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I remember reading about this somewhere, but I cannot find the source anymore and I'm not sure what the benefit of doing it would be. Do the results of the regressions have anything in common (e.g. standard errors of the coefficients, intercept, r-squared, SST)?

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The sum and difference are just special cases where the coefficients are fixed at particular values, either exactly equal or equal in absolute value, respectively. This is immediate from writing out the equation: $$ \begin{align} y&=\beta_0 + \beta_1 (x_1 + x_2) \\ &=\beta_0 + \beta_1 x_1 + \beta_1 x_2 \end{align} $$ and analogously for the difference $x_1 - x_2$.

There’s no general reason to believe that this is a good model, hence people fit regressions that allow the coefficients to vary. It's possible to contrive some special scenario where there's a strong reason to believe this (perhaps the laws of physics or chemistry imply such a constraint), but that's not a general problem.

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