Why continuous treatments matter

Many marketing levers are continuous: ad spend, discount depth, loyalty rewards, and prices. Treating them as binary loses information and can distort the causal target. Continuous treatment designs keep the intensity scale and focus on dose-response functions.

The estimand: dose-response

Let $D_{it}\in \mathbb{R}$ denote treatment intensity. Potential outcomes are $Y_{it}(d)$, and the conditional dose-response is:

$$ \mu(d\mid x)=\mathbb{E}[Y_{it}(d)\mid X_{it}=x]. $$

The average dose-response is $\mu(d)=\mathbb{E}[Y_{it}(d)]$, obtained by averaging over the covariate distribution. Marginal effects and elasticities are derived from the slope of $\mu(d)$.

Two core assumptions

1) Conditional independence (unconfoundedness).

$$ Y_{it}(d) \perp D_{it} \mid X_{it}, \alpha_i, \lambda_t, \quad \forall d \in D. $$

After conditioning on covariates and fixed effects, treatment intensity is as-good-as-random.

2) Positivity (overlap).

$$ 0 < r(d\mid X_{it}, \alpha_i, \lambda_t) \quad \text{for all } d \text{ in the support of interest.} $$

If no units ever have low (or high) intensity, causal effects at those levels are not identified.

Why panels help

Unit fixed effects control for time-invariant confounders, and time fixed effects control for common shocks. This makes unconfoundedness more plausible, but only if the remaining confounders are time-varying and observed.

Practical identification strategies

  • High-dimensional controls and DML: flexibly adjust for many confounders without overfitting.
  • Factor models: capture common demand shocks and identify effects from unit-specific deviations.
  • Generalized propensity scores: check overlap across intensity levels.

Common pitfalls

  • Treating intensity as binary when the dose-response is nonlinear.
  • Ignoring overlap violations (e.g., zero-spend cells never occur).
  • Confounding from feedback loops where spend responds to recent outcomes.

Takeaway

Continuous treatment designs target dose-response functions, but they require stronger assumptions than binary designs. Credible inference depends on overlap, rich controls, and diagnostics that probe sensitivity to treatment intensity modeling.

References

  • Shaw, C. (2025). Causal Inference in Marketing: Panel Data and Machine Learning Methods (Community Review Edition), Section 3.2.5.
  • Hirano, K., and Imbens, G. W. (2004). The propensity score with continuous treatments.