Mathematical and Statistical Methods for Actuarial Sciences and Finance

Energy markets: crucial relationship between prices 29

Ta b le 3 .Cointegration rank test (trace and maximum eigenvalue)

Nr. of coint. vec. Eigenvalue λtrace λ^0 trace.^05 λmax λ^0 max.^05

r= 0 0.121 237.2 29.79 240.3 21.13

r≤ 1 0.020 32.91 15.49 32.24 14.26

r≤ 2 0.000 0.672 3.841 0.672 3.841

A rejection of the null ‘no cointegrated’ relationship and ‘rat most 1’ in favour of
‘rat most 2’ at the 5% significance level is provided. This provides evidence of the
existence of two cointegrating relationships among the three commodity price series.
In a VECM framework, the presence of two cointegrating vectors,r=2, on a set of
n=3 variables allows the estimation of an−r=1 common (stochastic) trend [16].
The common trend may be interpreted as a source of randomness which affects the
dynamics of the commodity prices. In this case we may assume oil prices represent
the leading risk factor in the energy market as a whole.
To better analyse the dynamics of the markets we use the Engle-Granger [5] two-
step methodology. This method consists in estimatingeach cointegrating relationship
individually using ordinary least squares (OLS) and then including the errors from
those cointegrating equations in short-rundynamic adjustment equations which allow
the explanation of adjustment to the long-run equilibrium. The first step is to estimate
the so-called cointegrating regression

y 1 ,t=α+βy 2 ,t+zt (5)

wherey 1 ,tandy 2 ,tare two price series, both integrated of order one, andztdenotes
the OLS regression residuals. We perform the test twice for each couple of time series
using as dependent variable both of the series. For each couple of time series, using
both of the series as dependent variables. The results are reported in Table 4. The
null hypothesis of no cointegration is rejected at the 8% significance level for the
regression oil vs electricity, at the 1 % level in all the other cases. The coefficientsβ
in equation (5), which represent the factors of proportionality for the common trend,
are estimated by OLS.
According to the Granger representation theorem, if two series cointegrate, the
short-run dynamics can be described by the ECM. The basic ECM proposed in [5]
can be written as follows:

y 1 ,t=φy 2 ,t+θ(y 1 ,t− 1 −α−βy 2 ,t− 1 )+t (6)

where(y 1 ,t− 1 −α−βy 2 ,t− 1 )represents the error correction termzt− 1 of equation (5),
φmeasures the contemporaneous price response,^10 θrepresents the speed of the
adjustment towards the long-term cointegrating relationship, andt∼i.i.d.( 0 ,).

(^10) The parameterφapproximates the correlation coefficient between first differences in prices
(yi,tandyj,t) and it will be close to 1 when the two commodities are in the same market.
Therefore, a higher value ofφis a sign of a stronger integration of the market [3].