Skip to content

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

An analyst transforms PIT values u_t into z_t = Φ^-1(u_t) and wants to test the joint hypothesis of correct distribution and independence using a likelihood-ratio test, as in the Berkowitz approach. The unrestricted model is z_t - μ = ρ(z_{t-1} - μ) + ε_t with ε_t ~ N(0,σ²). What is the null hypothesis and the degrees of freedom of the LR statistic?

The Berkowitz likelihood-ratio test has a null that the transformed series is i.i.d. standard normal: mean zero, variance one and zero autocorrelation. Three parameters are restricted, so the statistic is chi-square with three degrees of freedom.

  1. Aμ = 0, σ² = 1, ρ = 0, with 3 degrees of freedomCorrect
  2. Bμ = 0 and σ² = 1 only, with 2 degrees of freedom
  3. Cρ = 0 only, with 1 degree of freedom
  4. Dμ = 0, σ² = 1, ρ = 1, with 3 degrees of freedom

Explanation

If forecasts are correct, z_t are i.i.d. standard normal: mean zero, variance one, no autocorrelation. The unrestricted model has three parameters (μ, σ², ρ) and the null fixes all three, so the LR statistic is chi-square with 3 degrees of freedom. Testing only mean and variance omits the independence restriction.

Did you get it right without looking?

One question tells you little. A timed set on Beyond Exceedance-Based Backtesting of Value-at-Risk Models shows your real accuracy, how long you take and where you lose marks.

Practise Beyond Exceedance-Based Backtesting of Value-at-Risk ModelsFree diagnostic mock

More Beyond Exceedance-Based Backtesting of Value-at-Risk Models questions