28 C. Bencivenga, G. Sargenti, and R.L. D’Ecclesia
to catch seasonal changes in correlations when interpreting the rolling correlation
coefficient. The unconditional correlation coefficients,^8 ρT, together with the main
statistical features of the rolling correlations,ρs,s= 1 ,...,T, between the energy
price series are reported in Table 2. It is interesting to notice that the rolling correla-
tions between gas and oil show some counterintuitive behaviour.
Ta b le 2 .Unconditional correlation and rolling correlations between log pricesMatrices ρT E(ρs)σ(ρs) Max(ρs) Min(ρs)
Oil/Elect 0.537 0. 0744 0.260 0.696 − 0. 567
Gas/Elect 0.515 0.119 0.227 0.657 − 0. 280
Oil/Gas 0.590 -0.027 0.426 0.825 − 0. 827These results do not provide useful insights into the real nature of the relationship
between the main commodities of the energy markets.
5 The long-run relationship
Table 1 confirms a stochastic trend for all the price series; a possible cointegration
relationship among the energy commodity prices may therefore be captured (i.e., the
presence of a shared stochastic trend or common trend). Two non-stationary series are
cointegrated if a linear combination of them is stationary. The vector which realises
such a linear combination is called the cointegrating vector.
We examine the number of cointegrating vectors by using the Johansen method
(see [10] and [11]). For this purpose we estimate a vector error correction model
(VECM) based on the so-called reduced rank regression method (see [12]). Assume
that then-vector of non-stationaryI( 1 )variablesYtfollows a vector autoregressive
(VAR) process of orderp,
Yt=A 1 Yt− 1 +A 2 Yt− 2 +...+ApYt−p+t (3)withtas the correspondingn-dimensional white noise, andn×nAi,i= 1 ,...,p
matrices of coefficients.^9 Equation (3) is equivalently written in a VECM framework,
Yt=D 1 Yt− 1 +D 2 Yt− 2 +···+DpYt−p+ 1 +DYt− 1 +t (4)whereDi=−(Ai+ 1 +···+Ap),i= 1 , 2 ,...,p−1andD=(A 1 +···+Ap−In).
The Granger’s representation theorem [5] asserts that ifDhas reduced rankr∈( 0 ,n),
thenn×rmatricesandBexist, each with rankr, such thatD=−B′andB′Ytis
I( 0 ).ris the number of cointegrating relations and the coefficients of the cointegrating
vectors are reported in the columns ofB.
The cointegration results for the log prices are shown in Table 3.
(^8) The unconditional correlation for the entire period is given byρT=cov(x,y)
̂σx̂σy.
(^9) In the following, for the VAR(p) model we exclude the presence of exogenous variables.