# Can an instrumental variable equation be written as a directed acyclic graph (DAG)?

Directed acyclic graphs (DAGs) are efficient visual representations of qualitative causal assumptions in statistical models, but can they be used to present a regular instrumental variable equation (or other equations)? If so, how? If not, why?

Yes.

For example in the DAG below, the instrumental variable $$ZZ$$ causes $$XX$$, while the effect of $$XX$$ on $$OO$$ is confounded by unmeasured variable $$UU$$.

The instrumental variable model for this DAG would be to estimate the causal effect of $$XX$$ on $$OO$$ using $$E(O|ˆX)E(O|\widehat{X})$$, where $$ˆX=E(X|Z)\widehat{X} = E(X|Z)$$.

This estimate is an unbiased causal estimate if:

1. $$ZZ$$ must be associated with $$XX$$.

2. $$ZZ$$ must causally affect $$OO$$ only through $$XX$$

3. There must not be any prior causes of both $$OO$$ and $$ZZ$$.

4. The effect of $$XX$$ on $$OO$$ must be homogeneous. This assumption/requirement has two forms, weak and strong:

• Weak homogeneity of the effect of $$XX$$ on $$OO$$: The effect of $$XX$$ on $$OO$$ does not vary by the levels of $$ZZ$$ (i.e. $$ZZ$$ cannot modify the effect of $$XX$$ on $$OO$$).
• Strong homogeneity of the effect of $$XX$$ on $$OO$$: The effect of $$XX$$ on $$OO$$ is constant across all individuals (or whatever your unit of analysis is).

The first three assumptions are represented in the DAG. However, the last assumption is not represented in the DAG.

Hernán, M. A. and Robins, J. M. (2020). Causal Inference. Chapter 16: Instrumental variable estimation. Chapman & Hall/CRC.