What fat tails do to your variance estimate

Question

For general i.i.d. data with kurtosis \kappa = \mu4/\sigma^4, the variance of the sample variance is, for large n, \operatorname{Var}(S^2) \approx \frac{\sigma^4}{n}\,(\kappa - 1). You estimate variance from n = 250 daily returns (about a year) whose kurtosis is \kappa = 7. How much noisier is this estimate than the Gaussian benchmark, and what effective sample size does that correspond to?

Accepted Answer

Fat-tailed value. With \kappa = 7 and n = 250: \operatorname{Var}(S^2) \approx \frac{\sigma^4}{250}\,(7 - 1) = \frac{6\sigma^4}{250} = 0.024\,\sigma^4. Gaussian benchmark. A normal has \kappa = 3, giving \operatorname{Var}(S^2){	ext{normal}} \approx \frac{\sigma^4}{250}\,(3 - 1) = \frac{2\sigma^4}{250} = 0.008\,\sigma^4. Ratio. The noise is inflated by \frac{\kappa - 1}{3 - 1} = \frac{6}{2} = 3, so your fat-tailed variance estimate has three times the variance of a Gaussian estimate on the same 250 points. Effective sample size. To achieve the same precision, a Gaussian sample would need n{\…

What fat tails do to your variance estimate

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