🌐 DS Causal Inference: PSM, Difference-in-Differences & Synthetic Control

Executive Summary: Causal inference separates senior Data Scientists and Product Analysts from junior data query roles. In real-world tech systems, ethical constraints, spillover dynamics, and marketplace network effects frequently prohibit randomized controlled trials (A/B testing). This guide systematically explores Rubin’s Potential Outcomes framework, Pearl’s Causal DAGs, Propensity Score Matching (PSM), Difference-in-Differences (DiD), Synthetic Control (SCM), and Instrumental Variables (2SLS) with mathematical derivations and pure Python implementations.

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💡 Interactive Mermaid Architecture

graph TD
    subgraph A["1. Core Causal Inference Arsenal"]
        A1["PSM & IPW: Propensity Score Matching & Weighting (Observed Confounders)"]
        A2["DiD: Difference-in-Differences (Time-Invariant Unobserved Fixed Effects)"]
        A3["SCM: Synthetic Control Method (Convex Combination for Single Unit Counterfactual)"]
        A4["IV & 2SLS: Instrumental Variables (Endogeneity & Unobserved Confounders)"]
        A1 --> A2 --> A3 --> A4
    end

    subgraph B["2. Industrial Validation & Robustness Suite"]
        B1["Parallel Trends & Event Study Specification"]
        B2["Covariate Balance & SMD Caliper Matching"]
        B3["In-Time & In-Space Placebo Permutation Tests"]
        B4["Weak Instrument F-Test (F > 10) & Overidentification"]
        B1 --> B2 --> B3 --> B4
    end

    A --> B

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Chapter 1: DiD Theory and the Parallel Trends Assumption

Difference-in-Differences (DiD) estimates the net causal effect by comparing the pre-post changes across treatment and control groups:
$$text{ATE} = (bar{Y}{T, text{post}} – bar{Y}}}) – (bar{Y{C, text{post}} – bar{Y})$$}

Its fundamental lifeline is the Parallel Trends Assumption: in the absence of treatment, the average outcome of the treatment group would have evolved parallel to that of the control group.

The Two-Way Fixed Effects (TWFE) regression is parameterized as:
$$Y_{it} = alpha_i + lambda_t + beta D_{it} + epsilon_{it}$$
where $alpha_i$ absorbs individual time-invariant heterogeneity, $lambda_t$ absorbs macro temporal shocks, $D_{it} = text{Treat}_i times text{Post}_t$, and $beta$ captures the causal Average Treatment Effect (ATE).

💡 Intuitive Understanding: DiD performs a “double subtraction”: first computing the within-unit change (canceling baseline differences), then computing the across-group delta (canceling macroeconomic trends).

🎤 Interview Pitch: “DiD identifies causality through dual subtractions under parallel trends: $text{ATE} = (T_{text{post}} – T_{text{pre}}) – (C_{text{post}} – C_{text{pre}})$. When pre-trends diverge, Event Study specifications diagnose anticipation or differential growth paths.”

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Chapter 2: Pure Python DiD Estimator

def pure_python_did_estimate(y_treat_post: float, y_treat_pre: float, y_ctrl_post: float, y_ctrl_pre: float) -> float:
    delta_treat = y_treat_post - y_treat_pre
    delta_ctrl = y_ctrl_post - y_ctrl_pre
    return delta_treat - delta_ctrl

if __name__ == "__main__":
    ate = pure_python_did_estimate(15.0, 10.0, 11.0, 9.0)
    print("✅ DiD Causal ATE Estimate:", ate)  # (15-10) - (11-9) = 5 - 2 = 3.0

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Chapter 3: Rubin Potential Outcomes & Pearl’s Causal DAGs

Causal inference is the science of uncovering missing Counterfactuals. For any unit $i$:
* $Y_i(1)$: Potential outcome under treatment ($T=1$).
* $Y_i(0)$: Potential outcome under control ($T=0$).

The Fundamental Problem of Causal Inference: we only observe $Y_i = T_i Y_i(1) + (1-T_i) Y_i(0)$. One of the potential outcomes is always unobservable.

1. Three Core Identifiability Assumptions

1. SUTVA (Stable Unit Treatment Value Assumption): No cross-unit interference and no hidden variations of treatments.
2. Unconfoundedness (Conditional Ignorability): $(Y(1), Y(0)) perp T mid X$. Given covariates $X$, treatment assignment is as good as random.
3. Positivity (Common Support / Overlap): $0 < P(T=1 mid X) < 1$ for all support points of $X$.

2. Causal DAGs and the Backdoor Criterion

• Chain ($X to M to Y$): Conditioning on mediator $M$ blocks causal flow from $X$ to $Y$.
• Fork / Confounder ($X leftarrow C to Y$): Common cause $C$ induces spurious correlation. Must condition on $C$ to block the backdoor path.
• Collider ($X to C leftarrow Y$): Common effect $C$ blocks paths by default; conditioning on collider $C$ erroneously opens a spurious dependency (Berkson’s bias).

Backdoor Criterion: A covariate set $Z$ identifies the causal effect if: (1) no node in $Z$ is a descendant of $T$; (2) $Z$ blocks every backdoor path between $T$ and $Y$.
$$P(Y mid do(T=t)) = sum_z P(Y mid T=t, Z=z) P(Z=z)$$

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Chapter 4: PSM & Doubly Robust Estimation (AIPW)

High-dimensional covariates create the curse of dimensionality. Rosenbaum & Rubin proved the Propensity Score Theorem:
$$e(X) = P(T=1 mid X) implies (Y(1), Y(0)) perp T mid e(X)$$

1. PSM Production Workflow

graph LR
    P1["1. Logistic Regression for e(X)"] --> P2["2. Common Support Trimming"]
    P2 --> P3["3. Caliper Nearest-Neighbor Matching"]
    P3 --> P4["4. Covariate Balance SMD < 0.1 Check"]
    P4 --> P5["5. Post-Match ATE / ATT Estimation"]

1. Score Estimation: Train Logistic Regression / GBDT to predict probability $e(X)$.
2. Common Support Trimming: Discard treatment units with $e(X) > 0.95$ and control units with $e(X) < 0.05$.
3. Caliper Matching: Restrict matches within distance threshold $text{Caliper} le 0.2 sigma_{text{logit}(e)}$.
4. Balance Verification: Compute Standardized Mean Difference (SMD). Balance is achieved when $text{SMD} < 0.1$ across all features.

2. Augmented Inverse Probability Weighting (AIPW)

$$hat{tau}{text{AIPW}} = frac{1}{N} sum right) right]$$}^N left[ left( hat{mu}_1(X_i) + frac{T_i (Y_i – hat{mu}_1(X_i))}{hat{e}(X_i)} right) – left( hat{mu}_0(X_i) + frac{(1-T_i)(Y_i – hat{mu}_0(X_i))}{1-hat{e}(X_i)

Doubly Robust Guarantee: If either the outcome model $mu(X)$ or the propensity score model $e(X)$ is correctly specified, $hat{tau}_{text{AIPW}}$ is an unbiased and asymptotically normal estimator.

import numpy as np

def pure_python_psm_nearest_neighbor(
    X_treat: np.ndarray, y_treat: np.ndarray,
    X_ctrl: np.ndarray, y_ctrl: np.ndarray,
    caliper: float = 0.05
) -> dict:
    ps_treat = 1.0 / (1.0 + np.exp(-X_treat.mean(axis=1)))
    ps_ctrl = 1.0 / (1.0 + np.exp(-X_ctrl.mean(axis=1)))

    matched_treat_y = []
    matched_ctrl_y = []

    for i, p_t in enumerate(ps_treat):
        diffs = np.abs(ps_ctrl - p_t)
        best_idx = np.argmin(diffs)
        if diffs[best_idx] <= caliper:
            matched_treat_y.append(y_treat[i])
            matched_ctrl_y.append(y_ctrl[best_idx])

    att = np.mean(matched_treat_y) - np.mean(matched_ctrl_y)
    return {
        "matched_pairs": len(matched_treat_y),
        "ATT_estimate": float(att),
        "mean_treat": float(np.mean(matched_treat_y)),
        "mean_ctrl": float(np.mean(matched_ctrl_y))
    }

if __name__ == "__main__":
    np.random.seed(42)
    Xt = np.random.randn(100, 3) + 0.5
    Xc = np.random.randn(300, 3)
    yt = Xt.sum(axis=1) + 2.5 + np.random.randn(100) * 0.2
    yc = Xc.sum(axis=1) + np.random.randn(300) * 0.2

    res = pure_python_psm_nearest_neighbor(Xt, yt, Xc, yc)
    print("✅ PSM Result:", res)

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Chapter 5: Synthetic Control Method (SCM)

When policies occur at an aggregate unit level (single city, country, or app platform), Synthetic Control (SCM) models the counterfactual trajectory via a convex combination of untreated donor pool units:
$$W^* = argmin_W (X_1 – X_0 W)^T V (X_1 – X_0 W) quad text{s.t.} quad w_j ge 0, sum w_j = 1$$

Dynamic causal treatment effect:
$$hat{tau}{1t} = Y, quad forall t > T_0$$} – sum_{j=2}^{J+1} w_j^* Y_{jt

Placebo Permutation Testing

Run placebo trials on each donor pool unit iteratively. Evaluate the Post/Pre RMSPE ratio:
$$text{RMSPE Ratio} = frac{text{RMSPE}{text{post}}}{text{RMSPE}$$}}
Significance is attained if the treated unit ranks at the extreme top percentile of the empirical permutation distribution ($p = 1/(J+1) < 0.05$).

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Chapter 6: Instrumental Variables (IV) & 2SLS

When unobserved confounders $U$ induce endogeneity, Ordinary Least Squares (OLS) produces severe bias. Instrumental variable $Z$ breaks endogeneity.

graph LR
    Z["Instrument Z (Randomized Promo Push)"] --> T["Endogenous Treatment T (Subscribed)"]
    T --> Y["Outcome Y (Long-term LTV)"]
    U["Unobserved Confounder U (User Loyalty)"] --> T
    U --> Y

Core Assumptions

1. Instrument Relevance: $text{Cov}(Z, T) neq 0$. First-stage $F$-statistic must exceed 10 ($F > 10$) to rule out weak instruments.
2. Exclusion Restriction: $Z$ affects $Y$ solely through $T$, meaning $Z perp Y mid (T, U)$.

Two-Stage Least Squares (2SLS)

• Stage 1: $T = gamma_0 + gamma_1 Z + gamma_2 X + v implies hat{T}$
• Stage 2: $Y = beta_0 + beta_{text{IV}} hat{T} + beta_2 X + epsilon$

The Wald Estimator identifies the Local Average Treatment Effect (LATE) for the complier subpopulation:
$$beta_{text{IV}} = frac{text{Cov}(Y, Z)}{text{Cov}(T, Z)} = frac{bar{Y}{Z=1} – bar{Y}}}{bar{T{Z=1} – bar{T}$$}

def pure_python_wald_iv_estimate(y_z1: float, y_z0: float, t_z1: float, t_z0: float) -> float:
    numerator = y_z1 - y_z0
    denominator = t_z1 - t_z0
    if abs(denominator) < 1e-8:
        raise ValueError("Weak instrument: first-stage uptake delta is near 0.")
    return numerator / denominator

if __name__ == "__main__":
    late = pure_python_wald_iv_estimate(120.0, 80.0, 0.6, 0.2)
    print("✅ 2SLS LATE Estimate:", late)  # 100.0

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Chapter 7: Causal Method Selection Decision Tree

graph TD
    Q0{"Can you run a randomized A/B test?"}
    Q0 -->|Yes| A0["Randomized Controlled Trial (RCT) + CUPED"]
    Q0 -->|No| Q1{"Is there an explicit cutoff score / threshold?"}

    Q1 -->|Yes| M1["Regression Discontinuity Design (RDD)"]
    Q1 -->|No| Q2{"Is treatment applied to an aggregate region or individual users?"}

    Q2 -->|Single Region / Full Rollout| M2["Synthetic Control (SCM) or CausalImpact"]
    Q2 -->|Multiple Panel Units over Time| Q3{"Do you have pre/post longitudinal panel data?"}

    Q3 -->|Yes| M3["Difference-in-Differences (DiD) + Event Study"]
    Q3 -->|No (Cross-Sectional Only)| Q4{"Are there severe unobserved confounders?"}

    Q4 -->|Unobserved Confounders Exist| M4["Exogenous Shocks & Instrumental Variables (IV / 2SLS)"]
    Q4 -->|All Confounders are Observed| M5["PSM Matching / IPW Weighting / Doubly Robust (AIPW)"]