Mathematical and Statistical Methods for Actuarial Sciences and Finance

Energy markets: crucial relationship between prices 27

whereyt=yt−yt− 1 and the lag lengthkis automatic based on Scharwz information
criterion (SIC). The results of the unit root test are reported in Table 1.

Ta b le 1 .Unit root test results for the logged price series

Series tγ τ 0 τ 1 τd Decision

Oil 1 .17 (1) − 0 .70 (1) − 2 .73 (1) − 43 .2(1) I(1)

Gas − 0 .30 (6) − 3. 37 ∗(6) − 5. 58 ∗∗(2) − 21 .1(1) I(1)

Elect − 0 .06 (14) − 3 .41 (14) − 4. 36 ∗∗(14) − 18 .4(1) I(1)

The 5% significance levels are− 1 .94 for ADF without exogenous variables,− 2 .86 for ADF
with a constant and− 3 .41 for ADF with a constant and trend.(∗)denotes acceptance of the
null at 1%,(∗∗)denotes rejection of the null at the conventional test sizes. The SIC-based
optimum lag lengths are in parentheses.

We run the test without any exogenous variable, with a constant and a constant plus
a linear time trend as exogenous variables in equation (1). The reportedt-statistics are
tγ,τ 0 andτ 1 , respectively.τdis thet-statistic for the ADF tests in first-differenced data.
tγis greater than the critical values but we reject the hypothesis in first-difference,
hence we conclude that the variables are first-difference stationary (i.e., all the series
areI( 1 )).

4 The short-run relationship

Alexander [1] presents the applications of correlation analysis to the crude oil and
natural gas markets. Correlation measures co-movements of prices or returns and can
be considered a short-term measure. It is essentially a static measure, so it cannot
reveal any dynamic causal relationship. In addition, estimated correlations can be
significantly biased or nonsense if the underlying variables are polynomials of time
or when the two variables are non-stationary [18].
To analyse a possible short-run relationship among the variables, we estimate a
rolling correlation overτj=100 days^7 according to:

ρs[x,y]=

1

τj− 1

∑s+τj

i=s(xi−̂x)(yi−̂y)

̂σx̂σy

, s= 1 ,...,T−τj, (2)

whereT =1580 (the entire period 2001–2007), and̂σxand̂σyare the standard
deviations ofxandy, estimated on the corresponding time window.
Correlation changes over time, as expected, given the non-stationarity of the un-
derlying processes. Volatilities of commodity prices are time dependent and so are the
covariance and the unconditional correlation. This means that we can only attempt

(^7) This window period is suggested in [6]. We also perform the analysis with larger windows
(τj= 100 , 150 ,200 days), getting similar results.