Why the canonical 2x2 still matters

Section 4.1 uses the two-group, two-period DiD as the clean benchmark. The logic is simple, the estimator is transparent, and it clarifies the Parallel Trends assumption that underpins all DiD variants.

The 2x2 DiD contrast

For treated units, the observed change is their treated outcome post minus pre. For control units, the observed change is their untreated outcome post minus pre. The population DiD contrast is:

$$ \tau_{\text{DiD}} = E[Y_{i,\text{post}}(1) - Y_{i,\text{pre}}(0) \mid i \in \text{treated}] - E[Y_{i,\text{post}}(0) - Y_{i,\text{pre}}(0) \mid i \in \text{control}]. $$

Under Parallel Trends, the control-group change stands in for the treated group’s counterfactual change, so $\tau_{\text{DiD}}$ equals the ATT.

Regression form and interpretation

With two groups and multiple periods, the DiD regression is:

$$ Y_{it} = \alpha_i + \lambda_t + \tau D_{it} + \varepsilon_{it}. $$

Here $\alpha_i$ absorb time-invariant differences, $\lambda_t$ absorb common shocks, and $D_{it}$ turns on in post-treatment periods for treated units. With two periods, $\tau$ equals the 2x2 DiD estimator. With multiple periods, this interpretation still hinges on parallel trends and constant effects within the two-group design.

Parallel trends is about unobservables: what treated units would have done without treatment. We cannot verify it directly. We can, however, bring indirect evidence:

  • Pre-trend checks in multiple pre-periods should be near zero.
  • Placebo tests that treat earlier periods as “post” should be near zero.

These are diagnostics, not proof. Pre-trend tests have low power and can lead to selection bias if used to choose specifications.

Marketing settings often violate parallel trends:

  • Treatment assigned to fast-growing units (selection on trends).
  • Endogenous timing based on unit-level shocks.
  • Competitive responses that contaminate control outcomes.
  • Seasonality correlated with treatment timing.

If treatment assignment is related to observables that also drive trends, conditioning can restore identification:

$$ E[Y_{i,\text{post}}(0) - Y_{i,\text{pre}}(0) \mid i \in \text{treated}, X_i] = E[Y_{i,\text{post}}(0) - Y_{i,\text{pre}}(0) \mid i \in \text{control}, X_i]. $$

Covariates should be chosen for substantive reasons (store size, local income, competitive density). Over-conditioning, missing confounders, or post-treatment controls can bias estimates.

Doubly robust DiD

When relying on conditional parallel trends, analysts can model outcomes, model treatment assignment, or do both. Doubly robust DiD combines regression adjustment and propensity score weighting and remains consistent if either model is correctly specified. This is useful in marketing settings where neither model is likely to be perfect.

Where canonical DiD applies

Canonical DiD fits discrete events with a clear on/off timing: regulatory changes, national ad launches, or platform updates rolled out at a single date. The identifying variation comes from the timing of the event and the availability of credible controls.

Why it is only a benchmark

Most marketing panels feature staggered adoption, evolving effects, and heterogeneity across cohorts. The 2x2 design is a clear benchmark, but it does not solve those complexities. Section 4.2 turns to estimands that handle staggered adoption directly.

References

  • Shaw, C. (2025). Causal Inference in Marketing: Panel Data and Machine Learning Methods (Community Review Edition), Section 4.1.
  • Sant’Anna, P. H. C., and Zhao, J. (2020). Doubly robust difference-in-differences estimators.
  • Roth, J. (2022). Pre-trend testing in DiD.