Two portfolios, unequal return spreads: Welch's t-test

Question

An analyst compares monthly returns (in percent). Portfolio X: 36 months, mean 1.2, sample standard deviation 2.0. Portfolio Y: 24 months, mean 3.9, sample standard deviation 5.0. The volatilities differ sharply, so use Welch's test.

Run Welch's two-sample t-test at the 5% level (two-sided): do the mean returns differ? Report the t-statistic and its approximate degrees of freedom.

Accepted Answer

Hypotheses: H0: \muX = \muY versus H1: \muX 
eq \muY. Standard error of the difference: 	ext{SE} = \sqrt{\frac{sX^2}{nX} + \frac{sY^2}{nY}} = \sqrt{\frac{4}{36} + \frac{25}{24}} = \sqrt{0.1111 + 1.0417} = \sqrt{1.1528} = 1.0737. Test statistic: t = \frac{1.2 - 3.9}{1.0737} = \frac{-2.7}{1.0737} = -2.52. Welch degrees of freedom: 	ext{df} = \frac{1.1528^2}{\dfrac{0.1111^2}{35} + \dfrac{1.0417^2}{23}} = \frac{1.3289}{0.0003527 + 0.0471770} \approx 28. The two-sided 5% critical value for about 28 degrees of freedom is roughly 2.048. Since -2.52 = 2.52 2.048, reject H0. The p-value is about 0.…

Two portfolios, unequal return spreads: Welch's t-test

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