1. Causal Inference Models Overview

A. Potential Outcomes Framework (Rubin Causal Model)

• Concept: Each unit has a potential outcome for each possible treatment.
• Key Methods:
  • Randomized Controlled Trials (RCTs) – Gold standard for causal inference.
    • In here, CUPED (Controlled Experiment Using Pre-Experiment Data) is a variance reduction technique. We compute a pre-experiment covariate X that is related to the outcome Y. Usually, choose X to be the pre-experiment Y. Then use X to compute an adjusted Y, Y’, and then use Y’ to compare between T and C groups (instead of Y).
    • The result? The test becomes more sensitive, meaning we can detect treatment effects with a smaller sample size or get more precise confidence intervals.
  • Matching (Propensity Score Matching, Nearest Neighbor Matching) – Matches treated and control units with similar characteristics.
  • Inverse Probability Weighting (IPW) – Weights units to balance treatment and control groups.
  • Difference-in-Differences (DiD) – Compares pre/post-treatment differences between treated and control groups.
• Assumptions:
  • Stable Unit Treatment Value Assumption (SUTVA) – No interference between units.
  • Unconfoundedness (Conditional Independence) – No unmeasured confounders.
  • Common Support (Overlap Condition) – Sufficient overlap in covariates between treated and control groups.
  • For DiD:
    • Parallel Trends: T and C groups would have evolved similarly without treatment (i.e. must have similar pre-treatment trends).
      • Example: If you’re studying the impact of a minimum wage increase on employment, but the treated regions already had a declining employment trend before the policy change, the results may be misleading.
    • No Anticipation: Treatments doesn’t affect the outcome before implementation.
      • Example: If businesses preemptively reduce hiring before a new minimum wage law takes effect, this would violate the assumption.
    • No Simultaneous Confounding Shocks: No other event affects T/C groups differently at the same time.
      • Example: If a new tax policy was introduced at the same time as the minimum wage increase, it would be hard to separate their effects on employment.
    • Stable Composition of Groups: No selection bias or differential attrition or migration. This ensures that differences are due to treatment, not changes in group characteristics.
      • Example of violation: If user self-select in to T group. If higher-skilled workers move out of states with a minimum wage increase, the observed employment effects might be due to worker mobility, not the policy itself.

B. Structural Causal Models (SCMs, Pearl’s Causal Graphs)

• Concept: Uses Directed Acyclic Graphs (DAGs) to model causal relationships explicitly.
• Key Methods:
  • Do-Calculus – Adjusts for confounding using causal graphs.
  • Instrumental Variables (IV) – Uses external factors to estimate causal effects when treatment is endogenous.
  • Front-Door Adjustment – Identifies causal effects using mediators.
• Assumptions:
  • Causal Sufficiency – The DAG correctly represents the causal relationships.
  • Instrument Validity (for IV) – Instrument affects the outcome only through treatment. This entails 2 assumptions below. We say that an IV is valid if it satisfies both.
    • Relevance - The IV must be strongly correlated with the treatment. This is so that the IV can generate enough variation in the treatment. If not, we have a “weak IV bias”.
    • Exclusion Restriction – No direct or other indirect path from instrument to outcome.
    • Explain these two by example: Study the effect of education on income (target). Here, education is expressed by number of years for schooling (treatment).
      • If choose IV = distance to the nearest college: Exclusion assumption is violated, since living near a college also improves networking opportunities or access to better jobs independent of schooling.
      • If choose IV = education reform laws: Might be possible, since it can affect the treatment and not likely to affect income directly.

C. Regression-Based Approaches

• Concept: Uses statistical models to control for confounders and estimate causal effects.
• Key Methods:
  • Ordinary Least Squares (OLS) with Covariate Adjustment – Assumes no unobserved confounders.
  • Fixed Effects Models – Removes unit-specific biases by differencing within units over time.
  • Synthetic Control Method – Creates a weighted combination of control units to estimate counterfactual outcomes.
• Assumptions:
  • Linearity (for OLS) – Relationship between treatment and outcome is linear.
  • No Time-Varying Confounders (for Fixed Effects) – Confounders do not change over time.
  • Parallel Trends (for Synthetic Control) – Control units evolve similarly to the treated unit before treatment.

D. Machine Learning-Based Approaches

• Concept: Uses ML to improve causal effect estimation.
• Key Methods:
  • Causal Forests (Generalized Random Forests) – Uses random forests to estimate heterogeneous treatment effects.
  • Double Machine Learning (DML) – Uses ML to flexibly control for confounders while maintaining valid inference.
  • Uplift Modeling – Predicts differential treatment effects at the individual level.
  • Deep Causal Models – Uses deep learning for estimating complex causal relationships.
• Assumptions:
  • Unconfoundedness (for Causal Forests, DML) – No unobserved confounders.
  • Correct Model Specification (for ML methods) – Models must be properly tuned and validated.
  • Sufficient Data (for Deep Learning) – Large datasets are needed to avoid overfitting.

2. Comparison of Causal Inference Models

3. Choosing the Right Method

• If you have randomized data → RCTs.
• If dealing with observational data:
  • Matching/IPW if confounders are well-measured.
  • DiD/Fixed Effects if longitudinal data is available.
  • SCMs (DAGs, IV) if confounding is complex.
  • Machine Learning (Causal Forests, Double ML) if high-dimensional data and heterogeneity matter.

4. Additional Notes

• CUPED is a part of RCT framework, rather than observational methods like DiD or IV.