Can I ignore multicolinearity problem if all the regression coefficients are highly significant? Can I ignore multicolinearity problem if all the regression coefficients are highly significant?
My data is large enough (i.e. I have several regression models where each of the data points for them ranges from 2958 to 11646 data points for every each 6 independent variables. so it is 6 times of these 2958 - 11646 data points for each independent variable to count the total number of data points) and all the resulting coefficients are significant enough in less than 0.01 level. The only thing I see is that one of the variable has the correlation of 0.9 (i.e. the correlation value of one variable to another one is 0.9 but I do not want to remove either of them.).
I am trying to see on unit increase effect of this variable while keeping all other variables constant. Can I keep this variable?
Besides, if I delete one of the variable with high VIF which is between 13 anad 14, all the other VIF are safe but the intercept becomes insignificant for all cases
I am also referring to the following website comment:
http://www.researchconsultation.com/multicollinearity-multiple-regression.asp
So, in sum, my ultimate goal is to use the final output from the logistic regression model generated from the independent variables and one binary dependent variable. If so, do you think I can ignore the multicolinearity problem?
 A: You can ignore multicollinearity for a host of reasons, but not because the coefficients are significant. In fact, it's one of the issues and manifestations of the multicollinearity issue when you have two or more variables which highly significant when put into the regression together, and not significant at all when added one by one.
For instance, in econometrics you'll get very significant coefficients if you have cross exchange rates, e.g. British pound to dollar, Euro to Dollar and Swiss mark to Dollar, while individually they may not be significant in your model.
A: Luckily, there is a diagnostic tool called the variance inflation factor (VIF) that allows you to assess whether you have a multicollinearity problem. Usually, VIF scores > 10 are a cause for concern. 
A: What is your ultimate goal? Are you going to use the model to make predictions on the mean response? If so, correlation is not a problem. However, if you want to make inference then you have to think about it.
I woulg suggest to look up VIF (variable inflation factor) and see whether there is really multicollinearity.
p.s. I could not comment since I've just signed up and have to reach minimum number of reputations.
