Advanced Causality

Topic 07 covers experimental design and quasi-experimental methods (DiD, RDD, IV). This topic goes deeper into the causal inference framework: DAGs for formalizing causal assumptions, the potential outcomes framework for defining causal effects, and modern ML-based causal methods (double ML, causal forests) for estimating heterogeneous treatment effects at scale. These tools are essential for senior practitioners who must design, critique, and defend causal claims from observational data.

1. DAGs and Structural Causal Models

A DAG (directed acyclic graph) has nodes = variables and directed edges = direct causal relationships. It is acyclic — no feedback loops in the static model.

Three Fundamental Structures

Chain (A → B → C): B mediates A's effect on C. Conditioning on B blocks the path.

Fork (A ← B → C): B confounds the A–C relationship. Conditioning on B blocks the spurious association.

Collider (A → B ← C): B is caused by both A and C. B is a collider. The path is blocked — A and C are independent. But conditioning on B opens this path and creates spurious dependence. This is collider bias.

d-Separation

Two nodes are d-separated (conditionally independent) given a set Z if every path between them is blocked by Z. Chains and forks are blocked by conditioning on the middle node; colliders are blocked by not conditioning on them — and are opened by conditioning.

Confounders vs. Mediators

• Confounder (fork): creates spurious association — should be adjusted for
• Mediator (chain): transmits the causal effect — should not be adjusted for if you want the total effect

2. 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 backdoor path between X and Y (paths with an arrow into X)

If Z satisfies the backdoor criterion, the causal effect of X on Y is identified by the adjustment formula:

You can estimate the causal effect from observational data by conditioning on Z — via stratification, regression, or inverse probability weighting (IPW).

Worked Example: Education and Earnings

Does education (X) cause earnings (Y)?

• Confounder: parental income (C) affects both education access and earnings. Backdoor path: X ← C → Y. Solution: condition on C to block this path.
• Mediator: skills (M) is on the path X → M → Y. Do not condition on M if you want the total effect of education on earnings.

3. Potential Outcomes Framework

The Rubin Causal Model defines causal effects via potential outcomes:

• $Y_i(1)$: potential outcome for unit $i$ if treated
• $Y_i(0)$: potential outcome for unit $i$ if untreated
• Individual treatment effect: $\tau_i = Y_i(1) - Y_i(0)$
• Fundamental problem of causal inference: we only observe one potential outcome per unit; the other is counterfactual

Estimands

• ATE: $\tau_{\text{ATE}} = E[Y(1) - Y(0)]$ — average over the full population
• ATT: $\tau_{\text{ATT}} = E[Y(1) - Y(0) \mid T=1]$ — average among those who were treated
• CATE: $\tau(x) = E[Y(1) - Y(0) \mid X=x]$ — heterogeneous effect for subgroup $x$

Identification Assumptions

• Unconfoundedness / ignorability: $(Y(0), Y(1)) \perp T \mid X$ — no unmeasured confounders given X
• Overlap / positivity: $0 < P(T=1 \mid X) < 1$ — every unit has positive probability of treatment and control
• SUTVA: stable unit treatment value assumption — no interference between units, no hidden versions of treatment

4. Double/Debiased Machine Learning (DML)

Motivation: if we regress Y ~ T + X with LASSO, regularization shrinks the T coefficient toward zero along with the X coefficients — biasing the treatment effect estimate.

DML (Robinson 1988 / Chernozhukov et al. 2018) uses the partialing-out approach:

The residuals $\tilde{Y} = Y - \hat{E}[Y \mid X]$ and $\tilde{T} = T - \hat{E}[T \mid X]$ are computed by any ML model using cross-fitting (k-fold). Then $\theta$ is estimated by OLS regression of $\tilde{Y}$ on $\tilde{T}$.

Why it works: Neyman orthogonality — the target parameter $\theta$ is orthogonal to errors in the nuisance functions $E[Y \mid X]$ and $E[T \mid X]$ in a first-order sense. Small estimation errors in those functions don't propagate to bias in $\hat{\theta}$. Cross-fitting prevents overfitting from inflating nuisance predictions on the same data used to estimate $\theta$.

from sklearn.linear_model import LassoCV
from sklearn.model_selection import cross_val_predict
import numpy as np

# Step 1: Partial out confounders X from Y and T
Y_resid = Y - cross_val_predict(LassoCV(), X, Y, cv=5)
T_resid = T - cross_val_predict(LassoCV(), X, T, cv=5)

# Step 2: OLS on residuals — this is the DML estimate of theta
theta = np.dot(T_resid, Y_resid) / np.dot(T_resid, T_resid)
print(f"DML ATE estimate: {theta:.4f}")

# Using EconML for full inference with confidence intervals
from econml.dml import LinearDML
dml = LinearDML(model_y=LassoCV(), model_t=LassoCV(), cv=5)
dml.fit(Y, T, X=X)
print(dml.effect_inference(X).summary_frame())

5. Causal Forests and Heterogeneous Treatment Effects

Standard DML estimates the ATE — a single number. But treatment effects often vary by subgroup (heterogeneous treatment effects / CATE).

Causal forests (Wager & Athey, 2018) adapt random forests to estimate CATE. Each tree is grown to maximize heterogeneity in treatment effects rather than prediction accuracy. Key property: honesty — each tree uses a held-out subsample to fit leaf-level treatment effects, providing asymptotically valid confidence intervals.

from econml.grf import CausalForest

cf = CausalForest(n_estimators=500, min_samples_leaf=20, random_state=42)
cf.fit(X_train, T_train, Y_train)

# Estimate CATE for each test observation
cate = cf.predict(X_test)             # shape: (n_test,)
cate_lb, cate_ub = cf.predict_interval(X_test, alpha=0.05)  # 95% CI

# Find most responsive subgroup
top_quartile = X_test[cate > np.percentile(cate, 75)]
print(f"Average CATE (top quartile): {cate[cate > np.percentile(cate, 75)].mean():.4f}")

Business application: a promotional offer has ATE = +$5 revenue lift. But a causal forest shows CATE ranges from −$2 (high-value customers who would have bought anyway) to +$15 (price-sensitive mid-tier customers). Targeting only the mid-tier segment triples ROI.

6. Sensitivity Analysis

No observational study is immune to unmeasured confounding. Sensitivity analysis asks: how strong would an unmeasured confounder need to be to explain away the observed effect?

Rosenbaum bounds: parameterize by $\Gamma$ (odds ratio of treatment selection due to unmeasured confounder). At $\Gamma = 1$, there is no hidden bias. Find the smallest $\Gamma$ at which the conclusion reverses. If the effect is robust to $\Gamma = 2$, unmeasured confounders would need to double the odds of treatment selection to nullify the finding.

E-value: the minimum association (on the risk ratio scale) that an unmeasured confounder would need with both treatment and outcome to explain away the effect:

Large E-values indicate robust findings — a confounder would need implausibly strong associations to explain away the result.

Practice Questions