Identification by Comparing Interrupted Trajectories

Why Compare Interrupted Trajectories?

An outcome can change after an intervention for reasons unrelated to the intervention: a secular trend, a seasonal fluctuation, or a shock that affects both treated and untreated units. A simple before-and-after comparison in the treated group cannot separate these explanations from a treatment effect. A control series helps by showing how outcomes changed over the same calendar time without the intervention.

The comparative interrupted time-series (CITS) contrast compares the treated group’s pre-to-post change with the control group’s change over that same interval. Its causal interpretation depends not on the two groups having identical outcome levels, but on a more specific counterfactual claim: without the intervention, their outcome changes would have been comparable.

For clarity, write \(\bar Y_{G,t}\) for the observed mean outcome in group \(G\) at time \(t\). If treatment starts after baseline time \(s\), then under consistency and no anticipation, the treated group’s observed mean at \(s\) is its untreated mean, while its observed mean at post-intervention time \(t\) is its treated mean. The control group is observed without intervention at both times. The contrast is therefore

The first parenthesis is the treated group’s observed change; the second is the control group’s observed change. Subtracting the latter removes the change that serves as an estimate of what the treated group would have experienced without the intervention, provided the identification condition below is credible.

The Identification Condition

The causal target at post-intervention time \(t\) is the treated group’s average effect at that time: its mean outcome under the intervention minus its mean outcome if it had not received the intervention. In potential-outcomes notation, this is

The difficulty is that \(Y_{T,t}(0)\), the treated group’s post-intervention outcome without the intervention, is unobserved. The control group can stand in for the missing untreated change only under an assumption about how the two groups would have changed.

Under consistency, no anticipation, an untreated control group, and this parallel-untreated-trends condition, the CITS contrast identifies \(\tau_t\). The algebra shows exactly why.

The first equality uses the observed outcomes implied by the design: treated units have untreated outcomes at baseline and treated outcomes after the intervention, while controls remain untreated. The second equality adds and subtracts the treated group’s unobserved untreated post-intervention mean. The bracketed term is the difference between the groups’ untreated changes; the identification condition sets that term to zero. The remaining term is the treated group’s average effect at time \(t\).

Assumptions That Give the Contrast a Causal Meaning

The arithmetic can always be carried out; causal identification depends on the design and assumptions. The parallel-untreated-trends condition is central, but several related conditions are needed to connect observed data to the potential outcomes in the derivation.

• Consistency and a well-defined intervention. The observed outcome for a treated group after intervention corresponds to the potential outcome under the intervention regime, and the observed outcome under no intervention corresponds to the untreated potential outcome. The intervention and its start time must be sufficiently well specified for these outcomes to be meaningful.
• No anticipation. Before the intervention begins, the treated group’s outcomes are not already affected by expecting it. If behavior changes in advance, a supposedly untreated baseline may already reflect the intervention.
• The control remains untreated and is not affected by treatment. The control series must represent outcomes under no intervention. Direct exposure, spillovers from treated units, or a related policy applied only to the control can make it a poor representation of the untreated trajectory.
• Comparable untreated changes. Had the intervention not occurred, the treated and control groups would have experienced the same mean change over the relevant interval. The condition need not hold for observed, treated outcomes after intervention; it concerns the counterfactual outcomes under no intervention.
• No differential concurrent shock that substitutes for the intervention. If another event at the intervention date affects only the treated group, the contrast combines that event’s effect with the intervention effect. A shared shock may cancel in the comparison, but a group-specific shock generally will not.
• Comparable measurement and group composition. Changes in outcome definitions, recording practices, or which units contribute to each group’s mean can create apparent changes unrelated to the intervention. The measured outcome should be comparable across groups and time.

Several Post-Intervention Times

When outcomes are observed at several post-intervention times, a CITS contrast can be calculated separately for each time using a specified pre-intervention baseline \(s\). Under the corresponding parallel-untreated-trends condition for each post time, each contrast identifies the treated group’s effect at that time. An average over post-intervention times can summarize a period, but it should be defined explicitly: it is an average of the time-specific effects, with the chosen time weights.

What Pre-Intervention Trajectories Can and Cannot Show

Multiple pre-intervention observations are useful because they allow investigators to inspect whether treated and control outcomes were changing comparably before intervention. Similar pre-intervention changes support the plausibility of the identifying assumption, while pronounced differences raise concern. But observed pre-intervention similarity cannot prove that the groups’ unobserved changes would have remained comparable after the intervention date. A new group-specific shock can arise later, and pre-intervention data cannot rule it out.

Nor does the condition require equal pre-intervention outcome levels. For example, a treated group can have a mean outcome consistently above the control group and still satisfy parallel untreated trends if the gap would have remained stable in the absence of intervention. Conversely, similar levels at baseline do not guarantee comparable future changes.

The identifying claim is specific to the interval and comparison being used. If untreated trends differ over a longer horizon, a contrast over that horizon may fail even if a shorter-interval contrast is plausible. Researchers should therefore state the baseline, post-intervention period, outcome scale, target effect, and substantive basis for treating the control trajectory as a counterfactual for the treated group.

Common Mistakes

• Wrong reasoning: “The groups began at different levels, so the comparison is invalid.” The CITS contrast subtracts changes, not levels. Different baseline levels are allowed when the identifying claim about untreated changes is reasonable.
• Wrong reasoning: “The outcome changed after intervention, so the intervention caused the change.” A before-and-after change includes secular movement and other events. The control change is subtracted to address shared movement, and causal interpretation still requires the assumptions stated above.
• Wrong reasoning: “The pre-intervention lines look similar, so parallel trends is proven.” Similar observed pre-intervention changes are evidence relevant to plausibility, not proof about the unobserved post-intervention counterfactual.
• Wrong reasoning: “The control increased by four units, so the treated group would certainly have increased by four units.” That is the substantive counterfactual assumption, not an arithmetic fact. A treated-specific concurrent shock or a different untreated trend would break it.