Two desks with very different vols: Welch or pooled?

Question

Desk A traded 60 days with mean daily return 0.10\% and standard deviation 0.5\%. Desk B traded 40 days with mean 0.30\% and standard deviation 2.0\%. Do their mean returns differ? Should you pool the variances or use Welch's test, and what does each give?

Accepted Answer

Choose the test. The standard deviations are 0.5\% and 2.0\%, a 4	imes ratio, so the equal-variance assumption behind the pooled test is badly violated. Use Welch's test, which does not assume equal variances. Welch. 	ext{SE} = \sqrt{\frac{sA^2}{nA} + \frac{sB^2}{nB}} = \sqrt{\frac{0.25}{60} + \frac{4.0}{40}} = \sqrt{0.00417 + 0.10} = \sqrt{0.10417} \approx 0.323\%. t = \frac{0.30 - 0.10}{0.323} = \frac{0.20}{0.323} \approx 0.62. The Welch (Satterthwaite) degrees of freedom are 	ext{df} \approx \frac{(0.10417)^2}{\frac{(0.00417)^2}{59} + \frac{(0.10)^2}{39}} = \frac{0.01085}{2.9	imes10^{-…

Two desks with very different vols: Welch or pooled?

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