The hypothesis test
If the theoretical tail risk TR0 predicted by the model is more negative than the sample tail risk (TR) ̂ then the risk model is said to “‘capture’ the tail risk” or to “provide sufficient risk coverage”. Otherwise, we may describe the risk model as “having failed to capture the tail risk”, based on the empirical tail risk backtest. So in the yellow case would require additional scrutiny of the VaR risk model.
Accordingly, a one-tailed complementary regulatory backtest to check whether the risk model provides sufficient risk coverage may be formulated in terms of z, with the null and alternative hypotheses
We wish to assess the evidence in the observed statistical value z ̂ for H0.
The test should measure how surprising the observed value z ̂ (or, equivalently〖x〗_T=αz ̂) is when H0 is assumed to be true. It is accepted that z ̂ is surprising whenever z ̂ lies in a region of low probability of the distribution of X ̅ in the model for which H0 is true; in terms of the distribution of X ̅ , it means that we should examine the left tail of the distribution.
Explicitly, the null hypothesis is rejected (i.e., the data provides sufficient evidence to reject the assumption of sufficient tail risk coverage) if the realized value of the sample statistic z ̂=α^(-1) 〖x〗_T is significantly less than z0 = -1 under the exponential tail assumption. This is equivalent to x ̅_T=α z ̂ being significantly less than - 0.01.
In this case, we will tend to see at least one of two occurrences in the sample data:
a large number of (not necessarily extreme) negative xt sample values, each representing one of the (not necessarily extreme) VaR violations;
a (not necessarily large) number of extreme negative xt sample values, i.e., extreme VaR violations.
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