# Solved – What advantages do “internally studentized residuals” offer over raw estimated residuals in terms of diagnosing potential influential datapoints

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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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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