Adding up regression coefficients in the presence of multicollinearity and reversal of signs? My time series model suffers from multicollinearity between two independent variables. 
When taking one of the variables out of the model I obtain a coefficient of 0.08 for variable 1 but when including  both variables simultaneously the sign reverses and I get a coefficient of -0.33 for variable 1 and 0.45 for variable 2. 
My professor said that if both variables were perfectly correlated the sum of both coefficients would be 0.12 and that 0.08 was therefore a sign for multicollinearity. Does anyone know a paper confirming what she said? 
Also when running the regression exluding variable 1, I was expecting the coefficient for variable 2 to be somewhat similar to the coefficient of variable 1 (using the same logic as above, i.e. adding up the coefficients should yield a coefficient that is similar to running the regression alone) but the coefficient is very different (0.3). Can somepne explain why this might be the case? 
 A: Why don't you check the variance inflation factors, if you are worried about multicollinearity? To me, your results indeed sound like multicollinearity, but there is no way to check for this by eyeballing, i.e. you cannot simply state that multicollinearity is an issue due to the behaviour of coefficients or standard deviations. Neither will your correlation coefficients as I don't know any person who agree what an acceptable correlation coefficient should be. You must check the variance inflation factors to judge whether multicollinearity is a serious issue in your model. Any decent statistical software has an option of inspection the variance inflation factors. Typically, these need to be below 5, at least for your interest variable. Otherwise, multicollinearity could be problematic. If you find that multicollinearity is, indeed, present in your model, then check the Wikipedia-page for Multicollinearity, in particular the "Remedies for multicollinearity"-section. There are many strategies for dealing with multicollinearity, once you have realised that multicollinearity is a problem in your statistical model. 
A: Strange behavior in coefficients and SE can be quite indicative of problems with multicollinearity, particularly when sign-switching of coefficients occurs upon adding new predictor variables (see here for a similar discussion).
If you think about it, we are actually fitting a plane through the cloud of data points when we have 2 predictors. If $x_1$ and $x_2$ are highly collinear, then the plane is fitting to a cloud of points that resembles a line or a tube instead of a dispersed cloud (as would be the case with less correlated variables). This results in instability and sign-switching!
I'm not sure about what your professor said regarding summing coefficients to as a sign of collinearity. However, I see three ways to diagnose multicollinearity in this case (two "art", one "science"):


*

*Plotting $x_1$ ~ $x_2$ and checking correlation coefficients. This should be your first step in exploring the relationship among predictors. (Although as another user alluded to, there is no hard cutoff for what is acceptable; this varies by field.)

*Sign-switching of coefficients when a new predictor is added, or large changes in SE. This is evidence of instability in the model which could be caused by collinearity. This might lead you to #3.

*Look at variance inflation factor (VIF) for a more quantitative measure of multicollinearity. (Although there is still some "art" to deciding where the cutoff is.)

