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

In the Berkowitz test, PIT values are transformed with the inverse standard normal CDF to give z_t. The z_t are modeled as z_t - mu = rho(z_{t-1} - mu) + epsilon_t, with epsilon_t having variance sigma^2. A likelihood ratio test compares this to the null mu = 0, sigma = 1, rho = 0. How many degrees of freedom does the LR statistic have, and what does rejection imply?

The LR statistic has 3 degrees of freedom, since mu, sigma and rho are each restricted under the null. Rejection means the transformed series is not iid standard normal: its mean, variance or autocorrelation differs, indicating a misspecified forecast distribution or dependence.

  1. A3 degrees of freedom; the mean, variance or serial dependence of the transformed series differs from the standard normal iid nullCorrect
  2. B2 degrees of freedom; only the mean and variance differ from the null
  3. C1 degree of freedom; only the exceedance frequency is incorrect
  4. D3 degrees of freedom; the model's VaR is conservative, never too low

Explanation

Three parameters (mu, sigma, rho) are restricted under the null, so the LR statistic is asymptotically chi-square with 3 degrees of freedom. Rejection means at least one of the mean, volatility or autocorrelation departs from the iid N(0,1) null, so the forecast distribution or its independence is misspecified. It does not indicate the direction of VaR bias.

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