Parallel Trends Across Cohorts and Calendar Time

Why Parallel Trends Must Be Stated by Cohort and Calendar Time

With staggered adoption, different groups begin treatment in different calendar periods. An effect for a particular adoption cohort and period therefore depends on a particular counterfactual: how that cohort’s outcome would have changed by that calendar time if it had remained untreated. Parallel trends supplies a condition for replacing that unobserved change with an observed change in an appropriate comparison group.

The condition is about changes in untreated outcomes, not equality of outcome levels. It must also be stated for the cohort and calendar-time comparison being used. A trend that is plausible for one adoption cohort, or over one interval, need not be plausible for another.

Definition: Let $G_i=g$ mean unit $i$ first adopts treatment in calendar period $g$, and let $G_i=\infty$ denote a never-treated unit. Write $Y_{it}(0)$ for unit $i$'s potential outcome in calendar period $t$ if untreated. For a treatment cohort $g$ and a post-adoption period $t\geq g$, the cohort-time average treatment effect is $$ ATT(g,t)=E[Y_{it}(g)-Y_{it}(0)\mid G_i=g], $$ where $Y_{it}(g)$ is the potential outcome under adoption beginning in period $g$. A cohort-specific parallel-trends condition equates the cohort’s mean change in untreated potential outcomes from a pre-treatment baseline $b_g$ to $t$ with the corresponding mean change for an untreated comparison group.

A common baseline is $b_g=g-1$, the calendar period immediately before adoption. A comparison group can consist of never-treated units, or, for a given $t$, units that have not yet adopted by $t$. The latter comparison group is defined by $G_i>t$, so its membership can change with calendar time. Whichever group is used, the condition must refer to its untreated outcomes over the same calendar interval as the treated cohort’s interval.

The Cohort-Specific Condition and Its Identification Logic

Using $b_g=g-1$, a parallel-trends condition for cohort $g$, period $t$, and comparison group $C_t$ is

E[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i=g] = E[Y_{it}(0)-Y_{i,g-1}(0)\mid i\in C_t].

For never-treated controls, $C_t$ is the set with $G_i=\infty$. For not-yet-treated controls, it is the set with $G_i>t$. In either case, the controls must remain untreated through $t$, and the assumptions for the baseline must ensure their outcomes there are untreated as well. The condition can hold for one cohort and fail for another; it is not a single statement that automatically applies to every adoption group.

To interpret the observed difference in changes as an ATT, also require that observed outcomes agree with the relevant potential outcomes, that treatment does not affect the baseline outcome (no anticipation), and that the comparison outcomes are observed for the relevant periods. These conditions make the pre-adoption outcome for cohort $g$ a valid untreated baseline and the comparison group’s change a valid estimate of the treated cohort’s missing untreated change.

Identification result: If the baseline is unaffected by treatment, the comparison group remains untreated through $t$, and cohort-specific parallel trends holds, then $$ ATT(g,t) = \{E[Y_{it}\mid G_i=g]-E[Y_{i,g-1}\mid G_i=g]\} - \{E[Y_{it}\mid i\in C_t]-E[Y_{i,g-1}\mid i\in C_t]\}. $$ The first bracket is the treated cohort’s observed change; the second is the comparison group’s observed change. Their difference identifies the cohort-time effect under the stated conditions.

The justification follows by separating the treated cohort’s observed change into its treatment effect and its untreated change. With no effect at $g-1$, the first bracket equals

E[Y_{it}(g)-Y_{i,g-1}(0)\mid G_i=g] = ATT(g,t)+E[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i=g].

The comparison group’s observed change equals its untreated change because it is untreated over the interval. Subtracting that change removes the treated cohort’s untreated change precisely when parallel trends equates the two mean untreated changes. What remains is $ATT(g,t)$. Thus, parallel trends is not a claim that the treated and comparison groups have the same outcome level at baseline; it is the restriction that their relevant untreated changes agree.

Worked Example: Identifying One Cohort-Time Effect

Suppose a hypothetical program is adopted by cohort $G=3$. We want $ATT(3,4)$, using never-treated units as controls and period 2 as the baseline. Assume no anticipation, untreated outcomes are observed for the controls in both periods, and cohort-specific parallel trends is credible for this comparison.

The treated cohort’s mean outcome is 50 in period 2 and 63 in period 4. The never-treated group’s mean is 48 in period 2 and 55 in period 4. The treated cohort’s observed change is $$ 63-50=13. $$ The comparison group’s observed change is $$ 55-48=7. $$

Therefore, the difference in changes is $$ ATT(3,4)=(63-50)-(55-48)=13-7=6. $$

Under the stated assumptions, the program increases the mean outcome for cohort 3 in period 4 by 6 outcome units, relative to what that cohort’s mean would have been without adoption. The calculation does not establish that parallel trends is true; that is an assumption about the unobserved untreated change.

Why the Comparison Is in Calendar Time

The relevant interval is defined by calendar periods. For cohort $g$, compare its outcome from the chosen pre-treatment baseline to calendar period $t$ with the comparison group’s outcome over those same calendar periods. Calendar time matters because economy-wide shocks, weather, policy changes, and measurement changes occur at particular dates and can affect groups differently.

Event time, such as “two periods after adoption,” is useful for describing when an effect occurs, but it does not by itself identify the untreated change. Two cohorts observed at the same event time may be observed in different calendar years. Their outcomes can therefore reflect different calendar shocks. Likewise, comparing each cohort with a different interval or control group changes the parallel-trends condition being invoked.

If the comparison group is not-yet-treated, its composition may vary with $t$. That is not automatically invalid, but for every target pair $(g,t)$, the untreated trend condition must hold relative to the actual group used at that $t$. A statement that trends match against never-treated units does not, without more, establish that they match against a changing mixture of later-adopting cohorts.

Worked Example: Same Calendar Interval, Different Comparison Cohorts

Consider a hypothetical early-adopting cohort with $G=4$. To study its period-5 effect, use period 3 as the baseline. The early cohort’s mean outcome rises from 70 in period 3 to 82 in period 5, a change of $82-70=12$. Never-treated units rise from 68 to 76, a change of $76-68=8$. If parallel trends holds relative to the never-treated units, the estimate is $$ (82-70)-(76-68)=12-8=4. $$

Now suppose a later-adopting cohort has $G=7$, with mean outcomes of 60 in period 3 and 66 in period 5. It is untreated through period 5, so it could be used as a not-yet-treated comparison. Its change is $66-60=6$, giving $$ (82-70)-(66-60)=12-6=6. $$

The two calculations use the same calendar interval, periods 3 to 5, but different comparison groups. The values 4 and 6 do not represent two arithmetic versions of the same assumption: the first requires parallel trends relative to never-treated units, and the second requires it relative to the later cohort. The later cohort’s outcome levels need not equal the early cohort’s levels, but its untreated change must be a credible stand-in for the early cohort’s untreated change. The data alone do not determine which identifying condition is more plausible.

Common Trends Across Cohorts: A Substantive Claim

A stronger claim sometimes considered is that all relevant cohorts share a common mean untreated trend over the calendar interval of interest. In a simple form, this says that for every pair of relevant groups $g$ and $h$,

E[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i=g] = E[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i=h],

with the comparison framed over a common calendar interval where both groups are untreated. More generally, cohort-specific parallel trends only requires the target cohort’s change to match its selected comparison group’s change. The stronger common-trends claim says the untreated changes align across a broader set of cohorts, not merely for one chosen pair.

Equal trends do not require equal levels. For example, groups can start at different outcome levels and still have the same mean increase. The substantive question is whether the forces that determine untreated outcome changes are similar across the groups and periods being compared. Adoption timing is often related to those forces: units may adopt earlier precisely because their outcomes are changing faster, because they face different local shocks, or because resources and eligibility change over time.

Worked Example: Evaluating a Common-Trends Story

Suppose, hypothetically, regions adopt a service expansion in different years. The outcome is annual clinic visits per 100 residents. Before any of the three groups adopts, the early-adopting regions have means of 40, 43, and 47 in years 1, 2, and 3. Later-adopting regions have means of 35, 36, and 37; never-adopting regions have means of 30, 31, and 32.

Across years 1 to 3, the early group’s change is $47-40=7$ visits per 100 residents. The later group’s change is $37-35=2$, and the never-treated group’s change is $32-30=2$. The average annual changes over the two-year interval are therefore $7/2=3.5$, $2/2=1$, and $2/2=1$ visits per 100 residents per year, respectively.

The early group’s pre-adoption increase exceeds each other group’s increase by $3.5-1=2.5$ visits per 100 residents per year. This pattern raises a substantive concern about using either comparison group to represent the early group’s untreated trend. It does not prove that the future untreated trends would differ by exactly 2.5, nor does it quantify a treatment effect. It does show why a common-trends claim needs support from knowledge of the rollout and outcome process, not just a convenient estimator.

A useful plausibility assessment asks why adoption dates differ, which groups face the same calendar shocks, and whether the outcome is driven by factors that evolve differently across cohorts. For example, if adoption is prioritized for regions with rapidly growing need, a common untreated trend may be doubtful even when the regions have similar baseline outcome levels. If rollout timing is administrative and unrelated to expected outcome changes, that can make the assumption more credible, though it does not guarantee it.

Pre-adoption outcome patterns can inform this assessment, but they cannot reveal the untreated outcomes that treated cohorts would have had after adoption. Similar observed pre-trends are therefore not proof of parallel trends; substantial differences can be a warning that a proposed comparison is poorly supported. The causal interpretation remains conditional on the trend assumption for the specific cohort, interval, and comparison group.

Common Mistakes

Assuming equal levels are required. Parallel trends concerns changes in untreated outcomes. Different baseline means are compatible with equal trends.
Treating “parallel trends” as one universal condition. The relevant condition depends on the adoption cohort, calendar period, baseline, and comparison group.
Mixing event time with calendar time. Two cohorts at the same event time can experience different calendar shocks. The identifying comparison must use the same calendar interval.
Assuming any not-yet-treated cohort is a valid control. Being untreated through $t$ makes a group eligible as a comparison, but does not ensure its untreated trend matches the target cohort’s.
Calling a pre-trend pattern proof. Similar observed pre-period changes cannot establish equality of unobserved post-period untreated changes.

Key takeaway: For each $ATT(g,t)$, state whose untreated change stands in for cohort $g$'s missing change, and over which calendar-time interval. The causal interpretation depends on the substantive plausibility of that cohort-specific parallel-trends condition.

Check Your Understanding

Use the assumptions and comparisons stated in each question.

In your own words, state cohort-specific parallel trends for cohort $g$, period $t$, and never-treated controls.
A treated cohort and its controls have different outcome levels at baseline but equal mean untreated changes. Which part of the parallel-trends condition is satisfied, and why are equal levels unnecessary?
Why can a not-yet-treated cohort be untreated through $t$ yet still be an implausible comparison group?
Suppose one cohort’s observed change from periods 2 to 4 is 9 units and the selected controls’ change is 5 units. Under the identification assumptions, what is the difference-in-differences estimate, and what does it measure?
Two adoption cohorts are observed at the same event time but in different calendar years. Give one reason their untreated changes might differ.