What does the parallel-trends assumption in difference-in-differences actually require?

## The statement Write `Y(0)` for the outcome a unit would have had without treatment. Parallel trends says that the average untreated change is the same in both groups: `E[Y(0)_after - Y(0)_before | treated] = E[Y(0)_after - Y(0)_before | comparison]` The left-hand side is not observable. For the treated group, after treatment starts, you see `Y(1)`, not `Y(0)`. That is exactly why the design needs an assumption at all — the whole point of difference-in-differences is to import the comparison group's change as a stand-in for a quantity you can never measure. ## What it does not say **It does not say the groups must start at the same level.** A treated market with 3x the baseline volume is not disqualified. A constant gap differences away. Candidates who claim otherwise are describing matching, not difference-in-differences. **It does not say the pre-period trends must have been parallel.** That is the evidence people offer for the assumption, and it is good evidence, but it is not the assumption. The assumption is about the *post-treatment* counterfactual. Two groups can move in lockstep for two years and diverge for a reason that has nothing to do with treatment in the third. **It does not say the treatment had no anticipation effect.** That is a separate assumption — no anticipation — and it fails whenever units change behaviour before the official start date because they knew it was coming. ## Ways it breaks **A differential shock.** Something hits one group and not the other during the window. A competitor launches in the treated market. A hurricane hits the comparison region. The double difference cannot distinguish that shock from the treatment; both show up as the treated group moving differently. **Selection into treatment on trajectory.** If units get treated *because* their outcome was moving in a particular way, parallel trends fails by construction. The classic case is the Ashenfelter dip: workers enrol in job-training programmes after a spell of unusually low earnings, so the treated group's earnings dip just before treatment and then recover for reasons of ordinary mean reversion. A naive difference-in-differences reads that recovery as programme impact. **Composition change.** If who is in the treated group changes between periods — new users join, heavy users churn — the group mean moves for compositional reasons. Panel data on the same units, or a fixed cohort, protects you here in a way repeated cross-sections do not. **Differential seasonality.** Two regions can both be seasonal but on different calendars. If the treatment starts in a month where their seasonal patterns diverge, the double difference picks up the divergence. ## Scale dependence This one catches people. Suppose the treated group starts at 200 and the comparison at 100, and both grow 10% over the window. In logs, the trends are exactly parallel. In absolute units, the treated group gained 20 and the comparison 10 — not parallel at all, and the difference-in-differences estimate on levels would report a spurious effect of 10. Because a monotone transformation of the outcome changes whether trends are parallel, you cannot say 'parallel trends holds' without saying in what scale. The practical rule is to pick the scale from the substance before you look at the estimate: if the effect you expect is multiplicative — a percentage lift in usage — model logs or a ratio; if the outcome is bounded or has zeros, logs may not even be available and you need a different plan. Deciding after seeing which scale gives the nicer pre-period is how a researcher fools themselves. ## Conditional parallel trends A weaker and often more defensible version: parallel trends holds *within* strata defined by observed covariates. Small restaurants may trend differently from large ones, but small restaurants in the treated area trend like small restaurants in the comparison area. You then estimate the effect within strata and aggregate, or reweight the comparison group so its covariate distribution matches the treated group's. This buys credibility when the raw groups are visibly different, at the cost of a stronger reliance on having measured the right covariates. It is still an assumption; it has just been made more plausible. ## How to talk about it in an interview Say the counterfactual sentence explicitly, name one concrete way it could fail in the scenario you were given, and name what evidence you would bring. Do not say 'I checked and it holds'. The honest register is: this is the assumption, here is why it is plausible here, and here is the specific thing that would break it.