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Consider a linear regression $$ y=X\beta+\varepsilon. $$ Residuals $e:=y-X\hat\beta$ are often used as substitutes for the unobserved model errors $\varepsilon$ for validating assumptions such as homoskedasticity of $\varepsilon$, normality of $\varepsilon$ and other.

When the model errors $\varepsilon$ are homoskedastic with variance $\sigma^2_\varepsilon$, the residuals $e$ have unequal variances: $\text{Var}(e)=\sigma^2_\varepsilon(I-H)$ where $I$ is an identity matrix and $H:=X(X^\top X)^{-1}X^\top$ is the hat matrix. (For the same reason, the residuals are also correlated.)

The heteroskedasticity in $e$ can be "corrected for"/"undone" by using (internally or externally) studentized residuals $\tilde{e}_{int}:=\frac{e}{\hat\sigma_{int}\sqrt{1-h_{ii}}}$ or $\tilde{e}_{ext}:=\frac{e}{\hat\sigma_{ext}\sqrt{1-h_{ii}}}$ where $\hat\sigma_{int}$ and $\hat\sigma_{ext}$ are internal and external estimates of error variance, respectively.

This heteroskedasticity correction seems to come at zero cost. No estimation or approximation errors seems to be introduced this way, aside from uniform (across data points) scaling due to imperfect estimates $\hat\sigma_{int}$ and $\hat\sigma_{ext}$, and the computational cost is low. This might suggest we should routinely use studentized residuals instead of raw residuals for regression diagnostics, because this way we remedy the problem of heteroskedasticity of $e$ without really sacrificing anything.

Question: Are there any reasons for not using $\tilde{e}_{int}$ or $\tilde{e}_{ext}$ instead of $e$ as a substitute for the true unobserved $\varepsilon$ whenever doing model diagnostics regarding homoskedasticity, normality and other common assumptions/conditions?

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$H_{ii}$ is small for large $n$

The magnitude of the diagonal of the hat matrix $H$ decreases quickly with the increase of the number of observations and scales as $1/n$. If we have the matrix $X$ such that the columns are perpendicular then

$$H_{ii} = \frac{X_{i1}^2}{\sum_{j=1}^n X_{j1}^2} + \frac{X_{i2}^2}{\sum_{j=1}^n X_{j2}^2} + \dots + \frac{X_{ip}^2}{\sum_{j=1}^n X_{jp}^2} $$

The mean of the diagonal will be equal to $p/n$*. So the size of the inhomogeneities that are due to the contribution of the diagonal of the hat matrix $H_{ii}$, is of the order of $\sim p/n$.


Diagnostic plots are often with large $n$

For a diagnostic plot one often has a large $n$ (because few points do not really show much of a pattern) and then the contribution of $H_{ii}$ to the variance of $e_i$ will be small and the variance of the $e_i$ will be relatively homogeneous. Or at least the $H_{ii}$ won't contribute much to inhomogeneity.

The effect of $H_{ii}$ is negligible and the reason to not use studentized residuals is simplicity.


*The trace of the projection matrix equals the rank of $X$, and since the rank is often the number of columns $p$ we have $$\sum_{i=1}^n H_{ii} = p$$ Which means that the average $H_{ii}$ is equal to $p/n$

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  • $\begingroup$ Thank you, that is helpful. $\endgroup$ Commented Sep 24, 2021 at 5:11

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