Statistical tests
A variety of statistical tests have been proposed to assess whether a reported probability integral transforms (p-values), , accord with the uniformity and independence properties. Just as in the case of assessing the accuracy of a single VaR measure these tests may be conducted individually, either the uniformity or independence property may be tested, or jointly.
The main advantage of these tests over tests based on a VaR measure at a single p level is that they, in principle, can provide additional power to detect an inaccurate risk model. By examining the entire range of probability integral transforms these tests can detect violations of the independence or unconditional coverage property across a range of different VaR levels.
This increased power to detect an inaccurate risk model, however, comes at some cost. One component of the cost comes in the form of an increased informational burden. In particular, in order to transform xt,t+1 to pt+1 one must have access to the entire conditional cumulative distribution function.
Risk models that assume a particular form of the distribution of portfolio losses and gains may be reasonable models of extreme outcomes but may not be useful for characterizing the frequency of more moderate outcomes.
If the underlying risk model is more focused on characterizing the stochastic behavior of extreme portfolio losses these models may be misspecified over a range of P&L’s which are not highly relevant from a risk management perspective. Accordingly, tests that employ the entire series of probability integral transforms may signal an inaccurate model due to this source of misspecification.
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