A VaR violation at any level should be independent from a VaR violation at any other level so that, for example, a violation of a portfolio's 5% VaR today should not portend a violation of the portfolio's 1% VaR tomorrow. In short, VaR violations at all levels should be independent from each other and follow the uniform distribution on the interval between 0 and 1.
It is worth comparing the observed (PIT), pt+1, to the hit indicator, It+1. The reported PIT provides a quantitative and continuous measure of the magnitude of realized P&L while the hit indicator only signals whether a particular threshold was exceeded. In this sense, the series of reported probability integral transforms provides more information about the accuracy of the underlying risk model. A series of reported PIT that accurately reflects the actual P&L distribution exhibits two key properties.
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 assume that 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.
A popular test is the Kolmogorov-Smirnov (K-S) test. It is a nonparametric test that can be used to compare a sample with a reference probability distribution. The Kolmogorov-Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution.
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