The following problem was brought to our attention by Randy Verbrugge:
Suppose the DAG is:
top row: unobserved U, with dotted arrows to X and Y.
bottom row: X –> W –> Y.
We can identify the causal effect of X on Y via the decomposition in the theorem. As I understand it, if we assume linearity, we regress Y on W and X and take the coefficient on W, call it b; then we regress W on X and take the coefficient, call it a; the causal effect is ab*x.
However, I wonder if we are (unavoidably) in a situation where X is a near-instrument for W, so that we will have a ton of bias.
Thanks for your help!
The danger of conditioning on an instrument or a near instrument only exists when we have some residual, uncontrolled bias. In our case, the W—>Y relation is completely deconfounded by conditioning on X, therefore, no bias-amplification can take place. The danger will resurface if there was a bi-directed arc between W and Y.
An important nuance: The capacity of an instrument to introduce bias where none existed (demonstrated in my paper) only pertains to situations where the zero bias is unstable, i.e., created by incidental cancellations. For example, if we had a bi-directed arc between W and Y that happened to cancel the bias created by the confounding path W<---X<--U-->Y, then the crude (regressional) estimate of the effect of W on Y would be unstably unbiased, and conditioning on W would introduce bias.
In general: The bias amplification phenomenon never conflicts with the rules of do-calculus.
Thanks for raising this question.