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