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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 risk score combines allele counts from several genetic variants, usually using specified weights. In Mendelian randomization it can act as a combined instrumental variable for an exposure; interpretation still requires instrument relevance, instrumental-variable independence, and the exclusion restriction.

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.

An instrumental variable $G$ supplies variation in an exposure $B$ that can be used for causal inference about its effect on $Y$. Its three core requirements are:

Relevance, independence, and exclusion are the core instrumental-variable assumptions. They do not by themselves identify an arbitrary population-average effect. For a binary instrument and treatment, instrumental-variable monotonicity and the usual consistency in causal inference assumptions can identify a local average treatment effect; stronger effect-homogeneity or structural-model assumptions are needed for other effect interpretations.

Under the displayed causal directed acyclic graphs, with nonzero effects along the relevant arrows and no exact cancellation of the effects on $B$, the valid instrumental variables are scenarios (i), (ii), and (iv). The path arguments are:

An upstream intermediate variable need not invalidate an instrumental variable if its routes to the outcome all pass through the target exposure; an outcome intermediate reached without that exposure does invalidate it.

The first model alone would suggest that more cycling is beneficial, but cycling was not randomized in that model: its coefficient can be distorted by confounding or reverse causality. The randomized incentive both substantially increased cycling and substantially increased asthma episodes.

If the incentive affects asthma only through cycling, it is an instrumental variable, and the corresponding Wald estimator of the effect per additional weekly mile isadditional episodes. Under that exclusion assumption the evidence therefore advises against cycling to work more often for this population. Without the exclusion assumption, the experiment establishes the harmful effect of the incentive itself but cannot by itself distinguish an effect of cycling from a direct effect of the monetary incentive.

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:

Two-sample Mendelian randomization combines genetic variant associations with an exposure estimated in one sample and their associations with an outcome estimated in another. It requires the instrumental variable assumptions and compatible associations across the populations; sample overlap can change the behaviour of weak instruments.

For a binary instrumental variable $Z$, exposure $X$, and outcome $Y$, the Wald estimator is the ratio of the instrument's effect on the outcome to its effect on the exposure:Under instrumental-variable assumptions it estimates a causal effect.

This weighted median of genetic variant ratio estimates consistently estimates a common causal effect when more than half of its asymptotic weight comes from valid instrumental variables, subject to the usual sampling regularity conditions. It is a sensitivity analysis for horizontal pleiotropy, not an unconditional guarantee of validity.