Differences-in-Differences

Summary

DD strategies use data with a time or cohort dimension to control for unobserved-but-fixed omitted variables. The key assumption is common trends: treatment and control groups would follow parallel paths in the absence of treatment.

Individual Fixed Effects

For panel data with individual $i$ observed at time $t$:

$$Y_{it}={\alpha }_i+{\lambda }_t+\rho D_{it}+X_{it}\delta +{\epsilon }_{it}$$

• ${\alpha }_i$: individual fixed effect (absorbs all time-invariant unobservables)
• ${\lambda }_t$: year effect (common time trend)
• Estimation: deviations from means (subtract individual averages) or first-differencing

Attenuation Bias

Fixed effects estimates are susceptible to attenuation from measurement error — year-to-year changes in mismeasured variables are mostly noise.

The DD Setup

When treatment varies at a group level (e.g., state policy changes):

$$Y_{ist}={\gamma }_s+{\lambda }_t+\beta D_{st}+{\epsilon }_{ist}$$

The DD estimator:

$${\hat{\beta }}_{DD}=({\bar{Y}}_{treat,after}-{\bar{Y}}_{treat,before})-({\bar{Y}}_{control,after}-{\bar{Y}}_{control,before})$$

Key Example: Minimum Wage (Card & Krueger, 1994)

• NJ raised minimum wage from $4.25to$5.05; PA did not
• DD estimate: employment increased by 2.76 FTE in NJ relative to PA
• Challenges the standard competitive labor market prediction

Common Trends Assumption

The identifying assumption: $E(Y_{0ist}|s,t)={\gamma }_s+{\lambda }_t$

• Treatment and control groups can have different levels but must share the same trend
• Testable with pre-treatment data (look for parallel pre-trends)
• State-specific trends as robustness check (requires 3+ periods)

Regression DD

$$Y_{ist}=\alpha +\gamma \cdot N{J}_s+\lambda \cdot d_t+\beta (N{J}_s\cdot d_t)+{\epsilon }_{ist}$$

Advantages:

• Easy to add covariates, additional states, and time periods
• Facilitates continuous “treatment intensity” designs
• Can include leads and lags to test for pre-trends

Fixed Effects vs. Lagged Dependent Variables

• If unobserved $a_i$ is the confounder → use fixed effects
• If past outcomes predict treatment → use lagged dependent variable
• These bracket the true effect: one tends to overestimate, the other underestimate

See Also

• The Experimental Ideal — the benchmark DD approximates
• Instrumental Variables — alternative quasi-experimental design
• Regression Discontinuity Designs — another strategy when treatment follows a rule
• Conditional Independence Assumption — what DD relaxes by using panel structure
• Omitted Variables Bias — the confounder fixed effects absorb
• Standard Errors and Clustering — clustering at the group level is essential for DD
• Research Questions in Econometrics — FAQ #3: identification strategies
• Mostly Harmless Econometrics - Overview
• Hierarchical Linear Models — Bayesian multilevel approach to varying intercepts and panel data
• Model Checking — posterior predictive checks for validating common trends assumptions
• Data Collection Models — Bayesian ignorability framework: DiD is the fix when standard ignorability fails
• Counterfactual Inference — explicit counterfactual framing of the treatment effect DiD estimates
• Randomization Inference - Overview — permutation/randomization inference for DiD designs