Test for unit root and estimations from model

As it has been explained in the methodological section, whether spot and futures prices contain a unit root will be tested first and whether they are cointegrated will be examined later.

The results of the Augmented Dickey-Fuller (ADF) and Phillips-Perron tests for unit root are reported in table 7. The ADF test statistics based on the Akaike’s information criterion show that none of the level series are stationary process, while for all the difference series the hypothesis of a unit root is rejected at the 1% level of significance, suggesting the presence of stationarity. The same null hypothesis is examined by the Phillips-Perron test which confirms that the logarithm of the prices are integrated I(1) and first differencing is sufficient to induce stationarity. This conclusion is in accord with the results of many previous researches regarding the non-stationary nature of the logarithmic price series.

As differencing once produces stationarity, we may conclude that each series is an I(1) process, which is necessary for testing for cointegration. This is done by means of Engle- Granger test that basically applies the ADF test, but with different critical values, to the residuals of the cointegrating regression5. The relevant results are presented in table 8. The null hypothesis that residuals contain a unit root is here clearly rejected in favour of the alternative hypothesis that they are stationary. Therefore spot and future prices result in being cointegrated and linked by a long run relationship with each another. The importance of these findings is that the change in spot return series is not just a function of change in futures

5 _{i.e. the regression of the logged spot and futures prices: log S}

returns, as assumed by the first differences model shown before, but also of lagged equilibrium errors and lagged value changes in spot and futures prices. Traditional models ignore this relationship producing a suboptimal hedge ratio.

Incorporating the error correction into the previous VAR model, the error correction model is estimated. The results are presented in table 9. They show that γs=-1.1 and γf=0.67 which means that the error correction term is correctly signed in both equations and implies that spot prices have a much greater speed of adjustment than the spot prices. Moreover the error correction of futures is highly significant in both the equation and spot and futures past prices lags seem able to explain the current movements of both the prices. Therefore a bi-directional causality seems to exist between the two markets. Besides the table shows that the cointegrating relationship is St-1-(0.985)*Ft-1=c which basically corresponds to the condition for long-run market efficiency6.

5.4 Estimations from model 4

In order to examine the efficiency of the VAR model, it could be useful to verify the features of the residuals. We have already seen that the VAR model has adequately considered the serial correlation examining the streams of the residuals.

On the contrary, the plot of the actual values of the residuals presented in figure 3 shows that even if the mean seems constant the variance is still changing through the time and the presence of autoregressive conditional heteroskedastic (ARCH) effects persists. This is also confirmed by the analysis proposed by McLeod and Li (1983), which examines the sample autocorrelation functions of the mean equation squared residuals for a significant Q-statistic at a given lag. The results, which show a high significance for the Q-statistic for each given lag, are reported in table 12. Since the presence of heteroscedasticity and ARCH effects is

detected, the assumption of constant variance over time and the estimation of constant hedge ratios may be inappropriate. The estimation of time-varying variances and covariances and as a consequence time-varying hedge ratios based on a GARCH model are therefore expected to give better results.

The bivariate GARCH (1,1) model in the Bollerslev, Engle, Kroner and Kraft (BEKK) specification is adopted here. The model is given by equations (4.12a\b-4.14) while table 10 displays the results of the estimation. The bivariate GARCH model has been estimated using a program for EViews, which uses the Marquardt algorithm to compute maximum-likelihood estimates. The estimated parameters for the mean equations seem to be statistically significant implying that the GARCH (1,1) error is able to capture the dynamics in the second moments of the joint distribution of returns. It is also interesting to note that the cross correlation of the variances P = 0.9717 is high but not perfect, just as we expected.

Table 13 shows that the GARCH model is really able to remove the serial correlation previously detected since uncorrelatedness in the vector of squared standardized residuals is now found for each given lag.

5.5 Hedging effectiveness

In this section, we evaluate and compare the hedging performances of the four hedging models considered in our study.

The hedge ratio for the conventional regression method is obtained as the estimated coefficient of the future price in the regression of spot on the future price. The ratio between the variance and the covariance of the residuals is instead used to obtain the optimal hedge ratios for the bivariate VAR model and the bivariate VEC model. The optimal hedge ratios for these three models are presented in table 11. It can be seen that the hedge ratio obtained from the VEC model is slightly greater than those obtained from the OLS and VAR models. This

result is consistent with those from Ghosh (1993) and Lien (1996) where it is noted that the hedge ratio results biased downward in size when the cointegrating relationship is ignored.

The dynamic hedge ratio obtained from the conditional variance and covariance between spot and futures price in the bivariate GARCH (1,1) with error correction model is plotted in figure 4. It shows signs of extreme volatility during the sample period. The sample mean of the hedge ratio is 0.933158 while the series ranges from a minimum of 0.59 to a maximum of 1.15.

Table 14 displays the in sample hedging performances of the various models. The naïve method is added just as a term of comparison. The results demonstrate that all hedging strategies permit achieving substantial risk reductions compared to the unhedged position. Indeed the unhedged portfolio suffers, not surprisingly, of the highest variance in the return. The naïve hedge, which assumes a unitary hedge ratio, follows with a variance reduction in relation to the unhedged portfolio equal to the 92,16%. The hedging performances for the remaining models do not differ very much. The bivariate GARCH (1,1) with error correction seems to offer performances slightly superior to the OLS, the VAR and the VAR with error correction model. In fact the variance reduction associated with the Garch model is 92.7451% against the 92.7442% of the OLS model, the 92.7438% of the VAR model and the 92.7309% of the VAR with error correction model. Also considering the results under a risk-return trade off basis the GARCH model is found to outperform the other models providing the greatest return and the lowest portfolio variance simultaneously.

The out-of-sample comparison conducted for the last thirty observations is shown in Table 15. All models provide lower portfolio returns and variance reduction than in the sample. In this case the GARCH method continues to achieve the best performances in variance reduction but no longer provides the highest portfolio return since in this it is outperformed by the OLS, which offers a greatest portfolio return of about 0.085%.