Causal Inference: Beyond Correlation

Correlation vs Causation

Causal inference is the process of determining whether one variable truly causes a change in another, as opposed to merely being correlated.

Three conditions for causation:

1. Temporal precedence: Cause precedes effect
2. Covariation: Cause and effect are related
3. No confounders: The relationship is not explained by a third variable

Potential Outcomes Framework

Rubin Causal Model

For individual i, define:

• Y_i(1) = outcome if treated (potential outcome under treatment)
• Y_i(0) = outcome if not treated (potential outcome under control)

Fundamental Problem of Causal Inference

Average Treatment Effect (ATE)

Conditional Average Treatment Effect (CATE)

SUTVA (Stable Unit Treatment Value Assumption)

1. No interference: One unit's treatment does not affect another's outcome
2. Consistency: Only one version of each treatment level

Propensity Score Matching

Key Property

Matching Methods

Implementation

Difference-in-Differences (DiD)

Two-Period Model

Parallel Trends Assumption

Implementation

Instrumental Variables (IV)

When to Use IV

When treatment is correlated with unobserved confounders (endogeneity):

Instrument Requirements

1. Relevance: Z is correlated with X
2. Exclusion: Z affects Y only through X
3. Exogeneity: Z is uncorrelated with confounders

Two-Stage Least Squares (2SLS)

Confounding:

    U (unobserved)
   / \
  v   v
  X   Y     Endogeneity: Cov(X, ε) ≈ 0

Instrumental Variable:

    U (unobserved)
   / \
  v   v
  X   Y     IV Z satisfies: Z ⊥ U, Z -> X
  ^
  |
  Z (instrument)

Regression Discontinuity Design (RDD)

Sharp RDD

Directed Acyclic Graphs (DAGs)

Pearl's Causal Hierarchy

1. Association: P(Y | X) — Seeing
2. Intervention: P(Y | do(X)) — Doing
3. Counterfactual: P(Y_x | X', Y') — Imagining

do-Calculus

Backdoor Criterion

A set Z satisfies the backdoor criterion relative to (X, Y) if:

1. No node in Z is a descendant of X
2. Z blocks every path between X and Y that contains an arrow into X

Key Takeaways

Practice Exercises

1. Propensity Score Matching: Perform PSM on job training data and check covariate balance before/after matching
2. DiD Analysis: Replicate the minimum wage study and test the parallel trends assumption
3. IV Estimation: Implement IV using quarter of birth as an instrument for education. Check instrument strength
4. RDD Design: Apply RDD to a dataset with a known cutoff. Estimate the effect at different bandwidths
5. Discussion: When would you prefer DiD over propensity score matching? How do you validate the exclusion restriction?