Multicollinearity in multiple regression I really hope you can help! I'm in the last stages of my PhD. My supervisor is keen on including all variables in the multiple regressions I am running. Some of the scales are intercorrelated (some with as high as r=.80). Is there a reason to still include them all? 
I've seen some other posts and they mention multicollinearity, but if I'm looking at finding the best predictors out of a related list of possible ones (that are correlated) can I still include them all in the regression to do that? I have mainly non-significant regressions, and some significant beta values. How am I supposed to interpret them meaningfully?
From what I can interpret from other posts, it looks like I should still include them regardless and maybe discuss it after? However, if the regression isn't significant then does it just mean I can't read too much into the significant predictors? Would they still be worth looking at in the future in a follow up? I seem to have low power (n=50). 
My standard errors are a maximum of .5 in some of the regression models, but usually around .2 or .3 in others. Any advice would be greatly appreciated. 
 A: Including variables in your multiple regressions is something that depends on your hypothesis and what you are testing. But you can check the variance inflation factor (VIF) that is used as an indicator of multicollinearity. If VIFs are less that 10, means  multicollinearity is not a problem. If VIFs for two variables is 10 or higher then you have to keep just one of those variables and eliminate the other one. 
Hope this help.  
A: There are several options in dealing with multicollinearity. Excluding highly correlated variables, using Principal Components Analysis to create uncorrelated predictors, using shrinkage methods (i.e. ridge regression), excluding common variance from one of the offending predictors but keeping it in another (by regressing one on another and keeping the residual as a predictor in the main model), and so forth. Using Bayesian methods is yet another approach. 
An excellent paper in this regard can be found here: http://srmo.sagepub.com/view/the-sage-encyclopedia-of-social-science-research-methods/n586.xml
The exact choice depends on the specifics of your research. The paper should be a good starting point. 
