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For my research I am looking into the relationship between an outcome (Y) and a predictor (X) as follows:

$$Y = X + e$$

where $e$ is the error term.

Because there might be reverse causality I am employing a 2 stage estimation strategy. My first stage being:

$$X = Z + u$$

where $u$ is an error term.

Now I was all concerned about the exclusion restriction and came up with a series of robustness checks. Instead the main critique I received concerned reverse causality in the first stage.

Of course I looked into this when I started this project and as far as I am aware there is two conditions that should be met for the instrument to be valid:

  1. relevance: Z should be correlated with X, or $corr(Z,X)= 0$;
  2. exogeneity: Z should not be correlated with unobserved factors influencing Y, or $corr(Z,e)\neq 0$.

Am I overlooking something? Or are there recent developments that I am not aware of maybe?

Any guidance much appreciated!

o_v
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    I have been wondering the same thing some time ago, and the answer is yes, it is a problem. IV estimator is biased. Take a look at the discussion under the question I posted above. – cure Feb 05 '21 at 14:08
  • Thanks a bunch. I remember seeing your question a while ago, but could not find it anymore. – o_v Feb 05 '21 at 14:18
  • But also please take a look at this question: https://stats.stackexchange.com/questions/472358/simultaneity-in-causal-diagrams Simultainety (or reversed causality) have not been solved directly in the literature of causal diagrams, so there is no operator as (as in my previous question), this has to be disentangled with the usage of the time, I think. – cure Feb 05 '21 at 14:39

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