Back out the hidden bias from MSE and variance

Question

An estimator of a portfolio's true volatility has mean squared error 29 and variance 20. You are not told its bias directly.

Mean squared error splits as $\text{MSE} = \text{Variance} + \text{Bias}^2$ .

What is the magnitude of the estimator's bias?

Accepted Answer

The idea. The decomposition is one equation with three quantities. Given any two, the third falls out. Here we know the total (MSE) and the scatter (variance), so whatever is left over must be the squared bias. Solve. Rearrange the identity: 	ext{Bias}^2 = 	ext{MSE} - 	ext{Variance} = 29 - 20 = 9. Take the square root: 	ext{Bias} = \sqrt{9} = 3. The estimator is off-center by 3 units. We can recover only the magnitude, not the direction: a bias of +3 or -3 both contribute 9 to MSE, and the decomposition cannot tell them apart. A useful sanity check: variance can never exceed MSE, because …

Back out the hidden bias from MSE and variance

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