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Let say I have a really simple model $y=x+controls+u$, x is the treatment and is endogenous dummy variable, and z is an instrument variable for x. I want to test the casual relationship between x and y(effect of treatment). I have tested that z is corrected with x, it increased people who are treated, it is also likely to be exogenous.

However, the problem is, I suspect that z will also increase the effect of treatment. That is, it not only affect y through increasing x, but also increase x’s effect. Also, z probably has nothing to do with people who aren’t treated, and it doesn’t affect y for the whole samples(not much people are treated).

Under this situation, will the instrument be invalid? I only took introductory econometrics, I guess maybe it is some sort of heteroscedasticity?

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That depends on the degree of heterogeneity in the treatment effect of x. If there is no heterogeneity in the treatment effect, your instrument is perfectly valid and you will get what you want. However, that doesn't seem to be the case here.

If there is large heterogeneity in the treatment effect, then the IV estimate converges to something irrelevant to what you're looking for, the average treatment effect of the whole population or the treated population.

One of the key assumptions for identification of something relevant(even if it is not exactly the average treatment effect) is that z is not correlated with the treatment effect. (if treatment effect is constant, there is no correlation by construction)

Search for some resources with keywords such as "local average treatment effect", "intention to treat" and so on.

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  • $\begingroup$ Sorry I am a little bit confused. I know large treatment effect heterogeneity will produce Late, but how does it related with “instrument correlated with treatment effect”? Do you mean that this will increase heterogeneity（ps:In my data, the instrument should have constant effect on treated populations’ treatment effect.)? Or that’s another thing and the estimation is not valid since the instrument positively correlated with treatment effect? Thank you for you time and suggestion. $\endgroup$
    – Key
    Commented Jun 13, 2018 at 21:58
  • $\begingroup$ What I mean is that if your instrument is correlated with treatment effect(which is random variable itself), you will not even get LATE. The bias(from the ATE) of IV can be even larger than OLS in such cases. $\endgroup$
    – Julius
    Commented Jun 13, 2018 at 22:16

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