Averaging correlated estimates hits a floor

Question

You average n = 10 analyst estimates of the same quantity. Each has variance \sigma^2 = 1, but the estimates are positively correlated with common pairwise correlation \rho = 0.3 (analysts read the same news). What is the variance of the average? How does it compare to the independent case, and what happens as you add more analysts?

Accepted Answer

Variance with correlation. For n equally correlated variables with common variance \sigma^2 and pairwise correlation \rho, \operatorname{Var}(\bar{X}) = \frac{\sigma^2}{n} + \frac{n-1}{n}\rho\sigma^2. Plugging in \sigma^2 = 1, n = 10, \rho = 0.3: \operatorname{Var}(\bar{X}) = \frac{1}{10} + \frac{9}{10}(0.3)(1) = 0.1 + 0.27 = 0.37, so the standard error is \sqrt{0.37} \approx 0.61. Versus independence. If the estimates were independent (\rho = 0), the variance would be \sigma^2/n = 0.1, standard error \approx 0.32. Correlation nearly quadruples the variance of the average: averaging correlate…

Averaging correlated estimates hits a floor

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