A funnel-shaped residual plot, what is actually broken?

Question

You regress monthly household spending on household income across 2,000 households. When you plot the residuals against the fitted values, the cloud is a tight sliver on the left and a wide fan on the right: low-income households sit almost on the line, high-income households scatter far above and below it.

What regression problem does this picture reveal, and does it bias your estimated slope?

Accepted Answer

Name it. The fan shape means the spread of the errors changes with the level of the fitted value. The error variance is not constant. That is heteroskedasticity (constant-variance errors would give an even band of the same width all the way across). Does it bias the slope? No. The least-squares slope only needs the errors to average to zero at each level of the regressor (the exogeneity condition). Non-constant spread does not touch that, so your slope and intercept stay unbiased and consistent. Point your finger at the right victim: what breaks is the standard errors, not the coefficients. W…

A funnel-shaped residual plot, what is actually broken?

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