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In simple linear regression, the variance of the residuals is given by $Var(e_i) = \sigma^2\Big(1 - \dfrac{1}{n} - \dfrac{(x_i-\bar{x})^2}{n\cdot s_x^2}\Big)$. This is obviously not constant.

A commonly used diagnostic plot is the residual vs predictor plot, in which we plot the $e_i$ versus $x_i$. And apparently, this diagnostic plot is satisfactory, if the plot is a (more or less) constant-width blur of points around a straight, flat line at height zero.

I get the 'height zero' part (reason being that $E[e_i]=0$ ). but i do not understand why residuals should roughly form a "horizontal band" around the $\text{residual}=0$ line. The residuals are not error terms. so just because the error terms have been assumed to be of constant variance does not mean that residuals will also have constant variance.

I mean it can happen that as i increases $x_i$ increases and hence $Var(e_i)$ decreases , in which case the residual vs predictor plot would be funnel shaped.

So, why should residual vs predictor plot form a "horizontal band" around the $\text{residual}=0$ line?

edit:- in response to @whuber's comment, to support my claim, i am posting an image that is an excerpt from the book applied regression analysis by draper and smith. enter image description here

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  • $\begingroup$ Roughly is the key word. The ideal behind this plot is that if you've captured the effect of the predictor in about the right kind of way, then it should be flat on average. But nothing is a free lunch here, as there could be other problems too. $\endgroup$
    – Nick Cox
    Commented Aug 28, 2023 at 15:01
  • $\begingroup$ What i am trying to argue is that assuming all assumptions of the simple linear regression model are satisfied and $x_i$ increases as $i$ increases then the residual vs predictor plot should not even roughly form a horizontal band. It should instead form a funnel around the line residual=0 $\endgroup$
    – abhishek
    Commented Aug 28, 2023 at 15:28
  • $\begingroup$ This question is based on an erroneous claim: the prediction bands are hyperbolae, not "flat lines." See, for instance, the plot at stats.stackexchange.com/a/18467/919. $\endgroup$
    – whuber ♦
    Commented Aug 28, 2023 at 15:39
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    $\begingroup$ That image from D&S is purely a schematic. That's why it's called an "impression." $\endgroup$
    – whuber ♦
    Commented Aug 28, 2023 at 16:24
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    $\begingroup$ @abishek It's common to adjust residuals for the effect of the "design" (the effect on variance as $x$ moves further from $\bar{x}$, and the equivalent effect in multiple regression) by dividing the residuals each by its own standard error. Most default displays of residuals incorporate this adjustment. $\endgroup$
    – Glen_b
    Commented Aug 28, 2023 at 22:01

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