The Berkowitz Test for Density Forecasts

Most VaR backtests, like Kupiec's, only ask one narrow question: did losses breach the VaR line about as often as promised? That test throws away almost all the information in a full predicted distribution — it only looks at one quantile. The Berkowitz test uses the whole predicted density instead. For each day, it takes the actual realized outcome and plugs it into the model's own predicted cumulative distribution function (CDF) for that day, producing a number between 0 and 1 — this is the probability integral transform. If the model's predicted distribution is exactly correct, these transformed values should be spread uniformly between 0 and 1, and independent across days, no matter how the underlying distribution's shape changed day to day.

Berkowitz then applies one more transform: passing each uniform value through the inverse normal CDF, so that a perfectly calibrated model produces a series that looks like independent draws from a standard normal distribution — mean 0, variance 1, no autocorrelation. A likelihood-ratio test then checks the actual transformed series against exactly that null hypothesis, catching not just miscalibrated tail probabilities but also autocorrelation the simpler VaR-breach count would miss (for instance, a model that gets the average breach rate right but breaches in clusters right before a crash).