Advances in Risk Management

RICCARDO BRAMANTE AND GIAMPAOLO GABBI 229

Table 12.1 Regression equation of correlation EUR–USD changes explained
by volatility differences

Variable Coefficient Std. error t-statistic Prob.

DEURV 1.062701 0.128748 8.254101 0.0000
DUSDV 0.926331 0.132735 6.978780 0.0000

MA(1) 0.136762 0.042159 3.243962 0.0012

R-squared 0.180499 Mean dependent var 0.065814

AdjustedR-squared 0.177546 S.D. dependent var 0.113572
S.E. of regression 0.102997 Akaike info criterion −1.702865

Sum squared residuals 5.887696 Schwarz criterion −1.679615
Log likelihood 478.0992 F-statistic 61.12074
Durbin–Watson stat. 1.971937 Prob(F-statistic) 0.000000

Notes: The dependent variable; differences of exponential correlations between the equity euro mar-
ket and the equity US market. Explanatory variables are DEURV: differences of exponential volatility
of the equity euro market; DUSDV: differences of exponential volatility of the equity US market; and
MA(1): moving average of first-order component. Number of observations=558 daily changes.

For all the three correlations we also estimated the model:

ρA,Bt=α+β 1 ·σAt− 1 +...+βn·σAt−n

+γ 1 ·σBt− 1 +...+γn·σBt−n+εt (12.2)

wherenis a time lag which was set up to 10 days during regression stepwise
search.
To model the regression equation, in many cases it was useful to introduce
a moving average component:

ρA,Bt=α+β 1 ·σAt− 1 +...+βn·σAt−n

+γ 1 ·σBt− 1 +...+γn·σBt−n

+θ 1 εt− 1 +...+θqεt−q+εt (12.3)

where the order of the MA term was generally set to one.
Table 12.1 shows a relative capability (R-squared is around 18 percent) to
explain correlation changes through volatility variables, even when a mov-
ing average factor was selected. All variables demonstrate a high value of the
studentt-test, that is a significant statistical contribution. The sum squared
of errors is by the way 80.5 percent lower than the complete time series.
As for the residuals, depicted in Figure 12.4, the Durbin–Watson (DW)
statistic which measures the serial correlation was very close to 2.^1
For the Euro area market and the USA we selected the first 10 percent of
higher correlation changes (56 data points). We then estimated the regres-
sion equation (12.2) as previously described and the results are displayed