How to deal with predictors which are not significant, although r-squared is significant? I did factor analysis and found three factors. To examine if which factors significantly affected a certain dependent variable, I added all three factors to a regression model. The correlation coefficients among the three factors are 0.7 to 0.8. The correlation between the factors and the dependent variable was about 0.30 at a 0.1 significant level. The result was that R-squared as the model fit was significant, but each standardization of the coefficients was not. However, using simple regression, all were significant. VIF was below 5 so there is no multicollinearity, I guess. The significant level was the same even after removing one of the factors. How should I analyze this? A stepwise method should be better?
 A: A stepwise method should not be better.  To a first approximation, stepwise regression should never be used (cf., Algorithms for automatic model selection).
You have multicollinearity.  The rule of thumb that you don't have problematic multicollinearity until the VIFs are >10 is just a rule of thumb.  Your variables are still strongly correlated with each other.  My guess is that you have relatively few data, such that the variables are not significant even though the VIFs are <5.  Let's say the VIFs = 4 for all variables.  That means the variances of the sampling distributions are 4x larger than they would be with uncorrelated variables.  That means that the standard deviations of the sampling distributions (i.e., your standard errors) are 2x larger than they would be.  If your N is low enough, a significant variable could become non-significant by doubling its SE.
This is not an uncommon situation.  It is harder to tell which variable is doing what when they are all very similar.  In your case, you can conclude that some combination of your variables (maybe just one, or some two of them, or all of them) is related to the dependent variable, but you don't know which.
