Also, the test isn’t defined when there are no consecutive exceedances at all, which is common. As these are inherently rare events, the test has limited power. The test largely depends on the frequency with which consecutive exceedances are experienced. The 0.95 quantile of the χ 2(1,0) distribution is 3.841, so we reject at the. The test is well known, since it was first proposed in an often-cited endorsement of testing for independence of exceedances.Įxtending our earlier notation q * for the coverage of a value-at-risk measure, we defineĪnd –2 log(Λ) is approximately centrally chi-squared with one degree of freedom-that is –2 log(Λ) ~ χ 2(1,0)-assuming. instances when both t –1 i = 1 and ti = 1 for some t. 14.5.1 Christoffersen’s 1998 Exceedence Independence TestĬhristoffersen’s ( 1998) independence test is a likelihood ratio test that looks for unusually frequent consecutive exceedances-i.e. Accordingly, backtests for independence can be judged, among other things, based on how broad their null hypotheses are. There is no way to directly test for independence, so null hypotheses address specific properties of independence-say exceedances not clustering or loss quantiles not being autocorrelated. If a value-at-risk measure fails an independence test, that can cast doubt on coverage or distribution backtest results obtained for that value-at-risk measure. We have already seen that coverage tests assume the former and most distribution tests assume the latter. Independence of exceedances tI and independence of loss quantiles tU are separate forms of independence that might be tested for. Independence tests are a form of backtest that assess some form of independence in a value-at-risk measure’s performance from one period to the next. 14.6 Example: Backtesting a One-Day 95% EUR Value-at-Risk Measure.14.5 Backtesting With Independence Tests.14.4 Backtesting With Distribution Tests.11.6 Shortcomings of Historical Simulation.11.5 Flawed Arguments for Historical Simulation.11.3 Calculating Value-at-Risk With Historical Simulation.11.2 Generating Realizations Directly From Historical Market Data.10.4 Monte Carlo Transformation Procedures.10.3 Quadratic Transformation Procedures.7.4 Unconditional Leptokurtosis and Conditional Heteroskedasticity.5.7 Breaking the Curse of Dimensionality.5.6 Implementing Pseudorandom Number Generators.5.5 Testing Pseudorandom Number Generators.4.8 White Noise, Moving-Average and Autoregressive Processes.3.17 Quantiles of Quadratic Polynomials of Joint-Normal Random Vectors.3.13 Quadratic Polynomials of Joint-Normal Random Vectors.3.8 Bernoulli and Binomial Distributions.3.5 Linear Polynomials of Random Vectors.2.3 Gradient & Gradient-Hessian Approx.1.6 Other Applications of Value-at-Risk.

Kupiec and christoffersen test