Start with the question
“Does the loyalty program work?” is not an estimand. You must define for whom, when, and how you want to aggregate effects. Section 2.3 lays out the core estimands that make those choices explicit.
ATE vs ATT
Using contemporaneous notation $Y_{it}(d)$:
- ATE: $\mathrm{ATE}=\mathbb{E}[Y_{it}(1)-Y_{it}(0)]$ answers the average effect if all units were treated in all periods.
- ATT: $\mathrm{ATT}=\mathbb{E}[Y_{it}(1)-Y_{it}(0)\mid D_{it}=1]$ focuses on the treated units and periods, which is often the ROI-relevant quantity in marketing.
ATE is useful for fully randomized settings; ATT is often the practical target when adoption is selective.
Cohort–time effects for staggered adoption
When adoption is staggered, effects can vary by cohort and calendar time. Define the cohort–time effect:
$$ \tau(g,t)=\mathbb{E}[Y_{it}(g)-Y_{it}(\infty)\mid G_i=g], \quad t\ge g. $$Aggregating $\tau(g,t)$ with different weights answers different policy questions. Traditional two-way fixed effects often mix these cells with unintuitive or even negative weights when effects are heterogeneous.
Event-time effects and dynamics
Event time is $k=t-G_i$. The dynamic response is captured by:
$$ \theta_k=\mathbb{E}[Y_{i,G_i+k}(G_i)-Y_{i,G_i+k}(\infty)\mid G_i<\infty]. $$For $k<0$, $\theta_k$ acts as a pre-trend check. For $k\ge 0$, it traces habit formation, wear-out, and delayed effects.
Long-run multiplier
Dynamic effects often matter more than the immediate lift. The long-run multiplier compares cumulative impact to the initial effect:
$$ \mathrm{LRM}=\frac{\sum_{k=0}^K \theta_k}{\theta_0}. $$If $\mathrm{LRM}>1$, carryover amplifies the total impact beyond the immediate lift.
Table 2.1 at a glance
| Estimand | Definition | Interpretation | When most relevant |
|---|---|---|---|
| ATE | $\mathbb{E}[Y_{it}(1)-Y_{it}(0)]$ | Average effect if all units treated | Randomized experiments; strong overlap |
| ATT | $\mathbb{E}[Y_{it}(1)-Y_{it}(0)\mid D_{it}=1]$ | Average effect on treated units | Policy evaluation; ROI |
| $\tau(g,t)$ | $\mathbb{E}[Y_{it}(g)-Y_{it}(\infty)\mid G_i=g]$ | Cohort–time effect | Staggered rollouts; heterogeneity |
| $\theta_k$ | $\mathbb{E}[Y_{i,G_i+k}(G_i)-Y_{i,G_i+k}(\infty)\mid G_i<\infty]$ | Effect $k$ periods post-adoption | Dynamic effects; pre-trends |
| LRM | $\sum_{k=0}^K \theta_k/\theta_0$ | Cumulative vs immediate effect | Carryover; habit formation |
Takeaway
Estimands are not technical details—they are the causal question. Precise definitions are what connect marketing decisions to the right identification strategy and estimator.
References
- Shaw, C. (2025). Causal Inference in Marketing: Panel Data and Machine Learning Methods (Community Review Edition), Section 2.3.
- Goodman-Bacon, A. (2021). Difference-in-differences with variation in treatment timing. Journal of Econometrics.