A colleague writes:
I have been enjoying your posts on the LATE. It got me thinking, as the number of instruments increases, does the LATE approach the ATE? Put another way, as the R2 of the instrumenting equation approaches 1, does the LATE converge to the ATE?
Historically there has been much discussion on the use of too many instruments (overidentification concerns), but if the above is true then there could be some serious upside to being grossly overidentified?
I’m no econometrician, so I don’t know the exact answer to your question, but let me take a stab at it by providing a simple example. My hunch is that having multiple instrumental variables (IVs) makes the problem worse, by which I mean “It could go either way.”
To see this, suppose you want to know the causal impact of D on Y and have IVs Z1 and Z2. And assume (heroically perhaps) that both instruments are valid.
A first LATE-ATE discrepancy is introduced because of compliance issues regarding Z1 — some people were induced to take up treatment D by Z1, others not, and the latter mess up your estimate of the ATE.
But then, suppose you introduce a second instrument Z2. That introduces a whole new compliance issue — some people were induced to take up treatment D by Z2, others not, and the latter mess up your estimate of the ATE, too.
I think your question relates to whether those who are compliers in light of Z2 were noncompliers in light of Z1. If Z2 allows making some Z1-noncompliers Z2-compliers, then I think your LATE is getting closer to the ATE. But if Z2 compounds the noncompliance issue, then the LATE is getting further away from the ATE. The difficulty, of course, is that this creates another identification problem — namely, that of identifying how Z2 compliance is related to Z1 compliance.
That said, please note that this is purely off the top of my head. As for the R-square of the first-stage regression, look up “overfitting.”
Could anyone who teaches econometrics or empirical methods chime in with whether my intuition is right?