I'm currently testing the homoscedasticity assumption of multiple linear regression. This can be done by plotting studentized residual plots. The problem with residual plots is even if the variance of the errors is constant, the variance of residuals is not.
My textbook says that using studentized residual-plots instead of residual plots solves this problem. But how?
I think that it is because when fitting the line, we see only the observations Y and not their true value. The fitted line has smaller residual for larger values of x, since larger x-vales have grater leverage. So, the variance of residuals at the end of my fitted line is smaller than in the middle. This leads to heteroscedasticity, even though with "true" Y-values my line would not lead to heteroscedasticity. Studentizing takes (somehow) the leverage of points into account and standardizes them.
Is my explanation correct and why is standardizing the variance needed?