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I have an initial regression of Y on X and Z. Both of my coefficients on X and Z are non-zero and strongly statistically significant. X and Z are correlated but I am told collinearity shouldn't be an overwhelming issue, OVB would be a stronger one, so I should keep both variables. I can interpret the coefficient on X as the effect of X on Y controlling for Z, and likewise can interpret the coefficient on Z as the effect of Z on Y controlling for X.

Now, at the same time, I am considering using Z as an instrument for a regression of Y on X. I have read on several posts on this forum that the fact that coefficient on Z is non-zero in the initial regression above doesn't invalidate the possibility that Z is a good instrument (the same DAG is often posted, which I have to say I do not fully understand).

For me there is a contradiction between these two assertions. Is there really a contradiction or have I misunderstood something? My worry is that as the coefficient on Z in the initial regression is non-zero and highly significant, it does reduce its validity as an instrument given I have already controlled for X in that regression, so the effect I'm estimating is in theory a direct effect of Z on Y independent of X, which seems fairly large. Anyone can help me build an intuitive argument (in words) about why the positive coefficient on Z is actually fine (if this is indeed the case)?

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If you are considering using Z as an instrument for X, then you think that a regression of Y on X is biased. The same goes for a regression of Y on X and Z, so you cannot trust the results that tell you that the coefficient on Z is non-zero.

If you have a second valid instrument for X, say W, then you can regress Y on X and Z, with W as an instrument for X. In this case, an non-zero coefficient on Z would indicate that Z has a direct effect on Y and is not a valid instrument.

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