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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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