How do you use an event-study plot to defend a difference-in-differences design?

## The specification An event study is difference-in-differences with the single post indicator replaced by a full set of period indicators measured relative to the adoption date. For a unit `i` in calendar period `t` that adopts at time `g`, define relative time `k = t - g`. You fit something of the form `Y = unit effects + period effects + sum over k of (beta_k * 1[relative time = k]) + error` with one value of `k` omitted. The `beta_k` for negative `k` are the **leads** — differences between treated and comparison units before anything happened. The `beta_k` for `k >= 0` are the **lags** — the effect path after adoption. **Why one period must be omitted.** The full set of relative-time indicators is collinear with the unit and period fixed effects, so the coefficients are only identified up to a normalisation. You drop one, conventionally `k = -1` (the period just before adoption), and every plotted coefficient is then read as *relative to that period*. This is not a modelling nicety you can skip; without it the model does not have a unique solution. It also means the plot is anchored, not absolute: shifting the reference period shifts the whole curve. **Binning the endpoints.** Far-out relative periods are estimated off very few units and become extremely noisy. The usual fix is to bin everything beyond some horizon into a single terminal lead and a single terminal lag, so the tails of the plot are estimated with enough data to mean something. ## Reading the leads The leads are the evidence for parallel trends. Three patterns matter. **Flat and tight.** Coefficients hovering near zero with narrow intervals across several pre-periods. This is what you want: before the treatment existed, the two groups moved together. **Sloping.** Coefficients drifting in one direction as adoption approaches. This is the Canada-versus-US case: usage in the treated market was already growing faster, and a two-period double difference would extrapolate that pre-existing divergence into the post-period and call it an effect. The direction matters — a rising pre-trend inflates a positive estimate. **A spike right before adoption.** Often anticipation: units changed behaviour once they knew the treatment was coming, or the treatment was targeted at units that had just moved. Either way, the period immediately before adoption is a bad reference, and you should re-anchor further back and see what the picture looks like. ## The honest caveats **Underpowered checks.** The leads are estimated on the same, often small, sample as the main effect, so their intervals are frequently wide enough to contain economically enormous pre-trends. 'No significant pre-trend' then carries almost no information. The right way to report it is to say how large a pre-trend the data could not have detected, not to say the test passed. **Selection on passing.** If you only publish results whose pre-trend check came out flat, you have selected on a noisy statistic. Conditioning on a passed pre-test can leave the surviving estimates *more* biased than the unconditional ones, because the cases that survive are those where noise happened to flatten a real trend. **Flat leads are not sufficient.** Parallel trends is about the post-period counterfactual. Nothing about pre-period agreement guarantees the groups would have kept agreeing. The leads make the assumption plausible; they do not deliver it. ## What to do when the leads slope In rough order of preference: 1. **Get a better comparison group.** A sloping pre-trend usually means the comparison unit is substantively wrong — a different growth stage, a different market maturity. Reselect on the mechanism rather than on the fit. 2. **Extend the pre-period.** A slope over three periods may be noise; over twelve it is a fact about the world. More pre-periods also make the check informative rather than decorative. 3. **Allow group-specific linear trends.** You can add a separate linear time trend per unit, which absorbs a steady pre-existing slope. The serious warning is that this also absorbs any treatment effect that grows linearly, so it can flatten a genuine effect to nothing. Report both with and without. 4. **Report a range rather than a point.** Ask how big a differential trend would have to be to explain the whole estimate away, and say whether that magnitude is plausible. 5. **Abandon difference-in-differences here.** Sometimes the honest answer is that the comparison group does not support a causal claim, and the readout should be labelled descriptive. ## Reading the lags The post-adoption coefficients are not just a robustness display; they are the substantive result. A single collapsed number hides whether the effect was a one-off jump that faded, a step change that persisted, or something that took three periods to appear. Product decisions differ sharply between those shapes, and the event-study plot is what lets you distinguish them.