Rubin Causal Model

Definition

The Rubin Causal Model (RCM) is a framework for causal inference built on potential outcomes: for every unit $i$ and every possible treatment value $d$, there exists a potential outcome $Y_i(d)$ representing what would be observed if unit $i$ received treatment $d$. The causal effect of treatment on unit $i$ is the comparison $Y_i(1) - Y_i(0)$. Because only one potential outcome is ever realized — the one corresponding to the treatment actually received — individual causal effects are unobservable. This "fundamental problem of causal inference" makes statistical assumptions unavoidable for identifying average causal effects.

Key Ideas

• Potential outcomes: The notation $Y_i(d)$ indexes all outcomes unit $i$ could have under all possible treatments. Only $Y_i(D_i)$ — the potential outcome under the actual treatment $D_i$ — is observed. The unobserved $Y_i(1-D_i)$ is the counterfactual.
• SUTVA (Stable Unit Treatment Value Assumption, Rubin 1978/1980): (a) no interference between units — unit $i$'s potential outcomes depend only on $i$'s own treatment, not others'; (b) no hidden treatment versions — there is a single, well-defined treatment $D=1$ and $D=0$. SUTVA is what allows writing $Y_i(Z, D)$ as $Y_i(D)$ and $D_i(Z)$ as $D_i(z_i)$.
• Ignorable assignment: Treatment is ignorable if assignment is independent of potential outcomes, possibly conditional on observed covariates. Random assignment makes this trivially true. In observational studies, ignorability requires the "no unmeasured confounders" assumption.
• Average treatment effect (ATE): $E[Y(1)-Y(0)]$ over the full population. Identified by random assignment.
• Average treatment effect on the treated (ATT): $E[Y(1)-Y(0)|D=1]$. Identified by matching, difference-in-differences (DiD), or specific natural experiments.
• Local Average Treatment Effect (LATE): $E[Y(1)-Y(0)|\text{complier}]$ — identified by instrumental variables (IV) under monotonicity. See Local Average Treatment Effect.
• Connection to structural equation models: The RCM embeds IV within potential outcomes (Angrist, Imbens, Rubin 1996), making the identifying assumptions transparent and substantively interpretable rather than cast in terms of unobservable disturbances.

How It Works

1. Define potential outcomes $Y_i(0), Y_i(1)$ for each unit before observing any data.
2. Choose a treatment assignment mechanism (random, observational, or instrument-based).
3. State assumptions sufficient to identify the desired estimand (ATE, ATT, LATE) — typically some form of ignorability, exclusion, or monotonicity.
4. Estimate the estimand from observed $(Y_i, D_i, Z_i, X_i)$ using the assumptions.
5. Perform sensitivity analysis to violations of key assumptions.

Why It Matters

The RCM provides a unified language for causal inference across statistics, economics, epidemiology, and social science. Before the RCM, causal claims from observational data were made informally or through structural equation models whose assumptions were opaque ("the disturbances are uncorrelated with the instrument"). The RCM makes every identifying assumption a statement about observable or potentially observable quantities — e.g., "holding $D$ fixed, the instrument $Z$ has no direct effect on $Y$" (exclusion restriction) — which is substantively evaluable even if not directly testable.

Heckman's Critique: R-1 to R-4 (Heckman 2008)

Heckman characterizes the Neyman-Rubin (NR) model by four postulates that distinguish it from the econometric structural approach:

• R-1 (No selection model): The NR framework postulates counterfactuals $Y_i(0)$ and $Y_i(1)$ without specifying a model for why individuals end up in the treatment they receive. The selection mechanism is treated as either randomized (experiment) or finessed via an instrument. The econometric approach explicitly models selection: an individual chooses treatment when the net benefit exceeds a threshold, which ties the selection index to the potential outcomes and enables identification from observational data without an instrument.
• R-2 (Objective ex post outcomes only): NR conditions on realized, objective outcomes — what actually happened. The econometric approach allows subjective evaluations — what individuals expected would happen before making a treatment choice (ex ante returns). Essential heterogeneity arises because agents act on ex ante expectations that may differ from ex post realizations (Cunha and Heckman 2007).
• R-3 (SUTVA): No interference between units; a single well-defined treatment. Shared by both frameworks as a working assumption, though both acknowledge it is violated in general equilibrium settings.
• R-4 (Recursive/triangular): The NR framework is inherently triangular: treatment is determined first (exogenously or by instrument), outcomes follow. Simultaneity — where $Y$ and $D$ jointly determine each other — has no analog in the NR framework. Haavelmo's (1943) simultaneous equations model handles this case through exclusion restrictions.

Three-Task Distinction (Heckman 2008)

Any causal analysis implicitly involves three separate tasks that must not be conflated:

1. Define the counterfactual: What would have happened to the treated units under the alternative? Holland's (1986) "no causation without manipulation" conflates this with identification — ruling out counterfactuals for attributes that cannot be manipulated. But defining counterfactuals and identifying them from data are logically separate steps.
2. Identify from ideal data: Given complete data on potential outcomes (e.g., a hypothetical randomized experiment), can the parameter of interest be computed? This is a mathematical question about estimands.
3. Identify from real data: Given the actual data available (observational, with selection), can the parameter be estimated? LATE conflates (2) and (3): it is simultaneously defined as the estimand and identified by the Wald ratio, so it is often unclear whether a researcher is making a definitional choice or an identification claim.

Open Questions

• SUTVA is often implausible in social settings with network effects, peer effects, or general equilibrium responses. How should LATE estimates from examiner IV (which assume no peer effects among applicants) be interpreted in the presence of spillovers?
• The RCM treats potential outcomes as fixed (frequentist) or as having a joint distribution (Bayesian). Do these two interpretations produce meaningfully different inference in finite samples?
• The three-task distinction implies that the choice of counterfactual is a modeling decision prior to any statistical analysis. For outcomes like "effect of race" or "effect of gender," how should analysts handle counterfactuals that cannot be physically realized?

Related

• Instrumental Variables
• Local Average Treatment Effect
• Marginal Treatment Effect
• Donald Rubin
• Joshua Angrist
• Guido Imbens
• James Heckman
• Trygve Haavelmo
• Paul Holland

Sources

• Angrist Imbens and Rubin 1996 — Identification of Causal Effects Using Instrumental Variables
• Angrist Imbens and Rubin 1996 — Identification of Causal Effects Using Instrumental Variables Rejoinder
• Heckman 1996 — Identification of Causal Effects Using Instrumental Variables Comment
• Heckman 2008 — Econometric Causality
• Moffitt 2005 — Remarks on the Analysis of Causal Relationships in Population Research