Blend when the biased forecast is quite tilted

Question

Two independent forecasts of the same quantity. Forecast A is unbiased with variance $\sigma_A^2 = 10$ . Forecast B is off by $b = 3$ on average and has variance $\sigma_B^2 = 3$ . You blend them as $w\,A + (1-w)\,B$ .

What weight $w$ on forecast A minimizes the mean squared error of the blend?

Accepted Answer

The idea. Judge each forecast by its total error. The unbiased forecast's total error is just its variance; the biased forecast's total error must include its squared bias. Effective error of each. - Forecast A: \sigmaA^2 = 10 (unbiased). - Forecast B: \sigmaB^2 + b^2 = 3 + 3^2 = 3 + 9 = 12. Weight inversely to effective error. With \beta = 12, wA = \frac{\beta}{\sigmaA^2 + \beta} = \frac{12}{10 + 12} = \frac{12}{22} = \frac{6}{11} \approx 0.545. Now A gets the larger share, because B's bigger bias (3, contributing 9 once squared) makes it the less accurate forecast overall despite its low va…

Blend when the biased forecast is quite tilted

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