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Say I have a binary treatment, X, and I'm interested in its (local average treatment) effect on a binary outcome variable, Y. I find (binary) instrument T that directly affects X but not Y.

How do I estimate this? I get that linear models like TSLS might not be ideal, but Angrist and Pischke's "Mostly Harmless Econometrics" seems to suggest that the effect estimates from something like a bivariate probit will be similar and maybe it isn't worth the trouble.

So I'm wondering: do I need to run something other than TSLS here? And if so, is there an R package for that? Thank you very much.

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What is our identifying assumption in IV? That we can use our instrument to break the treatment variable into an exogenous (predicted in the first stage) part and an endogenous (unpredicted) part. We put the exogenous part into our second stage model and the endogenous part becomes part of the error in that model. But, by the exclusion restriction, the exogenous part is uncorrelated with endogenous error. Hence, we assume that none of the predictors in our second stage model are correlated with the error---this is the identifying assumption.

Because we broke our treatment into exogenous and endogenous parts in a linear (additive) way, we need a linear model to maintain the identifying assumption. So, yes, a linear model (a linear probability model, if you want to sound fancy) is most appropriate here. Non-linear models need different identifying assumptions.

In R, you can use ivreg in the AER package to do IV estimation. See help here.

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The Wald and TSLS estimates coincide.

Also, when everything is binary and you have no covariates (you didn't say), you can actually bound the ATE or ATT. The bounds will be pretty informative if you have a strong instrument. Use the bounds from Pearl and Balke (1997) http://www.jstor.org/stable/10.2307/2965583.

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