Statisticsmedium

Treatment Effect of D on Y Where D Is Binary and Y Is Continuous

Question

Consider the treatment effect of

D

Y

where

D

is a binary treatment variable and

Y

is a continuous outcome. Discuss identification strategies, potential outcomes framework, and estimation methods.

Potential outcomes, identification strategies, and estimation methods for causal inference.

Average Treatment Effect

E[Y(1)−Y(0)]

ATE = E[Y(1) − Y(0)] under the potential outcomes framework

1Potential Outcomes Framework

For each individual

i

, define potential outcomes

Y_i(1)

(outcome if treated) and

Y_i(0)

(outcome if untreated). The individual treatment effect is

\tau_i = Y_i(1) - Y_i(0)

Fundamental Problem of Causal Inference

We can never observe both

Y_i(1)

and

Y_i(0)

for the same individual. This is the **fundamental problem** — we must estimate the counterfactual from data on other individuals.

  Potential Outcomes:

  Individual i:   D=1 → Yᵢ(1) (observed)
                  D=0 → Yᵢ(0) (counterfactual)

  τᵢ = Yᵢ(1) − Yᵢ(0) ← never observed directly

2Average Treatment Effect (ATE)

The ATE is

\tau_{ATE} = E[Y(1) - Y(0)]

. Under **random assignment** (

D \perp (Y(0), Y(1))

\tau_{ATE} = E[Y|D=1] - E[Y|D=0]

Randomization Solves Everything

With random treatment assignment, the simple difference in means is an unbiased estimator of the ATE. This is why **randomized controlled trials** (RCTs) are the gold standard for causal inference.

3Selection Bias in Observational Data

Without randomization:

E[Y|D=1] - E[Y|D=0] = \tau_{ATE} + \underbrace{E[Y(0)|D=1] - E[Y(0)|D=0]}_{\text{Selection Bias}}

Conditional Independence (CIA)

D \perp (Y(0), Y(1)) | X

— controlling for covariates

X

removes the selection bias. This is also called **unconfoundedness** or the **ignorability** assumption. It requires that all confounders are observed.

4Estimation Methods

Method	Key Assumption	When to Use	Strength
OLS with controls	CIA: D ⊥ Y(d)\|X	Rich covariate data	Simple, transparent
Propensity Score Matching	CIA + overlap	Many covariates	Reduces dimension
IPW	CIA + overlap	Flexible, semiparametric	Doubly robust variants
Instrumental Variables	Valid instrument Z	CIA fails, instrument available	Handles unobserved confounders
RDD	Continuity at cutoff	Treatment assigned by threshold	Quasi-experimental
Diff-in-Diff	Parallel trends	Panel data, policy change	Controls for fixed effects

ATE (randomized)E[Y|D=1] − E[Y|D=0]

ATE (observational, CIA)E_X[E[Y|D=1,X] − E[Y|D=0,X]]

LATE (IV)Cov(Y,Z)/Cov(D,Z)

GOLD STANDARDRandomized experiment

Quiz

Test your understanding with these questions.

What is the fundamental problem of causal inference?

Under random assignment, the ATE can be estimated by: