Why Draw a Graph for an Instrument?
An instrumental-variable argument concerns causal relationships that are not all directly observable. A directed acyclic graph (DAG) makes the proposed relationships explicit: it records which variables are assumed to cause which others, and it helps locate routes that may violate an instrument’s assumptions. Its value is not that a drawing proves an instrument valid, but that it makes the assumptions and their implications easier to inspect.
Let \(Z\) denote the instrument, \(D\) the treatment, and \(Y\) the outcome. The three central questions are whether \(Z\) changes \(D\) (relevance), whether assignment to \(Z\) is independent of the potential treatment and outcome responses (independence), and whether \(Z\) affects \(Y\) only through \(D\) (exclusion). A graph helps distinguish these questions: the first concerns a causal route from \(Z\) to \(D\), the second concerns noncausal open paths from \(Z\) to response-generating factors, and the third concerns directed routes from \(Z\) to \(Y\) that bypass \(D\).
An unmeasured common cause can be represented by a node such as \(U\), with arrows to each variable it causes. Leaving such a cause out of the drawing can conceal a backdoor path. The graph is therefore a causal model: its interpretation depends on whether its arrows and included causes are substantively credible.
When Is a Path Open?
A path can connect two variables without creating an active association between them under every conditioning choice. The crucial local configuration is whether a node on the path is a collider. On \(A\to B\to C\), \(B\) is a noncollider; it is also a noncollider on \(A\leftarrow B\to C\). On \(A\to B\leftarrow C\), \(B\) is a collider because both arrows point into it.
This rule applies to the entire path, not just one of its nodes. A path with several noncolliders is blocked if even one of them is conditioned on. Conversely, an unconditioned noncollider does not make a path open if another noncollider on that path is conditioned on or an unconditioned collider blocks it. A collider’s descendant can open the path even when the collider itself is not conditioned on.
A graph is said to d-separate two sets of nodes given a conditioning set if every path between the sets is blocked under this rule. Under the DAG’s Markov assumption, d-separation implies the corresponding conditional independence. The reverse implication need not hold: distinct causal routes can cancel statistically, so an observed independence does not by itself show that the graph has no open path.
In IV applications, the independence assumption is most directly about potential responses, commonly written \(Z\perp\!\!\!\perp\{Y(0),Y(1),D(0),D(1)\}\). An observed outcome \(Y\), by contrast, can be associated with \(Z\) through the intended causal route \(Z\to D\to Y\). Thus, simply asking whether \(Z\) and observed \(Y\) are d-separated is not a test of IV independence.
Three IV Conditions in a Graph
The graphical questions below are useful only when the graph represents the relevant causes and potential responses. In particular, an omitted common cause can turn an apparently closed or absent route into an open one.
- Relevance: \(Z\) must change the distribution of \(D\), so that the first stage is nonzero. A causal arrow \(Z\to D\) represents the proposed influence; an arrow alone does not guarantee a nonzero statistical first stage if effects cancel or the model is misspecified.
- Independence: \(Z\) must be independent of the potential responses. In a graph that represents their causes, look for open noncausal paths from \(Z\) to those causes. A common cause of \(Z\) and a response-generating variable is a typical threat.
- Exclusion: every directed causal path from \(Z\) to \(Y\) must pass through \(D\). A directed path from \(Z\) to \(Y\) that bypasses \(D\) is a forbidden direct or indirect effect of the instrument outside treatment.
These conditions are not interchangeable. A graph may show a strong causal route \(Z\to D\) and therefore make relevance plausible, while also showing a backdoor path that threatens independence. Or the instrument may be independent of response-generating causes but have a separate route to \(Y\) that violates exclusion. A valid argument requires all relevant conditions, along with the other IV assumptions established in the course.
Worked Graphical Analyses
Worked Example: A Randomized Treatment Offer
Suppose a hypothetical trial randomly assigns an offer \(Z\) of a counseling program. Some people offered counseling attend, so \(D\) records attendance; \(Y\) is a later health outcome. Let \(U\) denote unmeasured motivation, which may affect both attendance and health. The proposed graph contains \(Z\to D\), \(D\to Y\), \(U\to D\), and \(U\to Y\).
Relevance. In hypothetical data, 70% of those offered counseling attend, compared with 30% of those not offered it. The first-stage difference is
The 0.40 difference is a 40-percentage-point association between offer and attendance, consistent with a relevant instrument in this setting. The graph supplies a proposed causal route \(Z\to D\); the observed difference alone does not establish that assignment was independent.
Independence. Random assignment is the substantive reason to regard \(Z\) as independent of \(U\) and the potential responses, assuming assignment was implemented as described and was not undermined by selective enrollment or attrition. The path \(Z\to D\leftarrow U\to Y\) is closed without conditioning because \(D\) is a collider on that path. It would be incorrect to infer from that collider alone that conditioning on attendance is harmless: conditioning on \(D\) opens this path and can associate randomized offer with motivation.
Exclusion. The proposed graph has no route from \(Z\) to \(Y\) other than \(Z\to D\to Y\). But if the offer also provides advice that changes health behavior without attendance, the graph needs a route \(Z\to\text{Advice}\to Y\) that bypasses \(D\). That would violate exclusion. Randomization and the observed first stage do not rule out this possibility; it requires substantive knowledge of what the offer changes.
Worked Example: Distance as an Instrument for Education
Consider a hypothetical study using distance to a college \(Z\) as an instrument for college attendance \(D\), with later earnings \(Y\) as the outcome. Let \(F\) represent family resources and location history. Suppose the proposed graph contains \(Z\to D\to Y\), \(F\to Z\), \(F\to D\), and \(F\to Y\). It also contains \(Z\to M\to Y\), where \(M\) represents migration decisions directly affected by distance.
Relevance. Suppose 58% of people living near a college attend, compared with 40% of those living farther away. The hypothetical first-stage difference is
This is an 18-percentage-point difference in attendance. It is consistent with relevance, and \(Z\to D\) depicts the proposed influence of distance on attendance.
Independence. The path \(Z\leftarrow F\to Y\) is a fork at \(F\), a noncollider. If \(F\) is unconditioned, this path is open, so the graph does not support independence of distance from the causes of earnings. The path \(Z\leftarrow F\to D\) is also open. Measuring and conditioning on \(F\) would block these particular paths, but it would not establish that all relevant common causes have been represented or controlled.
Exclusion. The directed route \(Z\to M\to Y\) bypasses attendance \(D\). Under the stated graph, distance can affect earnings through migration even if college attendance were held fixed. This is a graphical violation of exclusion. A sizable first stage cannot repair either the open backdoor path or the forbidden bypass route.
Worked Example: Conditioning on a Collider Can Create a Path
Let \(Z\) be a randomized instrument, \(U\) an unmeasured cause of an outcome \(Y\), and \(C\) a post-assignment variable caused by both \(Z\) and \(U\). The graph includes \(Z\to C\leftarrow U\to Y\). Assume, hypothetically, that \(Z\) and \(U\) are independent binary variables, each equal to 1 with probability 0.5, and define \(C=1\) exactly when \(Z=U\).
Before conditioning, \(Z\) and \(U\) are independent: \(P(U=1\mid Z=1)=0.5\). The path from \(Z\) to \(Y\) through \(C\) is blocked because \(C\) is an unconditioned collider. Now restrict attention to \(C=1\). Since \(C=1\) exactly when \(Z=U\),
Within the selected group, \(Z\) and \(U\) are associated; if \(Y=U\), they are also associated with \(Y\). Conditioning on \(C\), or on a descendant of \(C\), opens the collider path. This example shows why adjusting for every measured variable is not automatically safe: a post-instrument variable can create rather than remove a route between the instrument and outcome causes.
Reading, Comparing, and Stress-Testing the Graph
A practical graph review can proceed in order: first mark the proposed \(Z\to D\) route; next trace noncausal paths from \(Z\) to causes of treatment and outcome responses; finally trace every directed path from \(Z\) to \(Y\) and check whether it passes through \(D\). For each path, apply the full opening rule to every node on it, not just the first common cause or collider noticed.
Graphs complement other approaches rather than replace them. A verbal design argument explains why arrows are plausible; a DAG makes the implied paths visible. Regression adjustment can block paths only under appropriate variable selection and assumptions; indiscriminate adjustment may open collider paths. An IV estimate or first-stage statistic can describe empirical associations, but neither can show from the data alone that the graph’s exclusion or independence claims are true. The introductory material on instrumental-variable assumptions gives the broader assumptions; here the graph is a way to make their causal structure explicit.
Several failure modes deserve particular attention:
- Confusing relevance with validity. A clear \(Z\to D\) route does not close a backdoor path or remove a direct route to \(Y\).
- Checking only direct arrows. Exclusion is violated by any directed route from \(Z\) to \(Y\) that bypasses \(D\), including a route through another mediator.
- Checking only one node on a path. A path is open only when every noncollider is unconditioned and every collider has itself or a descendant conditioned on.
- Treating conditioning as uniformly beneficial. Conditioning blocks paths through conditioned noncolliders but can open paths through colliders.
- Assuming a graph proves its own assumptions. A graph records a causal proposal; omitted causes, misclassified treatment versions, or misunderstood mechanisms can make it misleading.
The main limitation is fundamental: DAGs show the implications of assumed causal structure, not whether that structure is true. Statistical cancellations can hide associations even when paths are open, and measured data cannot by themselves verify an exclusion restriction. A useful analysis therefore considers plausible alternative graphs, asks which substantive facts would rule them out, and reports the IV interpretation conditionally on those assumptions.
Check Your Understanding
Answer each question by tracing the stated paths. Brief solutions follow the questions.
- On the path \(Z\leftarrow U\to V\to Y\), suppose \(U\) is unconditioned and \(V\) is conditioned on. Is the path open?
- In a graph with \(Z\to D\to Y\) and \(Z\to Q\to Y\), which IV condition is directly threatened by the route through \(Q\)?
- Suppose a graph contains \(Z\to D\), \(U\to D\), and \(U\to Y\), with no arrow into \(Z\). Is \(Z\to D\leftarrow U\to Y\) open when nothing is conditioned on? What changes if \(D\) is conditioned on?
- Why does a nonzero observed difference \(P(D=1\mid Z=1)-P(D=1\mid Z=0)\) not, by itself, establish all IV assumptions?
- In \(Z\to C\leftarrow U\to Y\), what is the effect of conditioning on a descendant of \(C\) on the path between \(Z\) and \(Y\)?
Solutions. (1) The path is blocked: \(V\) is a conditioned noncollider, even though \(U\) is unconditioned. (2) Exclusion is threatened because \(Z\to Q\to Y\) reaches \(Y\) without passing through \(D\). (3) It is closed when nothing is conditioned on because \(D\) is a collider; conditioning on \(D\) opens it. (4) The difference is evidence of an association consistent with relevance, but it does not establish independence from potential responses or rule out a directed path to \(Y\) that bypasses \(D\). (5) Conditioning on a collider’s descendant opens the path, potentially associating \(Z\) with the cause \(U\) of \(Y\).