How far to shrink a noisy alpha toward a benchmark

Question

You have a noisy estimate x of a fund's true alpha \mu, with variance \sigma^2 = 9. A benchmark says similar funds cluster near a value b = 5. You shrink your estimate toward the benchmark: \hat\mu = w\,x + (1-w)\,b. Suppose the true alpha is \mu = 8. What weight w on your raw estimate minimizes the mean squared error?

Accepted Answer

The idea. Pulling your estimate toward the benchmark trades one error for another. It calms the noise (good), but it also drags you away from the true \mu toward b, creating a bias (bad). The best weight balances these two. Write the MSE. Let d = \mu - b = 8 - 5 = 3 be the true distance from the benchmark. The shrunk estimate has variance w^2\sigma^2 and bias (1-w)\,d (fully shrinking to the benchmark, w=0, leaves you off by the full distance d). So 	ext{MSE}(w) = w^2 \sigma^2 + (1-w)^2 d^2. Minimize. Setting the derivative to zero gives w^\star = \frac{d^2}{d^2 + \sigma^2} = \frac{9}{9 + 9}…

How far to shrink a noisy alpha toward a benchmark

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