Blending a model forecast with an analyst estimate

Question

You combine two independent estimates of a company's true value. Estimate A (an analyst) is unbiased with variance $\sigma_A^2 = 4$ . Estimate B (a model) is off by $b = 1$ on average but precise, with variance $\sigma_B^2 = 1$ . You blend them as $w\,A + (1-w)\,B$ .

What weight $w$ on the analyst estimate minimizes the mean squared error of the blend?

Accepted Answer

The idea. Compare the two by their total error. The analyst is unbiased, so its total error is its variance. The model is biased, so add its squared bias to its variance. Effective error of each. - Estimate A (analyst): \sigmaA^2 = 4. - Estimate B (model): \sigmaB^2 + b^2 = 1 + 1^2 = 2. Weight inversely to effective error. With \beta = 2, wA = \frac{\beta}{\sigmaA^2 + \beta} = \frac{2}{4 + 2} = \frac{2}{6} = \frac{1}{3} \approx 0.333. The model gets two-thirds of the weight, because even after accounting for its small bias it is twice as accurate as the analyst (total error 2 versus 4). A sma…

Blending a model forecast with an analyst estimate

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