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

A validator applies the Berkowitz test to a VaR model. She transforms PIT values using the inverse standard normal CDF to get z_t. Under the null hypothesis of a correct model, which joint restriction is tested on z_t modeled as an AR(1): z_t − μ = ρ(z_{t−1} − μ) + σε_t, and what is the likelihood ratio's approximate asymptotic distribution?

The Berkowitz test jointly restricts the mean to 0, the standard deviation to 1, and the autocorrelation to 0, so the transformed series is i.i.d. standard normal. The likelihood ratio statistic is therefore approximately chi-square with 3 degrees of freedom under the null.

  1. Aμ = 0, σ = 1, ρ = 0; chi-square with 3 degrees of freedomCorrect
  2. Bμ = 0, σ = 1; chi-square with 2 degrees of freedom
  3. Cρ = 0 only; chi-square with 1 degree of freedom
  4. Dμ = 1, σ = 0, ρ = 0; chi-square with 3 degrees of freedom

Explanation

Berkowitz tests that the transformed series is i.i.d. standard normal: mean 0, variance 1 (σ = 1), and no autocorrelation (ρ = 0). Three restrictions give a chi-square with 3 degrees of freedom. Option B omits the independence restriction, and C tests only independence.

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