Measurement error in independent variable may not really matter? In OLS, since measurement error in an independent variable biases its coefficient toward 0, if the coefficient ends up significant (with the expected sign), then is it true that the measurement error doesn't really matter in this case?
 A: If I understand your question, you want to know whether you can safely fit a model you know to be wrong because it is conservative.
Your primary question is: is it conservative?
The coefficient of the OLS regression line will be smaller relative to the coefficient of the principle components regression line.  But that is not the only component of the significance test.  What will the SE of the coefficient do?  
It seems certain that the coefficient SE in the OLS case will shrink relative to the PC coefficient SE because you are ignoring a (potentially) huge chunk of error which does not get built into the SE estimate.
If the SE shrinks to zero faster than the coefficient shrinks to zero, you could get a significant result in the OLS and not the PC despite the shrunken coeffiecient.
Moreover, what will the results be relative to "truth"?  And how would it fare in power?
This looks like an easy simulation to write.
I always meant to read Wayne Fuller's book (yeah, yeah, I know there are several newer ones but I heard him give a lecture once and his voice is still in my head "Mayzurement Ehrrur Mawdulls"...that sounds like I'm mocking him when actually I love accents and had not been exposed to one like his before).
