The reason I ask this is because it seems that internally studentized residuals seem to have the same pattern as raw estimated residuals. It would be great if someone could offer an explanation.

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#### Best Answer

Assume a regression model $bf{y} = bf{X} bf{beta} + bf{epsilon}$ with design matrix $bf{X}$ (a $bf{1}$ column followed by your predictors), predictions $hat{bf{y}} = bf{X} (bf{X}' bf{X})^{-1} bf{X}' bf{y} = bf{H} bf{y}$ (where $bf{H}$ is the "hat-matrix"), and residuals $bf{e} = bf{y} – hat{bf{y}}$. The regression model assumes that the true errors $bf{epsilon}$ all have the same variance (homoskedasticity):

The covariance matrix of the residuals is $V(bf{e}) = sigma^{2} (bf{I} – bf{H})$. This means that the raw residuals $e_{i}$ have different variances $sigma^{2} (1-h_{ii})$ – the diagonal of the matrix $sigma^{2} (bf{I} – bf{H})$. The diagonal elements of $bf{H}$ are the hat-values $h_{ii}$.

The truely standardized residuals with variance 1 throughout are thus $bf{e} / (sigma sqrt{1 – h_{ii}})$. The problem is that the error variance $sigma$ is unknown, and internally / externally studentized residuals $bf{e} / (hat{sigma} sqrt{1 – h_{ii}})$ result from particular choices for an estimate $hat{sigma}$.

Since raw residuals are expected to be heteroskedastic even if the $epsilon$ are homoskedastic, the raw residuals are theoretically less well suited to diagnose problems with the homoskedasticity assumption than standardized or studentized residuals.

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