The funnel disappears after you take logs

Question

You regress company revenue on marketing spend for 1,500 firms and see a residual plot that fans out badly as fitted revenue grows. On a hunch you re-run the regression with the natural log of revenue as the response. Now the residual plot is a clean, even band with no fanning.

What was going on, and what did the log transform accomplish?

Accepted Answer

The original errors were heteroskedastic. The fanning plot showed the error spread growing with the level of revenue, the signature of non-constant variance. Concretely, the noise was roughly proportional to the size of revenue: big firms had big dollar errors, small firms had small dollar errors. That is multiplicative (proportional) heteroskedasticity. Why logs flatten it. When the error scales with the level, taking logs converts proportional swings into roughly additive ones. A 10 percent miss is the same distance in log space whether the firm is small or large, so the log residuals have …

The funnel disappears after you take logs

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