Weighting aggregated data by known population sizes

Question

You regress a per-region average outcome (say average customer spend) on regional characteristics. Each region's average is computed over its population $P_i$ , which ranges from tiny rural regions to huge metros. A region's average has variance roughly $\sigma^2 / P_i$ .

Explain how you would fit this, why an unweighted regression is inefficient, and what weighting achieves.

Accepted Answer

The problem with unweighted OLS. An unweighted regression gives a small rural region, whose average is based on a few hundred people and therefore very noisy, the same influence as a metro region whose average is rock-solid over millions. That is inefficient: the fit is swayed by the noisiest rows as much as by the most reliable ones. OLS stays unbiased, but its standard errors are needlessly wide (and the classical ones are also wrong, since the variance is not constant). Weight by population. The variance of region i's average is \sigma^2 / Pi, so weight each region by the inverse of its va…

Weighting aggregated data by known population sizes

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