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Figure 13.2: Histogram of the Transform Probability

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



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Figure 13.2: Histogram of the Transform Probability

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