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:
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.”