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I am only interested in the causal and unbiased effect of x on y and I have used additional covariates in my model to control for other effects. I have a pretty decent instrument for my potentially endogenous explanatory variable x. Do I have to instrument for other covariates, if I suspect them to be endogenous too. In fact, what about if I just leave the other endogenous variables out of the regression, while just use the instrument for my endogeous variable in consideration?

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I have just given an answer here to a related question with a bit more maths, but here is just the gist of it applied for your case.

In practice I have come across many high grade papers where the authors just instrument for the one variable in consideration, while ignoring a few other dubious other covariates. However, from a purely theoretical standpoint, they should have controlled for all other explanatory factors that are under the suspicion to be endogenous--as long as you think that these explanatory factors are necessary to control for to satisfy the validity assumption of your instrument.

Let's rehash: A proper instrument is one that is correlated with your endogenous regressand after partialling out any other explanatory variable. Further it is uncorrelated with the error term of the original relationship. In your case, one of the explanatory variables related to the instrument is endogenous and potentially biased.

Assuming that your explanatory variable is endogenous too, as long it is relevant in determining Y, it adds to your instrument being informative, which is a good thing. The question is, if the instrument remains valid when controlling for some explanatory variables that are endogenous.

The validity assumption of your instrument practically a question of logic (unless you find a way test it do a thorough identification strategy). Insofar, if you believe that your edogenous other explanatory variable is necessary to partial out the effects in your instrument, then it may harm the validity assumption of your instrument.

Card (1992) instruments "years of education" with "travel distance to school" to determine the return on wages. He assumes that "intelligence" or "ability" are confounding factors that have not been controlled for. "Education" is therefore assumed endogenous as it relates to "ability". It is wickedly conceivable that "distance to school" is related with "age" which is assumed to be related with "wage" but also "ability", as for example, older people live farther away from cities where schools were built but may be less/more able as they are a different generation.

If age is endogenous for some reason and thus biased, it might invalidate your instrument, since you assume you have controlled for age that was necessary to do so to have your instrument being valid, in other words, you wanted distance to be unrelated with ability when keeping age constant.

By how much the validity of the instrument suffers, it is difficult to say. My intuition is that this is not a big thing if one or two other explanatory variables among a vector of several other determinants are endogenous; however, before you jump into conclusions, if you have established instruments for your other explanatory variables, use them by all means. It will add to the credibility of your hypothesis.

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