Causal Effect

Definition

A causal effect describes the change in an outcome (dependent variable) that is directly caused by a change in a treatment or independent variable, holding everything else constant.

• Put simply: “If X changes, how much does Y change because of X, not because of other factors?”
• This is stronger than correlation. Correlation only shows association, while causation means X actually produces a change in Y.

---

Example

• Medicine: Does taking a new drug (treatment) cause lower blood pressure (outcome)?
• Economics: Does increasing minimum wage (policy change) cause higher unemployment (outcome)?
• Education: Does attending extra tutoring (treatment) cause higher exam scores (outcome)?

---

Formal Representation

In causal inference, we often use the potential outcomes framework (Rubin Causal Model):

• $Y(1)$: Outcome if the unit receives the treatment.
• $Y(0)$: Outcome if the unit does not receive the treatment.
• Causal effect for a unit = $Y(1) – Y(0)$.

Problem: We can only observe one of them for each person (either treated or not). This is called the fundamental problem of causal inference.

---

Average Causal Effect (ACE)

Since individual causal effects are unobservable, we estimate the average across many units:

$ACE = E[Y(1) – Y(0)]$

• $E[\cdot]$: Expectation (average).
• Interpretation: On average, how much the treatment changes the outcome.

---

Identifying Causal Effect

To estimate causal effects, we need strategies that control for confounders (other factors that influence both treatment and outcome).

Common methods:

1. Randomized Controlled Trials (RCTs): Random assignment ensures fairness.
2. Matching / Stratification: Compare treated and control groups with similar characteristics.
3. Regression Adjustment: Control for covariates statistically.
4. Instrumental Variables (IV): Use an external factor correlated with treatment but not directly with the outcome.
5. Difference-in-Differences (DiD): Compare before/after between groups.
6. Causal Graphs (DAGs): Formal framework to reason about confounding.

---

Why Causal Effect Matters

• In science: To test hypotheses about mechanisms.
• In policy: To know whether interventions really work.
• In business: To measure the true impact of marketing, pricing, product changes, etc.

---

In short:
A causal effect is the true impact of one variable on another, not just correlation. It’s what would change in the outcome if we intervened and changed the treatment.