Figure 13.2: Histogram of the Transform Probability
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Backtesting the Entire Distribution
51
• Figure 13.2 shows the histogram of a
sequence,
obtained from taking Ft(RPF,t+1) to be normally distributed
with zero mean and variance d/(d-2), when it should have
been Student’s t(d), with d = 6
• Thus, we use the correct mean and variance to forecast the
returns, but the shape of our density forecast is incorrect
• The histogram check is of course not a proper statistical
test, and it does not test the time variation in
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Backtesting the Entire Distribution
52
• If we can predict
using information available on day t,
then
is not i.i.d., and the conditional distribution
forecast, Ft(RPF,t+1) is therefore not correctly specified either
• We want to consider proper statistical tests here
• Unfortunately, testing the i.i.d. uniform distribution
hypothesis is cumbersome due to the restricted support of
the uniform distribution
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Backtesting the Entire Distribution
53
• We therefore transform the i.i.d. Uniform
to an i.i.d.
standard normal variable
using the inverse cumulative
distribution function, -1
• We write
• We are now left with a test of a variable conforming to the
standard normal distribution, which can easily be
implemented
• We proceed by specifying a model that we can use to test
against the null hypothesis
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Backtesting the Entire Distribution
54
• Assume again, for example, that we think a variable Xt may
help forecast
• Then we can assume the alternative hypothesis
• Then the log-likelihood of a sample of T observations of
.
under the alternative hypothesis is
• where we have conditioned on an initial observation
• Parameter estimates
can be obtained from
maximum likelihood or from linear regression
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Backtesting the Entire Distribution
55
• We can then write a likelihood ratio test of correct risk model
distribution as
• where the degrees of freedom in the 2 distribution will
depend on the number of parameters, nb, in the vector b1
• If we do not have much of an idea about how to choose Xt,
then lags of
itself would be obvious choices
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Backtesting Only the Left Tail of the
Distribution
56
• In risk management, we often only really care about
forecasting the left tail of the distribution correctly
• Testing the entire distribution as we did above, may lead us
to reject risk models which capture the left tail of the
distribution well, but not the rest of the distribution
• Instead we should construct a test which directly focuses
on assesses the risk model’s ability of capturing the left tail
of the distribution which contains the largest losses
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Backtesting Only the Left Tail of the
Distribution
57
• Consider restricting attention to the tail of the distribution
to the left of the VaRpt+1—that is, to the 100 . p% largest
losses
• If we want to test that the
observations from, for
example, the 10% largest losses are themselves uniform,
then we can construct a rescaled
variable as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Backtesting Only the Left Tail of the
Distribution
58
• Then we can write the null hypothesis that the risk model
provides the correct tail distribution as
• or equivalently
• Figure 13.3 shows the histogram of
corresponding to
the 10% smallest returns
• The data again follow a Student’s t(d) distribution with d =
6 but the density forecast model assumes the normal
distribution
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 13.3: Histogram of the Transform Probability
from the 10% Largest Losses
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
59
Backtesting Only the Left Tail of the
Distribution
60
• We have simply zoomed in on the leftmost 10% of the
histogram from Figure 13.2
• The systematic deviation from a flat histogram is again
obvious
• To do formal statistical testing, we can again construct an
alternative hypothesis as in
• for t+1 such that RPF,t+1 < -VaRpt+1
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen