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I know that to check the homoscedasticity assumption in OLS regression, we plot residuals vs predicted values. However, Excel provides plots of residuals vs each independent variable. What is the purpose of these graphs, what would be considered an abnormal finding, and what would we do about it?

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Simply: Residuals vs. fitted and residuals vs. independent variables have a similar purpose. While you can catch some forms of heteroscedasticity seeing a typical "funnel" shaped distribution of residuals, that's not all these plots are for. With residuals versus fitted values or predictors, you can also catch "curvilinearity" when the smoothed trend in the mean residual has clear departures from 0, a "swiggly" shape if you can call it that. This is model misspecification.

According to the assumptions of OLS, the residuals are conditionally independent of the fitted values and of the individual predictors. Swiggles mean this is not the case. The problem is that if you identify a particular swiggle vs. a predictor, changing the adjustment of that predictor to match what you see may not solve the issue, because the residual is in the null space of all other predictors.

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  • $\begingroup$ Unless I am misunderstanding (and please let me know if I am), I think this answer is wrong in its present form. The residual vector is not independent of the predictors. Under standard OLS and model assumptions you have $\mathbf{R} \sim \text{N}(\mathbf{0}, \sigma^2 (\mathbf{x}^\text{T} \mathbf{x})^{-1})$. This dependency means that it is preferable to use added-variable plots instead of comparisons of the residuals and individual predictors. Can you please review and correct if necessary, or let me know why I'm wrong. $\endgroup$ – Ben Feb 19 at 4:31

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