Total error of predicting a new observation

Question

A model forecasts tomorrow's return. From how it was fit, its estimated response has bias 0.2 and variance 0.05. On top of that, tomorrow's actual return has its own irreducible randomness with variance $\sigma^2 = 0.8$ around the true mean.

The expected squared error of predicting a brand-new observation is $\text{bias}^2 + \text{variance} + \sigma^2$ .

What is the expected squared prediction error?

Accepted Answer

The idea. Two different things can go wrong when you predict a new outcome. First, your model's estimate of the average outcome may be off (that is estimation error: bias and variance). Second, even a perfect estimate of the average cannot predict the specific new draw, because the outcome itself is random (that is the irreducible noise). Both add to the total. Add the three parts. 	ext{Prediction MSE} = 	ext{bias}^2 + 	ext{variance} + \sigma^2 = (0.2)^2 + 0.05 + 0.8 = 0.04 + 0.05 + 0.8 = 0.89. Notice how much of the error, 0.8 out of 0.89, is the irreducible noise of the future itself. Ev…

Total error of predicting a new observation

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