Formalizing the Event-Time Estimand

The fundamental building block in an event-study design is the average treatment effect at event time $k$, denoted $\theta_k$. For $k \ge 0$, $\theta_k$ measures the average difference between treated and untreated potential outcomes exactly $k$ periods after adoption.

$$ \theta_k = E[Y_{i, G_i+k}(G_i) - Y_{i, G_i+k}(\infty) \mid G_i < \infty, G_i + k \le T] $$

For $k < 0$, we produce “placebo leads.” Under the assumptions of no anticipation and parallel trends, these coefficients should hover close to zero.

Heterogeneity and Aggregation Weights

In staggered rollouts, $\theta_k$ is rarely a single monolithic effect. It is an aggregation of smaller cohort-specific time effects $\tau(g, t)$, where $t = g+k$.

$$ \theta_k = \sum_{g} w_{gk} \tau(g, g + k) $$

The weighting scheme $w_{gk}$ inherently defines your final estimand. Do you weight uniformly? Do you weight by the cohort’s sample size? The choice of weights is a design decision that fundamentally shapes the interpretation of the results, so pre-specify them transparently.

Cohort-Specific vs. Calendar-Time Approaches

When assessing staggered rollouts (e.g., launching a new pricing policy in distinct waves), treatment responses frequently vary across cohorts.

  • Cohort-Specific Event-Time Profiles ($\theta_{g,k}$): Tracks the response to treatment for a designated cohort $g$ directly across its own event timeline.
  • Calendar-Time Aggregation: If your goal is to forecast Q4 quarterly revenue rather than studying the abstract treatment dynamics, you can aggregate $\tau(g, t)$ strictly on calendar time $t$. This answers “What was the total impact in Q4 across all cohorts currently active in the program?”

Cumulative Effects and the Long-Run Multiplier (LRM)

In marketing, evaluating the carryover or persistent impact of an intervention is vital. A standard measure for this is the Long-Run Multiplier (LRM). Assuming the immediate adoption effect $\theta_0 \neq 0$:

$$ \text{LRM} = \frac{\sum_{k=0}^{K} \theta_k}{\theta_0} $$
  • LRM > 1: The cumulative impact exceeds the immediate hit. This is common in campaigns building brand awareness, loyalty structures generating habit formation, or networks achieving critical mass.
  • LRM < 1: The effect decays or erodes over time. Promotional discounts and price cuts often exhibit LRM < 1 because they merely pull demand forward (stockpiling) instead of creating net new demand.

Edge case note: If an intervention has a delayed onset and the immediate effect $\theta_0 \approx 0$, the LRM formula becomes unstable or undefined. In these specific cases, forgo the LRM summarize metric entirely and present the plotted cumulative profile $\sum \theta_k$ instead.

Determining the Reference Baseline

Treatment effects must be measured against a baseline. The standard convention is to normalize the period exactly prior to adoption to zero (omitting $k = -1$ from the regression).

This approach ensures that all subsequent post-treatment effects $\theta_0, \theta_1, \dots, \theta_k$ represent the isolated causal shift explicitly relative to the untreated state just prior to activation. Occasionally, $k=0$ is omitted if anticipation thoroughly contaminated the “launch” period, but this structurally complicates how one interprets subsequent lags, so $k=-1$ remains the heavily preferred standard.