A city implements a minimum wage increase in 2018. Researchers compare employment in this city (treated) to a neighboring city (control) that did not raise its minimum wage. During 2018, both cities experienced employment growth because of a regional economic boom. How does difference-in-differences handle this confound?
AIt excludes the boom period from the analysis to avoid contamination
BIt subtracts the control city's change in employment from the treated city's change, removing the common trend
CIt adds the control group's growth to the treated group's post-period value to adjust for the boom
DIt uses the treated city's pre-period data as its own control, so the control city is unnecessary
DiD works through double subtraction. The treated city's before-after change captures both the policy effect AND the economic boom. The control city's before-after change captures only the boom (since it wasn't treated). Subtracting the second from the first removes the common trend and isolates the policy effect. This is the whole point of DiD: using the control group as a 'thermometer' that measures how much both groups would have changed anyway. The key assumption required is that both cities would have followed the same employment trend absent the policy — parallel trends.
Question 2 Multiple Choice
A researcher runs a staggered difference-in-differences study where 50 counties adopt a health policy between 2010 and 2020, each at a different time. She uses a standard two-way fixed effects (unit + time fixed effects) regression. What is the key risk in this approach?
AThe standard errors will be too large, reducing statistical power
BIncluding time fixed effects removes the variation needed to identify treatment effects
CWhen treatment effects are heterogeneous across cohorts or time, the TWFE estimator can be severely biased
DThe parallel trends assumption cannot be assessed without a single common treatment date
This is the central problem identified in recent DiD methodology. With staggered adoption and heterogeneous treatment effects, the TWFE estimator uses already-treated units as implicit controls for later-treated units — these 'contaminated controls' can produce a weighted average where some weights are negative, yielding estimates that misrepresent or even reverse the true average treatment effect. The solution is to use modern staggered DiD estimators (Callaway-Sant'Anna, Sun-Abraham) that construct clean comparisons using not-yet-treated or never-treated units as controls.
Question 3 True / False
An event study showing that the treated and control groups had statistically similar trends in the three years before a policy was implemented proves that the parallel trends assumption holds in the post-treatment period.
TTrue
FFalse
Answer: False
A pre-trend test supports but cannot prove parallel trends post-treatment. The counterfactual — what the treated group would have looked like absent the treatment — is unobservable after treatment begins. Similar pre-trends show the assumption is plausible (the groups moved together before), but a common pre-trend could diverge post-treatment for reasons unrelated to the policy. The test is a necessary plausibility check, not a proof. The honest framing is: similar pre-trends make parallel trends more credible; divergent pre-trends are a red flag that invalidates the design.
Question 4 True / False
In a difference-in-differences design with two groups and two time periods, the control group's post-treatment level directly estimates what the treated group's outcome would have been without the treatment.
TTrue
FFalse
Answer: False
DiD does not require that the treated and control groups have the same *levels* — only the same *trends*. The control group's *change* over time is used to estimate the counterfactual trend for the treated group. The treated group's pre-period level serves as its baseline, and the counterfactual post-period level is constructed by adding the control group's observed change to the treated group's pre-period level. The groups can start at completely different levels and still produce a valid DiD estimate, as long as they would have followed parallel trajectories in the absence of treatment.
Question 5 Short Answer
What is the parallel trends assumption in difference-in-differences, and why can it not be directly tested using post-treatment data?
Think about your answer, then reveal below.
Model answer: Parallel trends assumes that, absent the treatment, the treated and control groups would have followed the same trajectory over time. It cannot be tested post-treatment because once the treatment occurs, the treated group's actual post-period outcome reflects both the treatment effect and whatever underlying trend occurred — the counterfactual (what would have happened without treatment) is never observed.
The entire DiD strategy hinges on this assumption because it defines what the control group's trend is 'standing in for.' If the groups were on different trajectories to begin with — for example, the treated group was already improving faster before the policy — then the control group's change underestimates the treated group's counterfactual change, and DiD attributes some of that pre-existing trend to the policy. The most we can do is examine pre-treatment periods to assess whether the assumption is plausible. Event study designs make this diagnostic step transparent by plotting the treatment effect estimate at every pre- and post-period, showing whether the groups moved together before and diverged after.