Weighting when you are told standard errors

Question

Two independent unbiased estimates of a fund's true beta are reported with standard errors 2 and 3 (not variances). You blend them as $w \cdot (\text{first}) + (1-w)\cdot(\text{second})$ .

What weight $w$ on the first estimate minimizes the variance of the blend?

Accepted Answer

The idea. The inverse-variance rule works on variances, but precisions here are given as standard errors. A standard error is the square root of a variance, so square them first. Convert. Standard error 2 means variance 2^2 = 4; standard error 3 means variance 3^2 = 9. Minimize. Weight the first estimate inversely to its variance: w1 = \frac{	ext{Var}2}{	ext{Var}1 + 	ext{Var}2} = \frac{9}{4 + 9} = \frac{9}{13} \approx 0.692. A common mistake is to weight by 3/(2+3) = 0.6 using the standard errors directly. That is wrong: precision scales with the inverse variance, not the inverse standard …

Weighting when you are told standard errors

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