Funnel in the Residuals

You fit household spend on income with OLS. The residuals-versus-fitted plot is a funnel: residuals sit within about $\pm 2$ near fitted values of 20 and within about $\pm 40$ near fitted values of 500. A Breusch-Pagan test rejects its null that the residual variance does not depend on the regressors.

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A funnel means the relationship is non-linear, so adding a quadratic term is the fix.

The coefficients are biased toward the high-variance observations and the fit has to be discarded.

Refitting with heteroscedasticity-robust standard errors (HC0 to HC3) leaves the coefficients unchanged and repairs the inference.

The coefficient estimates are still unbiased: the assumption that broke describes the error variance, not the mean function.

The reported standard errors, t-statistics, p-values and confidence intervals are no longer trustworthy.

$R^2$ is undefined whenever the residual variance is not constant.