Q: How do adstock/carryover dynamics interact with the timing of sequential media experiments (delayed outcomes)?

How do adstock/carryover dynamics interact with the timing of sequential media experiments (delayed outcomes)?

Summary

Carryover means an intervention’s effect is spread over future periods, so a sequential experimentation loop faces delayed outcomes: you cannot read a test’s result — or start a clean next test — until the adstock has decayed. Three consequences follow. (1) Measurement windows must span the carryover tail (including the post-change period), or ROAS is undercounted. (2) Back-to-back experiments contaminate each other through lingering adstock, so you need washout periods or must model the overlap explicitly. (3) The cleanest treatment is to make carryover a latent state in a state-space model and let a Kalman filter/smoother attribute delayed sales to past exposures — which also lets adaptive BED account for the fact that an experiment’s information arrives with a lag when it schedules the next test.

Answer

1. Carryover = the effect outlives the exposure

The adstock transform replaces current spend with a normalized weighted average of current and past spend over $L$ periods; the geometric-decay form peaks at exposure and decays by a retention rate ${\alpha }_m$, while delayed adstock peaks ${\theta }_m$ periods later (Carryover (Adstock) Functional Forms). Equivalently, distributed-lag (Koyck/ADL/PDL) structures spread one period’s spend across many periods of response (Carryover Effects and Distributed Lags). So an experiment that changes spend in weeks $[t_0,t_1]$ keeps moving sales well past $t_1$ — a delayed outcome.

This is not a nuisance to ignore: ROAS, mROAS, and Optimal Media Mix computes attribution counterfactually “including the post-change period because carryover keeps affecting sales after the change.” Cut the window short and you systematically understate the channel’s ROAS.

2. Consequence A — size the measurement window to the carryover tail

Before reading a test result, wait for the adstock to substantially decay. A practical rule: the read-out window should cover the effective duration implied by ${\alpha }_m$ (e.g. weeks to reach <5% residual carryover, $L_{\text{eff}}\approx \log (0.05)/\log ({\alpha }_m)$). Reading at $t_1$ alone truncates the tail; the post-period must be part of the estimand.

3. Consequence B — washout to avoid cross-experiment contamination

In a sequential loop, if experiment $k+1$ starts while experiment $k$‘s adstock is still live, the two effects superpose and the design becomes confounded. Options:

• Washout period — a gap between experiments long enough for prior adstock to decay (simple, but slows the learning loop).
• Model the overlap — carry the residual adstock from prior tests as a known covariate so the likelihood $p(y\mid \theta ,\xi )$ correctly de-confounds concurrent effects. This keeps the loop fast and is the more information-efficient choice.

Either way, the incremental EIG of the next design (Sequential and Adaptive BED) must be evaluated against a posterior that already conditions on the in-flight carryover — otherwise the design overestimates how much new information the next test brings.

4. Consequence C — treat carryover as a latent state (the cleanest solution)

Adstock is literally a hidden accumulator that evolves over time and drives observed sales — a state-space model:

Carryover as a state-space model ( Linear-Gaussian State-Space Models)

A state-space model pairs a transition equation for a hidden Markov state ${\mathbf{x}}_k$ with an observation equation linking measurements ${\mathbf{y}}_k$ to that state. Encode the adstock accumulator as (part of) ${\mathbf{x}}_k$: it decays by ${\alpha }_m$ each period (transition) and, after saturation, contributes to sales (observation). Delayed outcomes are then just observations of a slowly-decaying state.

With this framing, smoothing attributes delayed sales back to the exposures that caused them, in $\mathcal{O}(N)$ time, and naturally handles the lag between acting and observing. The CausalImpact approach is exactly this: a state-space decomposition (trend + seasonality + regression) whose posterior predictive gives the counterfactual over the post-intervention period — the right tool to read a geo-holdout whose effect trails off gradually (see Q - Encoding a Geo-Holdout as a Bayesian Experimental Design and Computing Its EIG).

5. Consequence D — delayed information changes the adaptive schedule

In adaptive design, each step picks ${\xi }_t$ from the posterior after seeing history $h_{t-1}$. With carryover, the outcome of ${\xi }_{t-1}$ is not fully observed when it is time to choose ${\xi }_t$ — the informative sales are still accruing. A myopic greedy loop that assumes immediate feedback will mis-time tests. Remedies:

• Delay-aware scheduling — condition on partially-observed outcomes (filtered state), not on a not-yet-arrived final read.
• Non-myopic policies — a DAD-style policy trained to maximize total EIG over the horizon internalizes that information arrives late, spacing and sequencing tests accordingly.

Practical Implications

• Set the read-out window from the retention rate ${\alpha }_m$, and always include the post-change period in the estimand — otherwise ROAS is biased low.
• Insert washouts or model residual adstock between sequential tests to prevent contamination.
• Represent carryover as a latent state and use a Kalman filter/BSTS so delayed outcomes are attributed correctly and read out as counterfactuals.
• Make the adaptive schedule delay-aware (or use a non-myopic policy) so the next test isn’t chosen on stale, still-accruing feedback.

Source Notes

Note	Relevance
Carryover (Adstock) Functional Forms	Geometric vs delayed adstock; retention rate ${\alpha }_m$
Carryover Effects and Distributed Lags	Koyck/ADL/PDL distributed-lag structures
ROAS, mROAS, and Optimal Media Mix	Attribution must include the post-change carryover period
Sequential and Adaptive BED	Incremental EIG loop that scheduling must respect
Linear-Gaussian State-Space Models	Carryover as a latent decaying state
Bayesian Filtering and Smoothing · The Kalman Filter	$\mathcal{O}(N)$ attribution of delayed sales to exposures
Bayesian Structural Time-Series Model	Counterfactual post-period read-out (CausalImpact)
From Designs to Policies (Deep Adaptive Design)	Non-myopic policies that internalize delayed information

Related Concepts

• Design of Dynamic Response Models — dynamic model specification for lagged effects
• Local Linear Trend and Seasonality — the other state-space components a media time series needs
• Multivariate Persistence and Cointegration — long-run/persistent media effects beyond finite adstock
• Q - Continuous Learning in Media Measurement with Interaction Effects — the loop this timing analysis constrains
• Q - Encoding a Geo-Holdout as a Bayesian Experimental Design and Computing Its EIG — where the window $[t_0,t_1]$ is set

Gaps

• No note on experimental design under delayed/censored feedback specifically (e.g. delayed-reward bandits, surrogate outcomes). The synthesis is built from adstock + sequential-BED + state-space notes.
• No treatment of optimal washout length as a formal trade-off (information lost to waiting vs bias from contamination).
• Attribution windows / conversion lag from the ad-measurement literature are not ingested.

Follow-Up Questions

• What washout length minimizes total loss (waiting cost + contamination bias) for a given retention rate ${\alpha }_m$?
• How do you compute incremental EIG when the previous experiment’s outcome is only partially observed (filtered)?
• Can a single state-space model jointly host adstock, seasonality, and the experiment indicator so all tests are analyzed in one filter?