01:220:322 Chap.10 Pooled Cross Sections and Difference-in-Differences
Pooled Cross Sections and Difference-in-Differences
Pooled cross sections
Independent random samples drawn in different periods can be stacked into a pooled cross section.
Nobody is followed over time, but pooling increases the sample and allows questions about change.
Time effects
With several periods, include one fewer year dummy than periods; each coefficient shifts the intercept relative to the base year, and a joint F test asks whether the intercept changed at all.
Interacting a regressor with year dummies lets its slope change over time too.
Difference-in-differences
With a treated and a control group observed before and after a policy, the regression of the outcome on a treated dummy, an after dummy and their product reproduces the four group means.
The coefficient on the product is the difference-in-differences estimate: the change for the treated group minus the change for the control group. Robust standard errors provide inference.
Why not a single difference
A before-and-after comparison for the treated group mixes the effect with everything else that changed over time.
An after-period comparison of treated and control mixes it with pre-existing differences. Differencing twice removes both.
Parallel trends
The key assumption is that without treatment the two groups would have followed a common path of change. It concerns changes, not levels, so the groups may start far apart.
It can be probed with more pre-treatment periods or a second control group, and adding controls helps when the composition of the samples shifts over time.
Reading the regression output
In the difference-in-differences regression, the intercept is the control group before the policy, the treated dummy is the starting gap, the after dummy is the common change, and the product term is the effect.
With a log outcome, one hundred times the product coefficient is an approximate percentage effect. Adding controls can sharpen the estimate and guard against changes in who is sampled.
What the pooled lecture adds
Year dummies let the average outcome drift over time for reasons unrelated to the regressors, and interactions with year dummies let a slope change.
A joint F test on those interactions asks whether the slope changed at all.
What this chapter covers
- 01
Pooled cross sections
- 02
Year dummies and base years
- 03
Interacting regressors with time
- 04
The two-by-two difference-in-differences table
- 05
Regression form of DID
- 06
Parallel trends
Worked example · free
Difference-in-differences from four means
- 2Treated change: 17.9 - 19.5 = -1.6 hours; control change: 17.2 - 18.0 = -0.8 hours.
- 2DID = -1.6 - (-0.8) = -0.8 hours per week.
- 2Assumption: without the increase, teen hours in both counties would have moved alike.
Key terms
- Pooled Cross Section
- Independent random samples from different periods combined into one data set.
- Year Dummy
- An indicator for observations from a given period, measured against a base year.
- Difference-in-Differences
- The change in the treated group's mean minus the change in the control group's mean.
- Parallel Trends
- The assumption that treated and control groups would have changed alike without the treatment.
- Control Group
- Units not exposed to the policy, whose change stands in for the treated group's counterfactual change.
- Treatment Group
- Units exposed to the policy whose change over time is compared with the control group.
Pooled Cross Sections and Difference-in-Differences FAQ
Do the treated and control groups need the same starting level?
No. Different starting levels are absorbed by the treated-group dummy. What matters is that, without the treatment, both groups would have moved by the same amount over time.
Which coefficient is the treatment effect in the DID regression?
The coefficient on the product of the after-period dummy and the treated-group dummy. The other terms capture the starting gap and the change common to both groups.
How can parallel trends be checked?
With data from two or more periods before the policy, compare the groups' earlier movements; a second control group offers another check. A single pre-period cannot test the assumption.
Why are robust standard errors used in difference-in-differences?
The four groups defined by treatment status and period can have different outcome variances. Robust standard errors remain valid when they do, so the test on the interaction coefficient keeps its stated significance level.
Why not just compare the treated group before and after?
Everything else that changed between the two periods would be credited to the policy. The control group's change measures that common shift, which the difference-in-differences subtracts away.
Exam move
Redo every DID example twice, once from the four means and once from the regression coefficients, and confirm the two answers match.
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