FRM Part II · FRM Exam Part II · Backtesting VaR
A risk analyst applies the Kupiec proportion-of-failures likelihood ratio test to a bank's daily VaR exceptions. Under the null hypothesis that the model's true exception probability equals the stated tail probability, what is the asymptotic distribution of the test statistic, and what is the critical value at a 5% significance level?
The Kupiec statistic is asymptotically chi-square with one degree of freedom under the null, because it tests a single restriction on the exception probability. The 5% critical value is 3.84. The two-degree-of-freedom value of 5.99 applies to the conditional coverage test, not Kupiec.
- AStandard normal, with a critical value of 1.96
- BChi-square with 1 degree of freedom, with a critical value of 3.84Correct
- CChi-square with 2 degrees of freedom, with a critical value of 5.99
- DStudent t with T-1 degrees of freedom, with a critical value that depends on sample size
Explanation
The Kupiec unconditional coverage statistic is a likelihood ratio with one restriction (the exception probability equals p). It therefore follows a chi-square distribution with 1 degree of freedom. The 95% critical value is 3.841. The 2-degree-of-freedom value of 5.99 belongs to the conditional coverage test, which adds an independence restriction.
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