# Journal of Time Series Analysis

## Extending the Limits of Backtesting via the ‘Vanishing p’‐Approach

### Early View

We derive backtests of value‐at‐risk and expected shortfall forecasts for levels that vanish as a function of the sample size n. In the standard case, the level of the forecasts is assumed to be fixed, leading to χ 2‐limiting distributions of the Portmanteau‐type backtests. We show that for levels vanishing at the order of n −1/2, Poisson‐type limits arise instead. These mimic key features of the test statistics, such as discreteness. Simulations demonstrate that for forecast levels and sample sizes of practical interest, using the Poisson‐type limits leads to much improved size vis‐à‐vis the standard χ 2‐limits.

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