The evaluation of the predictive power
across all possible confidence levels
can be carried out with the help of
a transformation of the empirical distribution
. If
is the true distribution function of the loss
within the holding period
, then the random quantity
is (approximately) uniformly distributed on
.
Therefore we check the values
for
,
where
is the empirical distribution. If the prediction quality of the model is adequate, these values
should not differ significantly from a sample with size
from a uniform distribution on
.
The P-P plot of the transformed distribution against the uniform
distribution (which represents the distribution function of the
transformed empirical distribution) should therefore be located
as closely to the main diagonal as possible. The mean squared
deviation from the uniform distribution (MSD) summed over all
quantile levels
can serve as an indicator of the predictive power of a quantile-based
risk measure like VaR.
The
XFGpp.xpl
quantlet creates a P-P plot and calculates the
MSD indicator.