Econometrics Interactive

Difference-in-Differences

Estimate causal treatment effects by comparing changes over time between treatment and control groups. The gold standard for policy evaluation.

📊 The DiD Framework

DiD = (Y_T,post - Y_T,pre) - (Y_C,post - Y_C,pre)

Compare the change in treatment group to the change in control group. The difference removes time trends affecting both.

Key Insight: If both groups would have changed equally without treatment, any extra change in treatment group = causal effect.

	Pre	Post	Change
Treatment	66.8	110.1	+43.2
Control	57.4	88.6	+31.2
DiD			12.0

Parameters

-20 30

30 70

0 10

0 20

📊 Estimates

True Effect 15

DiD Estimate 12.0

Estimation Error -3.0

Treatment vs Control Over Time

Vertical gap after T5 between green line and dashed gray = treatment effect. DiD estimates this by comparing changes.

The Parallel Trends Assumption

✓ Valid DiD

Pre-treatment trends are parallel. Any difference post-treatment is attributable to the intervention.

✗ Invalid DiD

Pre-treatment trends diverge. Groups were already on different trajectories—DiD estimate is biased.

⚠️ Key Assumptions

Parallel Trends

Critical

Without treatment, groups would have followed same trajectory

No Spillovers

Critical

Treatment doesn't affect control group

Stable Composition

Group membership doesn't change due to treatment

No Anticipation

Behavior doesn't change before treatment

💰 Pricing Applications

What to Test

→ New pricing tier impact (launch in one region first)
→ Promotional campaign effectiveness
→ Feature launch impact on engagement
→ Regulatory change effects

When to Use DiD vs A/B

DiD: Can't randomize, observational data, policy changes
A/B: Can randomize, want cleanest causal estimate

R Code Equivalent

# Difference-in-Differences regression
library(fixest)

# Prepare data
panel_data <- data.frame(
  id = rep(1:n_units, each = n_periods),
  time = rep(1:n_periods, n_units),
  treated = rep(c(0, 1), each = n_units/2 * n_periods),
  post = as.numeric(time > treatment_period),
  outcome = y
)

# DiD regression with fixed effects
did_model <- feols(outcome ~ treated * post | id + time, 
                   data = panel_data)

# The coefficient on 'treated:post' is the DiD estimate
summary(did_model)

# Event study (test parallel trends)
event_study <- feols(outcome ~ i(time, treated, ref = treatment_period) | id + time,
                     data = panel_data)
iplot(event_study)  # Should be flat pre-treatment

✅ Key Takeaways

• DiD estimates causal effects from observational data
• Parallel trends assumption is critical
• Compare CHANGES, not levels

• Use when randomization isn't possible
• Validate with pre-treatment trend tests
• Works for policy/feature rollouts