Using a t-Test to Compare Bootstrapped VaR Estimates

Summary

The question asks whether two sets of bootstrapped Value at Risk estimates, generated from returns before and after a recession, can be compared with a t-test. The response says that non-normality in the underlying returns does not automatically rule out a t-test: with a sufficiently large sample, the central limit theorem can make the sampling distribution of the mean approximately normal. The proposed comparison is therefore framed as a test of the difference between average VaR estimates.

This answer is brief and does not assess whether the bootstrap replicates are independent, whether the two VaR samples are paired, or whether their dependence and tail behavior support the test’s assumptions. It also does not distinguish uncertainty in the estimated means from uncertainty in the VaR estimator itself, or suggest alternatives such as bootstrap confidence intervals for the difference. The central limit theorem argument is a useful starting point, not a complete validation of the test for this setup.

Key ideas

• A t-test concerns the distribution of sample means, not a requirement that underlying returns be normal.
• The central limit theorem can support approximate normality of means when the effective sample is sufficiently large.
• Dependence among bootstrap replicates can undermine the usual t-test assumptions.
• The choice of paired or independent comparison and the uncertainty in VaR estimates require further evaluation.

Tags

# Can you use a t-test on bootstrapped Value at Risk (VaR) figures?


# Can you use a t-test on bootstrapped Value at Risk (VaR) figures?












I need to compare VaR before and after the recession.

I have a series of market returns for a period before, and a series of market returns for the period immediately after.

Both have been bootstrapped 500+ times, allowing me to generate 500+ VaR's.

I have put these VaRs in a histogram, and I was wondering if I could do a T-Test to find out if the difference is significant?

I have a feeling I cannot, as the distribution of VaR's is not normal, however this doesn't matter as the T-test takes means which are normally distributed?

Can anyone clarify what I could to do?

Apologies for my lack of knowledge, I'm grateful for your patience.

Thanks!

## Answer by vonjd (score 1)

https://quant.stackexchange.com/a/16040

Using a t-test should be ok because even when the underlying distribution is not normal you have a large enough sample size which justifies the assumption that the distribution of the sample means should be approximately normal due to the Central Limit Theorem.

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