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FRM Part II · FRM Exam Part II · Beyond Exceedance-Based Backtesting of Value-at-Risk Models

In a Berkowitz test, PIT values are transformed with the inverse standard normal CDF to produce z_t. The model is z_t - mu = rho(z_{t-1} - mu) + epsilon_t, with epsilon_t having variance sigma^2. The null hypothesis is mu = 0, sigma^2 = 1, rho = 0. The log-likelihood is -1,000.0 under the unrestricted model and -1,004.2 under the restricted model. Using the likelihood ratio statistic, what is the statistic and the conclusion at 5% significance (chi-square critical values with 3 df: 7.81)?

The likelihood ratio statistic is 2 times the 4.2 difference in log-likelihoods, or 8.4. This exceeds the 5% chi-square critical value of 7.81 with three degrees of freedom, so the model is rejected. Omitting the factor of two would wrongly suggest it passes.

  1. ALR = 4.2; fail to reject the model
  2. BLR = 8.4; reject the modelCorrect
  3. CLR = 8.4; fail to reject the model
  4. DLR = 2,004.2; reject the model

Explanation

LR = -2(LL_restricted - LL_unrestricted) = -2(-1004.2 + 1000.0) = 8.4. With 3 restrictions, the critical value is 7.81, and 8.4 exceeds it, so the null of a correct model is rejected. Forgetting the factor of 2 gives 4.2, which would wrongly not reject.

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