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?
 A: Yes.
For example in the DAG below, the instrumental variable $Z$ causes $X$, while the effect of $X$ on $O$ is confounded by unmeasured variable $U$.

The instrumental variable model for this DAG would be to estimate the causal effect of $X$ on $O$ using $E(O|\widehat{X})$, where $\widehat{X} = E(X|Z)$.
This estimate is an unbiased causal estimate if:

*

*$Z$ must be associated with $X$. Edit: And, (as in the above DAG) this association itself must be unconfounded (see Imbens).


*$Z$ must causally affect $O$ only through $X$


*There must not be any prior causes of both $O$ and $Z$.


*The effect of $X$ on $O$ must be homogeneous. This assumption/requirement has two forms, weak and strong:



*

*Weak homogeneity of the effect of $X$ on $O$: The effect of $X$ on $O$ does not vary by the levels of $Z$ (i.e. $Z$ cannot modify the effect of $X$ on $O$).

*Strong homogeneity of the effect of $X$ on $O$: The effect of $X$ on $O$ 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.
A: Yes, they surely can.
As a matter of fact, the SCM/DAG literature has been working on generalized notions of instrumental variables, you might want to check Brito and Pearl, or Chen, Kumor and Bareinboim.
The basic IV dag is usually represented as:

Where $U$ is unobserved and $Z$ is an instrument for the effect of $X$ on $Y$. Although this is the graph you usually see, there are several different structures that would render $Z$ an instrument.
For the basic case, to check whether $Z$ is an instrument for the causal effect of $X$ on $Y$ conditional on a set of covariates $S$, you have to check two conditions:

*

*$Z$ is connected to $X$ in the original DAG;


*$S$ d-separates $Y$ from $Z$, in the DAG where the arrow $X\rightarrow Y$ is removed.
The first condition requires $Z$ to be associated with $X$  (relevance condition, otherwise the numerator of the IV estimand is zero). The second condition requires $Z$ to not be connected to $Y$ if but for its effect on  $X$ (that is, we cannot have violations of the exclusion and independence restriction, after conditioning on $S$).
For example, consider the graph below, with $W$ and $U$ unobserved. Here, $Z$ is, conditional on $L$,an instrument for the causal effect of $X$ on $Y$. We can create more complicated cases where it might not be immediately obvious whether something does qualify as an instrument or not.

One final thing you should have in mind is that identification using instrumental variable methods needs parametric assumptions. That is, finding an instrument is not enough for identification of the effect: you need to impose parametric assumptions, such as linearity or monotonicity and so on.
