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Suppose I am interested in studying the effect of schooling on earnings. I have two valid IVs for schooling namely:

  1. distance of residence from school, and
  2. quarter of birth.

I use each instrument separately and obtain two different IV estimates of the effect of schooling on earnings - b using (1) and c using (2). And it's found that b is not equal to c. Why do I get two different estimates of the effect of schooling on earnings despite both IVs being valid?

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  • $\begingroup$ Uhm... because you have different $Z$ variables in $\hat\beta_{\rm IV}=(X'Z(Z'Z)^{-1}Z'Z)^{-1} X'Z(Z'Z)^{-1}Z'Y$ ?? If your confidence intervals do not overlap, there's something seriously wrong with at least one instrument (may be both... and they are both fairly weak, anyway). I would throw in parental education for a good measure as yet another instrument. $\endgroup$ – StasK Dec 4 '15 at 21:01
  • $\begingroup$ Iv regression does not have finite sample results, it is based on asymptotics. The convergence can be very slow, especially if the instruments are weak which is likely the case. Unless you have many many thousands of observations, you cannot be sure. Also why estimate to different estimates? It is deficient compared to 2sls. $\endgroup$ – Repmat Dec 4 '15 at 22:21
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Under some reasonable assumptions, IV estimates give you the average treatment effect for "compliers", folks whose participation is changed by the instrument. With 2 distinct instruments, the two sets of compliers are going to be somewhat different. For example, if costs are the instrument, these units do not participate when costs/distance are high but do participate when costs are low would be the compliers. In a heterogeneous treatment effect world where earnings don't respond to schooling identically, the average causal effect will be different for two groups since they consist of different people (though there may be some overlap).

Also, there's a long tradition of people arguing that these two instruments are not valid for this research question. They could be biased in different ways, which is another potential reason why they could be different, even if there's no heterogenous treatment effect.

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