ECMT3120 Chap.6 Pooled Cross Sections and Difference-in-Differences
Pooled Cross Sections and Difference-in-Differences
Pooled cross sections combine independent samples drawn at different dates and use time indicators to describe aggregate shifts. Difference-in-differences adds a treatment group indicator and a treatment-by-post interaction to recover a relative change.
This chapter translates the four cell means into a regression coefficient, distinguishes repeated cross sections from genuine panels, and makes the parallel trends assumption explicit. It also covers composition changes, anticipation, clustered uncertainty and event timing as threats to a causal interpretation.
What this chapter covers
- 01
Independent cross sections pooled across time
- 02
Time indicators and changing intercepts
- 03
Group indicators and baseline gaps
- 04
Treatment by post interaction
- 05
Four cell mean difference-in-differences
- 06
Parallel trends and counterfactual change
- 07
Composition, anticipation and spillover threats
- 08
Inference aligned with treatment assignment
Worked example · free
Recover a policy effect from four means
- +1Compute the treated group change as 16 minus 12 equals 4.
- +1Compute the comparison group change as 11 minus 10 equals 1.
- +2Subtract the changes to obtain 4 minus 1 equals 3.
- +1Identify 3 as the coefficient on treated times post in the saturated two-by-two regression.
- +1State that the comparison change estimates the untreated counterfactual trend for the treated group.
- +2Condition the causal interpretation on parallel trends, stable composition, no anticipation and no spillovers.
Key terms
- Pooled cross section
- A dataset formed by stacking independently sampled cross sections from two or more dates.
- Post indicator
- A binary variable equal to one for observations measured after the policy or event.
- Treatment indicator
- A binary variable marking membership in the group exposed to the policy.
- Difference-in-differences
- The treated before-after change minus the comparison before-after change.
- Parallel trends
- The assumption that untreated outcomes for treated and comparison groups would have changed by the same amount.
- Anticipation
- Behavioural adjustment before formal treatment because agents expect the intervention.
Pooled Cross Sections and Difference-in-Differences FAQ
Is difference-in-differences just a before and after comparison?
No. A before and after comparison mixes the treatment effect with any common time shock. Difference-in-differences subtracts the comparison group's change, using it to estimate the counterfactual change for the treated group.
Can repeated cross sections identify the same two-by-two effect as a panel?
Yes, if each cross section represents the relevant population and composition remains comparable. Repeated cross sections do not track individual changes, however, so shifts in who enters each sample can mimic a group-time effect.
What evidence supports parallel trends?
Pre-treatment outcome paths, institutional timing and the absence of differential contemporaneous shocks can make the assumption more plausible. Similar pre-trends do not prove the counterfactual after treatment, but sharply different pre-trends are a serious warning.
Exam move
For every design, draw a two-by-two table with treated and comparison rows and before and after columns. Calculate the two within-group changes before subtracting them, and link the result to the interaction coefficient in the regression. Then write a one-sentence counterfactual: in the absence of treatment, the treated group would have followed the comparison group's change.
Stress-test that sentence against composition shifts, anticipation, spillovers and group-specific shocks. When discussing standard errors, identify the level at which treatment varies and whether observations share shocks within treatment units. Practise moving fluently among the table, the slope diagram and the regression specification.
Vary the outcome across exercises so a negative interaction sometimes represents improvement and sometimes deterioration. State the unit before judging the direction. For pooled cross sections, compare observable composition across dates and explain how selective entry or exit could change a group mean without changing any person's outcome.
For panels, note what following the same units resolves and what evolving confounders remain. Draw the untreated trend for the treated group as a missing counterfactual line, then list the evidence that makes the comparison line credible: pre-period paths, timing, institutional similarity and absence of competing interventions. When treatments occur at a group level, match the inference discussion to that assignment level.
End with one sentence containing the estimate, its units, the comparison group, the relevant period and the parallel-trends boundary.
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