When the mean is useless, match the second moment

Question

You model a return series as normal with mean fixed at $0$ and unknown variance $\sigma^2$ . The only unknown is the spread. From your data the average of the squares is $\frac{1}{n}\sum X_i^2 = 4$ .

Use the method of moments to estimate $\sigma^2$ .

Accepted Answer

The mean is a dead end. Normally the method of moments starts by matching the mean. But here the mean is known to be 0 no matter what \sigma^2 is, so the first moment equation reads "0 = 0" and tells you nothing about the unknown. You must climb to the next moment. Match the second moment. Because the mean is 0, the second moment equals the variance: \mathbb{E}[X^2] = \operatorname{Var}(X) + (\mathbb{E}[X])^2 = \sigma^2 + 0 = \sigma^2. Set that equal to the sample average of the squares: \hat\sigma^2 = \frac{1}{n}\sumi Xi^2 = 4, so \hat\sigma = 2. The general lesson. When a parameter does not…

When the mean is useless, match the second moment

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