The book Introduction to Statistical Learning by Gareth James et. al. states that a simple linear regression can be modeled as $\hat{y}_i=\hat{\beta}_0 + \hat{\beta_1}x_i$. The residuals are calculated $e_i=y_i-\hat{y}_i$.

They then go on to define the residual sum of squares as:

$$ RSS=e_1^2+e_2^2\dots e_n^2 $$

or equivalently as

$$ RSS=(y_1 - \hat{\beta}_0 - \hat{\beta_1}x_1)^2 + (y_2 - \hat{\beta}_0 - \hat{\beta_1}x_2)^2 +\dots+(y_n - \hat{\beta}_0 - \hat{\beta_1}x_n)^2 $$

My question is if $\hat{y}_i=\hat{\beta}_0 + \hat{\beta_1}x_i$, then why when they substitute into the RSS equation above do I see that it is equal to $\hat{y}_i=\hat{\beta}_0 - \hat{\beta_1}x_i$? Does it not matter because it is being squared? It just seems like for clarity you would substitute it exactly as you just previously defined it. I do not see anything in the errata. Is this a typo or does it just not matter?


As you said, the residuals are defined by :

$e_i = y_i - \hat{y}_i = y_i - ( \hat{\beta}_0 + \hat{\beta_1}x_i ) = y_i - \hat{\beta}_0 - \hat{\beta_1}x_i$

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  • $\begingroup$ Oh my gosh, duh. Sorry, I'm not thinking straight. Thanks for setting me right! Complete lapse in thought haha $\endgroup$ – Dylan Russell Oct 5 '18 at 8:20

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