Nested Counterfactuals and Cross-World Quantities

Why Nested Counterfactuals Need Care

A mediation question can ask what an outcome would be under one treatment value if the mediator were set to the value that same person would have had under another treatment value. The notation \(Y(a,M(a'))\) represents this construction. It combines an outcome response under treatment \(a\) with a mediator value generated under treatment \(a'\).

This combination is useful for defining natural effects, but it is not an ordinary observed outcome. For a given person, only one treatment and one resulting mediator are observed. The nested quantity instead joins features of multiple intervention scenarios for that person. Understanding how the quantity is defined is therefore distinct from asking whether its distribution can be learned from observed data.

Deriving the Nested Quantity From Structural Equations

Consider a structural causal model with treatment \(A\), mediator \(M\), and outcome \(Y\). For simplicity, write its mediator and outcome equations as

Here \(U_M\) and \(U_Y\) represent the unit-specific inputs to the mediator and outcome equations. They may include unmeasured attributes; the equations are not claims that these inputs are observed or independent. To define counterfactuals, retain the same unit-specific inputs while changing the intervention values. Setting \(A=a'\) in the mediator equation gives \(M(a')=f_M(a',U_M)\). Setting treatment to \(a\) and mediator to \(m\) in the outcome equation gives \(Y(a,m)=f_Y(a,m,U_Y)\).

The nested quantity follows by substituting the mediator value generated under \(a'\) into the outcome equation under \(a\):

The same \(U_M\) and \(U_Y\) belong to the same unit in both parts of this expression. That shared unit-level background is what makes the mediator value and outcome response parts of one person's counterfactual construction. The formula does not say that the mediator value under \(a'\) is observed alongside the outcome under \(a\), or that the two counterfactual components are statistically independent.

Structural Composition and Observed-Data Consistency

Consistency connects observed values with potential outcomes under the treatment and mediator values actually received. In the usual notation, if a unit receives \(A=a\), then its observed mediator equals \(M(a)\), and its observed outcome equals \(Y(a)\). If the outcome counterfactual is indexed by both treatment and mediator, then a unit observed with \(A=a\) and \(M=m\) has observed outcome \(Y(a,m)\). These statements presuppose that treatment and mediator interventions are sufficiently well-defined for the counterfactual labels to have a clear meaning.

Nested counterfactuals require a different operation: compose the mediator response under \(a'\) with the outcome response under \(a\), holding the unit's structural inputs fixed. The structural equations make that operation precise. Observed-data consistency alone does not provide the joint values of \(M(a')\) and \(Y(a,m)\) for different treatment worlds. For example, observing \(A=1\), \(M=4\), and \(Y=24\) in the preceding example is consistent with the observed-world counterfactuals under \(A=1\); it does not reveal that unit's \(M(0)\) or \(Y(1,M(0))\).

This distinction is important: defining the nested counterfactual is not the same as identifying its average from observed data. Identification requires additional assumptions, which concern how observed distributions relate to counterfactuals across treatment worlds. The structural substitution above defines the target; it does not establish that the target can be recovered from data.

Useful Checks on the Definition

The structural definition gives a basic check when the treatment values match. If \(a'=a\), then the mediator supplied to the outcome equation is exactly the mediator generated under that same treatment. Thus,

This is a consequence of the structural definitions: the nested expression reduces to the ordinary total outcome counterfactual under \(a\). It is not a claim that \(Y(a,M(a'))\) equals \(Y(a)\) when \(a'\ne a\). When the treatment values differ, the mediator inserted into the outcome equation may differ from \(M(a)\).

A second check is that the nested expression must use the same unit throughout. Combining one person's \(M(a')\) with another person's \(Y(a,m)\) would describe a different construction, not \(Y(a,M(a'))\) for a single unit. Individual-level response functions and their unit-specific inputs must be kept aligned.

Where Cross-World Quantities Arise

Any expression that combines a mediator counterfactual from one treatment setting with an outcome counterfactual under another can involve cross-world information. In \(Y(a,M(a'))\), the mediator's treatment input is \(a'\), but the outcome's treatment input is \(a\). When \(a\ne a'\), both pieces cannot generally be read off the same observed record: the record has one realized treatment, one realized mediator, and one realized outcome.

The term “cross-world” describes this combination of counterfactual scenarios. It does not mean that the quantity is incoherent. In a structural model with well-defined intervention equations, it is defined by substitution using the same unit-level inputs. Nor does the term, by itself, imply that the quantity is identifiable or unidentifiable. Those are separate questions about assumptions and observed data.

This distinction matters because mediation contrasts built from nested outcomes can have a clear causal definition while still requiring assumptions to connect them to data. The definition tells us exactly what response is being compared. It cannot, on its own, tell us whether the relevant counterfactuals have been observed or whether their distribution can be estimated.

Common Mistakes

• Treating a nested value as an observed outcome. Seeing \(Y(a,M(a'))\) does not mean a person received treatment \(a\) and was observed with mediator \(M(a')\). The expression is a counterfactual construction.
• Using observed-data consistency as a cross-world bridge. Consistency relates observed values to counterfactuals at the values actually received. It does not reveal a person's mediator under a different treatment or the outcome at a mediator value from another world.
• Changing the unit between components. \(M(a')\) and \(Y(a,m)\) in a nested quantity refer to the same unit. Pairing counterfactuals from different units does not define that person's nested outcome.
• Assuming every nested quantity is automatically identified. Structural equations can define the counterfactual, but defining it is not a proof that observed data determine its distribution.