Conditional Independence Assumption

Summary

The CIA states that, conditional on observed covariates $X_i$, potential outcomes are independent of treatment assignment. This is the key assumption that gives regression a causal interpretation — sometimes called “selection on observables.”

Formal Statement

For multi-valued treatment $s_i$ with potential outcomes $Y_{si}\equiv f_i(s)$:

$$Y_{si}\perp \perp S_i|X_i$$

This means that conditional on $X_i$, treatment is “as good as randomly assigned.”

From CIA to Causal Regression

Under the CIA with a linear constant-effects model $f_i(s)=\alpha +\rho s+{\eta }_i$:

1. Decompose ${\eta }_i=X_i^{\prime }\gamma +v_i$ (the part explained by observables + remainder)
2. The CIA ensures $v_i$ is uncorrelated with $s_i$ conditional on $X_i$
3. This gives the causal regression: $Y_i=\alpha +\rho s_i+X_i^{\prime }\gamma +v_i$

The coefficient $\rho$ has a causal interpretation as the average causal effect.

When Does the CIA Hold?

• In randomized experiments, by design (possibly conditional on stratification variables)
• In observational studies, when you believe all confounders are observed and controlled for
• The big question: what are the right control variables $X_i$?

Bad Controls

Not all controls are good. Variables that are themselves affected by treatment (“bad controls”) can introduce bias rather than remove it. Only include pre-treatment covariates or variables known to be unaffected by treatment.

Related Concepts

Concept	Relationship
Selection bias	What the CIA eliminates
OVB	What happens when CIA fails
IV	Alternative when CIA is implausible
Propensity score	Dimension-reducing tool under the CIA

See Also

• Regression and the CEF
• Omitted Variables Bias
• The Selection Problem
• Data Collection Models — Bayesian treatment of ignorability, the direct parallel to CIA
• Directed Acyclic Graphs — DAGs provide the algorithmic tool for finding the adjustment set that satisfies the CIA