0.07665z2 = 0 are -2.22 ± 2.85i, which lie well outside the unit circle. Thus the VAR(l) model is a stable representation of spot and futures returns in WTI crude oil and it may be assumed that the series are indeed jointly covariance-stationary.

Correlations between international equity indices can also vary considerably over time (Erb et al., 1994; Longin and Solnik, 1995). Exponentially weighted moving average correlations with X - 0.94 of daily returns to the S&P 500, the CAC and the FTSE 100 index from September 1997 to September 1998 are shown in Figure 11.9. These correlations appear to vary considerably over the year. Is it possible that the S&P 500, the CAC and the FTSE 100 indices are not jointly stationary?

The correlations are relatively stable over time, but could this be just an artefact of the historic correlation method being applied to series with stress events?