## October 29, 2015

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It would probably be helpful for naive readers if you clarified things in:

The claim is that if these three assumptions are met then causal effects can be estimated with coefficients for the exposure variables that are consistent and asymptotically unbiased.

A bit. Instrumental Variable Estimators are used to estimate the Local Average Treatment Effect - sometimes referenced as the Complier Average Treatment Effect in the medical literature. The estimator is local to the conditions specified in the first stage equation (e.g., it is an unbiased estimator but the generalizability of the estimate is reduced to a subset within the available sample and the analogous population). Additionally, IVE are used relatively frequently in the context of true randomized controlled trials when the intent - or offer - to treat is not always accepted or when there are issues with compliance to the assigned groups. In these cases a just identified first stage with the initial random assignment is sufficient to identify the causal effect of the treatment on the treated. A good reference for these types of estimators is Angrist and Pischke (2008). Mostly Harmless Econometrics. Princeton, NJ: Princeton University Press.

I'm confused about the AR significance values. It's stated as a test of the A3 assumption, but I don't know what that means. Is a significant results suggesting the assumption HAS NOT been violated or HAS?

@Billy_Buchanan: This is a great point and thanks for mentioning this. Typically, along with the three “core” assumptions (A1)-(A3), one needs to make additional assumptions to point-identify the treatment effect. Usually, these assumptions revolve around population homogeneity. For example, the one you mentioned, the local average treatment effect (LATE), reduces homogeneity in the population by making an assumption that no "defiers" exist in the study, e.g. in the education example, those who are encouraged to go to finish high school defy their encouragement and instead, don't finish high school. The other popular homogeneity assumption is based on some form of structural models, which typically identify the treatment on the treated (ToT) (see Hernan and Robins (2006) "Instruments for causal inference: an epidemiologist's dream?" for a survey of various assumptions to point-identify the treatment effect).

For better or for worse, most users of IV assume a homogeneous treatment effect, making the LATE equal to ToT, which equals the average treatment effect, and the presentation above takes this view. Also, while a semi-ideal use of IV methods would use RCT data where non-compliance is present, since under this setup, assumption (A3) is automatically satisfied, again, for better or for worse, it's common for users of IV to use observational data and use "exogenous variation" as instruments.

In fact, the instrumental variables (IV) literature is pretty extensive and the presentation, for better or for worse, only went through the most common usage of IV. It also focused on the users' awareness of the "core" assumptions of IV, the (A1)-(A3), under this common setting. But, it's definitely important to know the subtleties of the estimands in IV, specifically the additional layer of assumptions regarding homogeneity, and it's definitely worth reading more about it.

@Edomaniac. The AR test, roughly speaking, tests the null hypothesis of no treatment effect, i.e. H_0: beta=0 versus the two-sided alternative where beta is the treatment effect, although the null, to varying degree, also includes a test of exogeneity, i.e. Assumption (A3). In any case, the R printout above for the AR test is a robust procedure when the assumption (A1) is violated. That is, even if assumption (A1) is violated for this data, the p-value and the resulting confidence interval should still provide "honest" information about the true treatment effect, i.e. beta, in the model.

All the methods for violation of IV assumptions (A1)-(A3) reported in the R printout above are methods that will provide honest'' information about the treatment effect under the model, even if the violations of assumptions were to occur. More information about these methods can be found in the paper that accompanies this R software ivmodel by Jiang, Kang, and Small (2015) “ivmodel: An R Package for Inference and Sensitivity Analysis of Instrumental Variables Models with One Endogenous Variable.”

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