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$$Y_t = a + bX_{1,t} + cX_{2,t} + dX_{3,t} + e_t$$

A high $R^2$ in $X_{1,t} = \alpha + \beta X_{2,t} + \gamma X_{3,t} + \varepsilon_t$ will always result in a higher standard error of the $b$ estimator, all else held equal. This is an example of the multicollinearity problem. It does not result in a biased or inconsistent estimator.

Question: Assuming correct specification and the satisfaction of the other assumptions, if my statistical test is powerful enough to detect significance in $\hat{b}$ at an acceptable test size, do I need to continue to worry about multicollinearity?

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Jase
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    If you care that small changes in your input data may result in large changes in your regression, then you need to worry about collinearity. See the many other questions about collinearity on this site. – Peter Flom Dec 27 '12 at 13:31
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    If b and c are collinear and your test leads you to conclude that b is significant, you could still incorrectly fail to reject the null hypothesis for c; it is harder to find significant estimates of both parameters when they are correlated. – david25272 Feb 25 '14 at 23:32

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There are competing factors.

One the one hand, multicollinearity inflates standard errors. On the other hand, removing a variable to remove the multicollinearity can lead to omitted-variable bias, and it is not clear that you are better-off with a narrow standard error for a biased estimate than you would be with the multicollinear variable still in the model. It might even be the case that you wind up with a wider standard error, too, since removing a variable can result in a higher estimated error variance.

You are in an enviable position where the significance is able to shine through the variance inflation resulting from the multicollinearity. In some sense, it doesn’t matter that you had reduced power to reject a false null hypothesis: you were able to do so, so it doesn’t matter how likely such a finding was. However, rejecting in a low-power situation is likely to result in an inflated sense of effect size, so, in that sense, it does matter.

Dave
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