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If theory points to multiple possible instruments that can be used, how can I choose which one of them is the best for 2SLS?

I read that good instrument should be significant and correlated with the endogenous variable. Does that imply that when I am preforming an auxiliary regression to see whether the instruments are good, I can compare them on the basis of their significance and $R^2$ of the auxiliary regression?

Like for example if I have two arbitrary IV variables,(lets call them $Z_1$ and $Z_2$ for the lack of better names) and in the auxiliary regression $Z_1$ has $p-value: 0.001$, and the auxiliary regression has $R^2: 0.25$ and $Z_2$ has $p-value: 0.02$, and its auxiliary regression has $R^2: 0.19$, does it mean that $Z_1$ is better instrument than $Z_2$? Or is this interpretation wrong?

Are there any other IV quality criteria to consider when picking the right IV?

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Suppose you have several valid instrumental variables. Then you can think of the following when choosing among them:

  1. If they capture the same local average treatment effects (i.e. they generate the same set of compliers), you can perhaps choose both and conduct an overidentifying restrictions test, or choose the one that has the strongest first stage (largest F-statistic).

  2. If they do not capture the same local average treatment effect, then you should compute both IV estimators, and compare their value. Try to explain why they give different or the same answers by thinking about the local average treatment effects that they induce.

You cannot simply base your choice of instrument on the $R^2$.

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