Difference-in-Differences: Policy Effect Estimator

Policy Design

True treatment effect 15.0

Parallel-trends violation 0.00

Noise (σ) 4.0

Sample size / period 15

DiD Estimate

DiD estimator

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True effect

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Bias (DiD − true effect)

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Pre-trend gap (parallel-trends check)

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How it works

Difference-in-differences (DiD) estimates a causal treatment effect from panel data by comparing the change in outcomes for a treated group against the change for a control group across the same time span:

DiD = [ Ȳ(treat, post) − Ȳ(treat, pre) ]
    − [ Ȳ(control, post) − Ȳ(control, pre) ]

This differencing cancels out any fixed, time-invariant gap between the groups and any trend that affects both groups equally — it only needs the parallel trends assumption: absent treatment, the two groups would have moved in parallel. The dashed line is the counterfactual this assumption implies for the treated group: its own pre-period average, shifted by the control group's actual post-period change.

True treatment effect — the real causal bump added to the treated group from the policy period onward. Compare it to the DiD estimator above.
Parallel-trends violation — an extra slope applied only to the treated group, active in both pre- and post-periods. Even a small nonzero value breaks the assumption: the pre-trend gap stat turns non-zero and the DiD estimate becomes biased even though the estimator's formula hasn't changed.
Noise (σ) — per-observation random scatter, showing DiD is a statistical estimate, not an exact readout.
Sample size — more observations per period shrink the scatter of individual points around each group's mean trend.

Real-world use: DiD is the workhorse of policy evaluation — minimum-wage changes, program rollouts, natural experiments — precisely because it needs no randomization, only a comparable control group and (approximately) parallel pre-trends.

Treated group Control group Counterfactual (parallel-trends)

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