Two huge coefficients with opposite signs that nearly cancel

Question

Your two-factor regression returns coefficients of about $+180$ and $-176$ on two closely related growth factors, so the net effect is small. Adding a handful of new observations swings these to $+140$ and $-137$ . The fitted values barely move throughout.

What explains the enormous, offsetting, unstable coefficients while the fit stays steady?

Accepted Answer

The diagnosis: multicollinearity. Two closely related regressors share almost all their variation, so the model has a nearly unidentified direction, the difference between the two factors. Least squares can load up a huge positive coefficient on one and a huge negative coefficient on the other; their difference is what the data cannot pin down, so it takes tiny amounts of information to move both coefficients a lot. Why the fit is stable but the coefficients are not. The combined effect of the two factors (their sum, roughly) is well determined, which is why the fitted values and R^2 hardly c…

Two huge coefficients with opposite signs that nearly cancel

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