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The exclusion restriction requires an instrumental variable to affect the outcome only through the exposure whose causal effect is being studied.

A genetic variant is a heritable difference in a DNA sequence relative to a chosen reference. Genetic variants can be used as instrumental variables in Mendelian randomization when the required causal assumptions are credible.

Instrument relevance requires the instrumental variable to change the conditional distribution of the exposure. A nearly irrelevant instrument is a weak instrument and can produce unstable estimates.

Mendelian randomization uses inherited genetic variants associated with an exposure as instrumental variables. Its causal interpretation requires instrument relevance, instrumental-variable independence, and the exclusion restriction.

The core instrumental variable assumptions are:

For a local average effect one additionally uses instrumental-variable monotonicity. Because the valid instrument supplies variation in education independent of the confounders and has no direct route to the outcome, the instrument-outcome association can be attributed to the education changed by the instrument without measuring every confounder.

Conditionally on $X$, a valid instrumental variable $Z$ must satisfy three core conditions.

Together with consistency of potential outcomes and positivity in causal inference, these assumptions make variation in $A$ induced by $Z$ causally interpretable. In the displayed graph, relevance is the edge $Z\to A$, independence is the absence of a path from $Z$ to $U$ after conditioning on $X$, and exclusion is the absence of a direct $Z\to Y$ edge.

Definition 1 satisfies Property 1 but not Property 2. Let $S_1$ contain every non-descendant that blocks some backdoor path. Every backdoor path begins $A\leftarrow P$ for a parent $P$ of $A$; whenever that path can transmit confounding, $P\in S_1$. Conditioning on $S_1$ therefore blocks every such path at its first nonendpoint vertex, so $S_1$ is sufficient.

For failure of Property 2, consider the faithful graph with arrowsThe variable $X_i$ blocks the backdoor path $A\leftarrow C\to X_i\to Y$, so Definition 1 calls it a confounder. Every sufficient set containing $X_i$ must nevertheless contain $C$ to block $A\leftarrow C\to Y$. Once $C$ is included, deleting $X_i$ leaves a sufficient set. Thus $X_i$ can never be essential as Property 2 demands.

Definition 2 satisfies Property 2 but not Property 1. If $X_i$ belongs to every minimal sufficient adjustment set, choose one such set $I$ and put $J=I\setminus \{i\}$. By minimality, $J\cup \{i\}$ is sufficient and $J$ is not, proving Property 2.

For failure of Property 1, use the faithful chain-shaped backdoor pathBoth $\{C_1\}$ and $\{C_2\}$ are minimal sufficient adjustment sets. No variable belongs to every minimal sufficient set, so Definition 2 labels no variable a confounder, but the empty set is not sufficient.

Definition 3 satisfies Property 1 but not Property 2. Under faithfulness, its associational criterion contains enough non-descendants to block every open backdoor path. Indeed, if such a path remained open, its first unconditioned parent of $A$ would be D-connected to $A$ and, after a suitable conditioning set, to $Y$ given $A$; faithfulness would place that parent in the Definition 3 set, a contradiction. Thus adjusting for all variables selected by Definition 3 is sufficient.

For failure of Property 2, considerwith both $Z$ and $U$ observed and a faithful distribution. The instrumental variable $Z$ is associated with $A$. Conditioning on the collider $A$ opens $Z\to A\leftarrow U\to Y$, so $Z$ is associated with $Y$ given $A$ and Definition 3 calls it a confounder. Any sufficient set containing $Z$ must also contain $U$ to block $A\leftarrow U\to Y$, but $\{U\}$ is already sufficient. Hence removing $Z$ never destroys sufficiency, violating Property 2.

In potential outcome notation, a valid instrumental variable $Z$ must satisfy:

In the graph, $Z_1$ has open noncausal paths to $Y$ that do not pass through $A$, includingEquivalently, these paths remain after deleting the causal edge $A\to Y$. Thus $Z_1$ is associated with potential outcomes through the latent variables $C$ and $S$, violating instrumental-variable independence; it is not a valid marginal instrument.

Let $Z$ indicate a recent birthday event, $A$ indicate a social gathering, and $Y$ denote subsequent household COVID-19 infection. The standard instrumental variable conditions are:

The usual consistency in causal inference, well-defined exposure, and absence of relevant interference in causal inference are also needed to interpret the result causally.

The instrumental variable graph iswith no arrow $Z\to Y$ and no common cause of $Z$ with $A$ or $Y$.

In potential outcome notation, a valid instrument requires: