A structural causal model (SCM) describes how variables are generated by causal assignments, together with a probability distribution for the background disturbances. For a quant assessing a change in execution policy, it specifies which mechanism changes and how the consequences propagate. A regression fitted to historical policy choices supplies no such intervention rule by itself.
An assignment is a claim about a mechanism
Write an assignment as
where denotes the direct causes of inside the model and contains background influences whose causes the model leaves unspecified. The complete model includes the variables, functions and joint disturbance distribution ; a graph alone is a partial specification. Bareinboim and co-authors, §1.2, Definition 1.
The arrow means “generate the left side using the right side.” Algebraically rearranging an assignment does not reverse its causal direction. An ordinary regression equation summarises an association unless additional causal assumptions justify a mechanism interpretation. A regression residual being orthogonal to a fitted regressor does not establish those assumptions. Pearl (1995), §2.2.
Here is an invented, dimensionless three-variable model:
Assume the disturbances are mutually independent, integrable and mean zero; their full distribution shapes are left unspecified because these calculations do not need them. The endogenous directed acyclic graph is : no direct mechanism, feedback or additional shared causes are included. These are stipulated assumptions, not findings from market data. More general SCMs can have dependent or shared disturbances; independence is an additional restriction, not part of every SCM's definition. Bareinboim and co-authors, §1.2.
Setting X replaces its assignment
The intervention removes and installs . The other assignments and disturbance distribution remain intact. That selective invariance is a substantive assumption about the intervention. Pearl (1995), §2.2.
Substitution through this particular model gives
so its population response is
The total response per unit of is , carried entirely through . At the population mean is 6.
Conditioning performs a different operation. Observing keeps the original assignment and tells us . Setting overrides that assignment without selecting units by . In this unconfounded model, independence implies too, at supported values of . The numerical equality follows from the assumptions; it does not make conditioning and intervention interchangeable. With hidden [confounding, that equality can fail.
Nor should an analyst automatically hold fixed while changing . In this model, that would block the only causal route to . The separate post-treatment bias article explains why mediator controls can change the question being answered.
Replay the observed unit
A counterfactual outcome asks what would have happened to a unit under a different action. The structural calculation uses the same disturbances across the factual and hypothetical worlds, rather than drawing a new unit. Pearl's three-step procedure, §2, makes the sequence explicit.
Suppose the observed unit has , , .
- Abduction: infer its background disturbances from the factual record:
- Action: replace the first assignment with . Keep and the other assignments.
- Prediction: propagate the change:
The unit's outcome changes from 14 to 8, a decrease of 6. Its hypothetical outcome 8 differs from the population mean 6 because its disturbances contribute . Keeping is consistent with setting : the intervention has severed the assignment that formerly equated them.
Original stipulated model. Replace one assignment, recompute its descendants and preserve the observed unit's disturbances; a population average answers a different question.
Exact abduction is possible here because all three variables are observed without error and the additive assignments can be inverted. With incomplete observations or non-invertible mechanisms, abduction generally produces a posterior over compatible disturbances. Prediction then averages over that posterior, sharing each disturbance draw across worlds. Redrawing from the unconditional discards the unit-specific evidence. Pearl (2013), §2.
A computable model is not an identified model
The arithmetic above answers a question inside the stipulated SCM. Identification asks whether the available data and maintained assumptions uniquely determine that answer. Different compatible mechanisms can fit the same observations yet disagree under intervention; fitting associations alone does not recover a causal model. Pearl (2009), §§2.1 and 3.2.2.
Causal discovery addresses learning structure under extra assumptions. Given a graph, the backdoor criterion and frontdoor criterion offer particular routes to identifying effects from observations. They do not automatically identify every individual's counterfactual. Real estimation also needs suitable data and support for the intervention values; a structurally defined extrapolation is not empirical validation. Pearl (2009), §§3.2–3.4.
For a financial regression example, Endogeneity and Instrumental Variables develops omitted influences, simultaneity and the additional assumptions needed for instruments. The preview is public; the full lesson requires membership.
Sources
- Judea Pearl, “Causal diagrams for empirical research”, Biometrika 82(4), 1995, pp. 669–688, §2.2: structural assignments, autonomous mechanisms and interventions.
- Elias Bareinboim, Juan D. Correa, Duligur Ibeling and Thomas Icard, “On Pearl's Hierarchy and the Foundations of Causal Inference”, technical report R-60, March 2021 revision, §1.2: SCM components and disturbance assumptions.
- Judea Pearl, “Structural Counterfactuals: A Brief Introduction”, Cognitive Science 37, 2013, pp. 977–985, §2: counterfactual submodels and abduction, action and prediction.
- Judea Pearl, “Causal inference in statistics: An overview”, Statistics Surveys 3, 2009, pp. 96–146, §§2.1, 3.2–3.4: assumptions, identification and counterfactual analysis.