Two groups, two different residual spreads

Question

You regress daily returns on a sentiment score for a universe that mixes large-cap and small-cap stocks. When you color the residuals by group, the large-cap residuals form a tight band around zero while the small-cap residuals scatter three times as wide, though both groups are centered on the line.

What does the two-tier scatter indicate, and is your slope biased?

Accepted Answer

It is heteroskedasticity, the grouped kind. The error variance is not constant: it is small for large caps and large for small caps. Constant variance would require both groups to share the same spread. Because the variance differs by category, the errors are heteroskedastic (this specific pattern is sometimes called groupwise heteroskedasticity). The slope is unbiased. Both groups are centered on the line, so the zero-mean-error condition holds and the estimated sentiment effect is not biased. Only the standard errors are wrong, because the classical formula pretends both groups are equally …

Two groups, two different residual spreads

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