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If I'm to find an instrumental variable for an equation, am I getting this idea right? I have this regression:

$$\text{stndfnl}=\beta_0+\beta_1\text{atndrt}e+\beta_2\text{priGPA}+\beta_3\text{ACT}+\beta_4\text{priGPA}*\text{atndrte}+u$$

where $\text{stndfnl}$ is the standardized outcome on a final exam, $\text{atndrte}$ is attendance rate, $\text{priGPA}$ is prior college GPA, and $\text{ACT}$ (if not obvious given the context) is the score of the standardized test, the ACT.

I'm told that if $\text{atndrte}$ is correlated with $u$, then, in general, so is $\text{priGPA*atndrte}$. So what might make a good instrumental variable for $\text{priGPA*atndrte}$? Do you think, say, the mother's education level ($\text{motheduc}$) would be a good IV? It could have a significant effect on $\text{priGPA*atndrte}$ and it could also be uncorrelated with the error term in the main regression.

Is this the correct way to approach this?

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  • $\begingroup$ This question is identical to this one. $\endgroup$ – tchakravarty Nov 26 '12 at 8:58
  • $\begingroup$ I will keep an eye on it, because unfortunately there's no answer for that one $\endgroup$ – Kyle Nov 26 '12 at 13:39
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    $\begingroup$ Depending on how housing at the school works (for example, if all students live in dorms, and dorm assignment is random), you might be able to use distance as an instrument for attendance rate. You might be able to do something similar with roommate assignment, which could give you exogenous variation in GPA. Being randomly matched with a party animal might cause your grades to suffer. $\endgroup$ – Dimitriy V. Masterov Nov 27 '12 at 3:56
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From an epidemiologist's perspective:

Given exposure X and outcome Y, a key challenge is to deal with confounding of the X->Y relationship in order to make stronger inferences regarding causation. Instrumental variables (IVs) work because they are related to the X, and unrelated to Y, other than through their effects on X (i.e. X completely mediates the IV->Y relationship). This is often a strong assumption to say the least.

In health research, Mendelian-randomization is the most common IV approach to understanding causal relationships between behaviors and outcomes. For a recent example, see here.

Given your example, I don't see how motheduc would qualify, because there are likely several ways that it could affect stndfnl, independently of the other predictor variables. Thus it seems like you should just include it as another predictor.

I've also never seen an example where an instrument for a product-term interaction was being sought.

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