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I'm doing a regression with a couple of highly correlated variables. I don't really care about the significance of individual estimators because I'm only using them to build an index. As it is often the case when there is multicolinearity, i get a estimator with the wrong sign which is not a problem. The thing is I wasn't expecting that the estimators would be statistically significant. Does that mean that there is a problem with my variance estimation or is it possible to find statistically significant coefficients in my multiple regression and have completely different coefficients when I regress each variable individually?
I'm asking the questions because I don't want that people see my regression output and try to interpret the coefficients even if they are statistically significant.

Thank you,

is it possible to find statistically significant coefficients in my multiple regression and have completely different coefficients when I regress each variable individually?

Entirely possible. However, it will take some arguing to convince anyone that the single variable coefficients are preferred to the multiple regression.