# Can a instrument 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 instrument variable equation (or other equations)? If so, how? If not, why?

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:

1. $$Z$$ must be associated with $$X$$.

2. $$Z$$ must causally affect $$O$$ only through $$X$$

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

4. 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.

• ATE is average treatment effect, which is the effect for a randomly plucked person in the population. IV with a monotonicity assumption (or no defiers) recovers only the local average treatment effect for the people who comply with the assignment, which is typically different from the population ATE if there's any heterogeneity, but often more interesting from a policy perspective. – Dimitriy V. Masterov Mar 14 '18 at 23:00
• @JulianSchuessler When the policy option consists of moving the instrument, the LATE/CATE is the right effect. For example, if the policy is a tax credit for solar panels, the impact for those who install only with the credit in place is the relevant one. For policy, we are often interested in the marginal entrant. – Dimitriy V. Masterov Mar 15 '18 at 14:55
• Why is it sufficient that Z is only associated with X (criterion 1)? Is it sufficient that Z does not causally affect X but is correlated with X through som unmeasured variable U? If so, why? – Elias Dec 18 '19 at 16:06
• @Alexis Thanks. I checked fig 16.3, and, intuitively, I find that the instrument should be valid in this case (do they prove it? I haven't read the book). However, suppose there is an unmeasured confounder $V$ that affects $Z$ and $A$. Then $Z$ will still be correlated (associated) with $A$ - but will it be valid? No, according to Imbens (page 40, second key assumption, 2019): arxiv.org/pdf/1907.07271.pdf (also, see fig 9c-9d). The condition is, moreover, not testable, since we need a causal assumption to be able to say that $V$ is, in fact, not a potential confounder. – Elias Dec 18 '19 at 18:04
• @Alexis I note that even though the article is unreviewed, Imbens is a world-renowned econometrician and an expert in the field. I wanted to refer to an accessible article and argument. His view is also expressed in standard, modern textbooks in causal inference in econometrics, such as "Causal Inference for Statistics, Social, and Biomedical Sciences". I am positing $V\to Z$ and $V\to A$ here, in addition to the causal relations expressed in fig. 16.3. One may also consider $V\to U$ and $U\to A$. I'm not positing $U\to Z$, though it may be considered. I'd guess one needs to control for $V$. – Elias Dec 18 '19 at 18:17

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 two simple graphical conditions:

1. $(Z \not\perp X|S)_{G}$
2. $(Z\perp Y|S)_{G_{\overline{X}}}$

The first condition requires $Z$ to be connected to $X$ in the original DAG. The second condition requires $Z$ to not be connected to $Y$ if we intervene on $X$ (represented by the DAG $G_{\overline{X}}$, where you remove the arrows pointing to $X$). You might want to check Causality (page. 248).

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.

• Could you clarify how Z satisfies A1 in your second graph? – Dimitriy V. Masterov Mar 15 '18 at 0:41
• @DimitriyV.Masterov what is the $A1$ you are referring to? Is it $(Z \not\perp X|L)_{G}$? That holds because $W$ is a common cause of $Z$ and $X$. – Carlos Cinelli Mar 15 '18 at 1:27