Methodology: VaR Backtest (Kupiec & Christoffersen)

Hit sequence

For each day t the indicator I_t = 1 if PnL_t < −VaR_t (an exception), else 0. Expected exception rate is α = 1 − confidence.

Kupiec POF (LR_uc)

The proportion-of-failures test compares observed exception rate π̂ = x/n to the theoretical α under H0:

LR_uc = −2 · ln( ((1−α)^(n−x) · α^x) / ((1−π̂)^(n−x) · π̂^x) )
LR_uc ~ χ²₁ under H0

Rejecting H0 means the model under- or over-estimates the unconditional VaR rate.

Christoffersen Independence (LR_ind)

Counts the four transitions:

n00 = #{I_{t-1}=0, I_t=0}
n01 = #{I_{t-1}=0, I_t=1}
n10 = #{I_{t-1}=1, I_t=0}
n11 = #{I_{t-1}=1, I_t=1}

π_01 = n01 / (n00 + n01)
π_11 = n11 / (n10 + n11)
π    = (n01 + n11) / (n00 + n01 + n10 + n11)

LR_ind = −2 · ln( L_independent / L_markov )
       ~ χ²₁ under H0

Rejecting H0 means exceptions cluster in time — typical of GARCH-misspecified models that under-estimate risk in volatile regimes.

Joint Conditional Coverage (LR_cc)

LR_cc = LR_uc + LR_ind ~ χ²₂ under H0

A model can pass POF but fail independence (right average rate, wrong clustering), or vice versa. The joint test catches both.

References

Kupiec, P. (1995). "Techniques for Verifying the Accuracy of Risk Measurement Models." Journal of Derivatives 3(2): 73–84. DOI: 10.3905/jod.1995.407942.
Christoffersen, P. F. (1998). "Evaluating Interval Forecasts." International Economic Review 39(4): 841–862. DOI: 10.2307/2527341.
Jorion, P. (2007). Value at Risk: The New Benchmark for Managing Financial Risk, 3rd ed., McGraw-Hill. ISBN: 978-0-07-146495-6.

Limitations

Power is low for n < 250 days; small samples often fail to reject obviously bad models.
Christoffersen tests first-order Markov clustering only; higher-order clustering requires alternative tests (e.g. dynamic quantile by Engle-Manganelli).
Both tests assume the VaR series is exogenous to the PnL series.

How VaR Backtest (Kupiec & Christoffersen) works