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I am doing an IV regression after OLS. From OLS I get significant results but I want to control for endogeneity and check reverse causality. So when I do IV, the sign of my main explanatory variable changes. That I can explain in theory. But my result is insignificant. Any reason why the significance changes when going from OLS to IV and any ways to handle the problem?

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There are several possibilities as to why the significance might vanish.

  1. the effect you are looking for is simply not there

Sometimes this is also a result, especially when you started to build your regression model in order to test the predictions of a theoretical model.

  1. your instrument is not very strong

If your instrument is not very strongly correlated with the endogenous regressor then the IV estimates become more imprecise, i.e. the standard errors and confidence intervals will be larger.

  1. your instrument is weak

This is the extreme case of the previous point. When your instrument is only weakly correlated with the endogenous regressor (the typical rule of thumb is an F-statistic of 10 or more for the first stage IVs) then this will lead to bias in the IV estimates. The reference for this problem would be Bound et al (1995).

  1. the exclusion restriction fails

You know that a good instrument must be correlated with the endogenous variable but uncorrelated with the structural error. In case you have selected an instrument that does not satisfy the exclusion restriction then IV/2SLS will not give you unbiased and consistent estimates which might bias the IV coefficient towards zero.

What to do now?
Check if your instrument is strong enough and think a bit more about whether the exclusion restriction holds. If you are confident about both of these points, check again if also an insignificant result makes sense and merely rejects a given theoretical prediction (insignificance does not equal "bad"). Sometimes a hypothesized effect is just not there. To check this, try to plot the first stage and the reduced form. As a rule of thumb: if you cannot see the causal effect in those graphs then it's probably not there.

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Another reason why this happens is due to heterogeneous treatment effects. Since your IV is giving you a Local Average Treatment effect, that is, the effect on compliers, it might be that only in the population of compliers, the effect is insignificant or not there. For eg: If you use sex of first two children as an IV for number of children a couple has, that is, if the sex of first two is same, then the couple is more likely to have third child, and reverse if the sex is different, then compliers are those who have a third child when treatment is switched on and not when it is off. If there are reasons to believe that compliers represent only a small proportion in your population then it is doubtful that the result you get is applicable to the population in general. This may in fact work in your favor.

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