Benquo comments on Causal Diagrams and Causal Models - Less Wrong

61 Post author: Eliezer_Yudkowsky 12 October 2012 09:49PM

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Comment author: Benquo 18 October 2012 09:26:05PM 1 point [-]

This helped me understand what Instrumental Variables are, but Andrew Gelman's critique of instrumental variables has me confused again:

Suppose z is your instrument, T is your treatment, and y is your outcome. So the causal model is z -> T -> y. The trick is to think of (T,y) as a joint outcome and to think of the effect of z on each. For example, an increase of 1 in z is associated with an increase of 0.8 in T and an increase of 10 in y. The usual “instrumental variables” summary is to just say the estimated effect of T on y is 10/0.8=12.5, but I’d rather just keep it separate and report the effects on T and y separately.

In Piero’s example, this translates into two statements: (a) States with higher penalties for murder had higher penalties for defamation, and (b) States with higher penalties for murder had less reporting of corruption.

Fine. But I don’t see how this adds anything at all to my understanding of the defamation/corruption relationship, beyond what I learned from his simpler finding: States with higher penalties for defamation had less reporting of corruption.

If your model is z -> T -> y, and you show that z interacts with each of T and y, isn't the next step just to look at the relation between z and y, controlling for T? In other words, if it turns out that z still matters in predicting y once you have T in your model, then you don't have an instrumental variable. But if T screens off the effect of z in predicting y, then z is an instrumental variable, and only affects y through T.

Comment author: IlyaShpitser 31 October 2012 06:17:14PM *  1 point [-]

Sorry, what you are missing is T and Y could be confounded by unobserved variables. That is, the real graph is:

z -> T -> Y, with T <- U -> Y, with U unobserved. Then if you control for T, you will get an open path z -> T <- U -> Y which is not causal. In general if your graph is

T -> Y <- U -> T, the causal effect is not a functional of the observed data. However with some parametric assumptions you can obtain the causal effect as a functional of the observed data if there is an instrument z.

Comment author: Benquo 31 October 2012 06:56:56PM 1 point [-]

Oh... so the idea in your second paragraph is that when you hold T constant, a change in z suggests an equal and opposite change in U (measuring by their mean effect on T). Then that change affects Y.

Comment author: IlyaShpitser 31 October 2012 07:02:23PM *  1 point [-]

That's exactly right. The fact that for treatment T, and outcome Y, there is generally an unobserved common cause U of T and Y is in some sense the fundamental problem of causal inference. The way out is either:

(a) Make parametric assumptions and find instrumental variables (econometrics, mendelian randomization)

(b) Try to observe U (epidemiology, etc.)

(c) Randomize T (statistics, empirical science)

There are some other lesser known ways as well:

(d) Find an unconfounded mediator W that intercepts all causal influence from T to Y:

T -> W -> Y

Then use the "front-door criterion."

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