Glossary
What is Adstock?
By ElenIQ · Last updated
Formula
Adstockₜ = Impressionₜ + λ · Adstockₜ₋₁
- Adstockₜ
- — the total carried-over advertising effect in the current period t
- Impressionₜ
- — the new spend or impressions delivered in period t
- λ (lambda)
- — the decay / retention rate between 0 and 1 - the share of the previous period’s effect carried forward
- Adstockₜ₋₁
- — the carried-over effect accumulated up to the previous period
What is adstock?
Adstock is the modelling term for the carryover advertising effect - the observation that the influence of an ad does not stop when the impression ends. Someone who sees a campaign today might search, compare, and finally convert a week later. In aggregate, this means a given week’s conversions are driven partly by that week’s spend and partly by a fading memory of the spend that came before it. Adstock captures that memory mathematically, and it is a core building block of media mix modelling.
The mechanism is a simple decay. As the formula above shows, each period retains a fraction (λ) of the previous period’s accumulated adstock and adds the new spend on top. A high decay rate means impact lingers for many weeks; a low rate means it fades almost immediately. The shape varies by channel: brand video and upper-funnel display tend to carry over for longer, while high-intent search converts close to the point of exposure. Modelling this carryover is essential to understanding the true commercial contribution of each channel, and it pairs naturally with the diminishing-returns behaviour of the Hill function used to describe saturation.
How does advertising carryover work?
Advertising rarely produces an instant, one-to-one response. Exposure plants awareness and intent that ripen over time - a process media planners describe as the carryover advertising effect. Picture a single burst of spend in week one: a portion of the resulting conversions land that week, a smaller portion the next, fewer still after that, forming a decaying tail rather than a single spike. With a decay rate of λ = 0.5, that burst still contributes 50% of its effect in week two, 25% in week three, and 12.5% in week four.
Carryover with λ = 0.5
- Week 1 - the spend burst lands
- 100%
- Week 2 - carried over (× 0.5)
- 50%
- Week 3 - carried over again (× 0.5)
- 25%
- Week 4 - and again
- 12.5%
- Week 5 - the tail keeps fading
- 6.25%
A single week of advertising keeps working for weeks afterwards. With λ = 0.5 the effect halves every week, so its half-life is exactly one week. Conversions you record this week are a blend of recent and older spend - which is why naive, single-window analysis misallocates budget between fast and slow channels.
The practical consequence is that the conversions you record in any week are a blend of recent and older spend. If two channels deliver the same number of conversions but one creates them slowly and the other harvests them instantly, naive analysis will treat them as identical - and badly misallocate budget between them. Understanding carryover is therefore closely tied to thinking about incremental return on ad spend and to measuring incrementality, which isolates the revenue advertising genuinely caused from the revenue that would have arrived anyway.
Typical adstock half-lives
The half-life - the time it takes for carried-over impact to fall to 50% - is the most intuitive way to read a decay rate, and it follows directly from λ via half-life = ln(0.5) ÷ ln(λ). There is no universal value: it depends on the channel, the category and the buying cycle. The ranges below are a useful sense-check on an estimated decay.
Reading an adstock half-life
- ~2–5 weeksA common working range many practitioners use across digital and broadcast channels.
- ~2.5 weeksOften cited for fast-moving consumer goods (FMCG), where purchase cycles are short and frequent.
- ~7–12 weeksLonger half-lives seen in some academic studies, typically for high-consideration or brand-led advertising.
Treat any single number with caution. A half-life that is too long can make a model attribute current sales to spend from months ago; one that is too short collapses the carryover the model is meant to capture. In practice the decay is estimated from the data and validated against how each channel actually behaves.
Adstock vs saturation - two different distortions
Adstock is often confused with saturation, but they correct for different things. Adstock is about time: it spreads the effect of spend across later periods. Saturation is about volume: it caps how much extra response each additional pound can buy within a period. Media mix models apply both, and in a deliberate order - adstock first, to shift impact across time, then a saturation curve to model diminishing returns.
| Adstock | Saturation | |
|---|---|---|
| What it captures | Carryover of impact over time | Diminishing response to more spend |
| The relevant dimension | Time (periods) | Volume (spend within a period) |
| Typical function | Geometric / exponential decay (λ) | Hill function / S-curve |
| Applied | First, to transform the spend series | Second, to the adstocked series |
| Key parameter | Decay rate λ → half-life | Half-saturation point & slope |
The two effects work together: carryover stretches impact across time, while diminishing returns cap how much impact extra spend can buy. Seeing both is what lets you judge the marginal ROAS of the next pound rather than the flattering blended average.
Why do short reporting windows undervalue demand creation?
Most reporting and attribution windows are short - a few days, sometimes a single click session. Adstock guarantees that a meaningful share of a channel’s real impact lands after that window closes. The conversions are counted, but credited to whatever channel happened to be in front of the customer at the moment of purchase. Demand-creation channels, whose payoff is inherently delayed, are therefore chronically under-measured, while last-click harvesting channels collect credit for demand they did not create.
Left unchecked, this distortion compounds. Budgets shift toward channels that look efficient inside the window, starving the upper-funnel activity that fed them in the first place - and the harvesting channels then quietly weaken as the demand pipeline dries up. Planning that accounts for carryover avoids this trap. It is also why scaling a winning channel often disappoints: as spend rises, each additional pound reaches less responsive audiences and returns less, the dynamic explained by saturation curves and budget ceilings.
How ElenIQ models adstock
ElenIQ applies an adstock transformation to historic spend before fitting its response models, so each period’s effect carries a decaying contribution from earlier spend rather than treating every week in isolation. The decay is estimated from the data, channel by channel, so demand-creating media is credited for the delayed conversions it produces instead of having that value silently reassigned to last-click channels.
Layering adstock on top of saturation curves lets ElenIQ forecast the fuller, time-shifted impact of a budget change before the spend is committed - true to the platform’s forecast-led, marginal approach to media planning. Rather than reacting to what a short window happened to capture, you can compare the carryover-adjusted return of each channel and decide where the next pound works hardest. Test allocations across channels with the budget allocation simulator, then see how it fits a full plan with Dex, ElenIQ’s forecasting engine, as you create a media plan.
Related terms
- Hill function - the curve used to model how response flattens near saturation, applied after adstock.
- Saturation curve - how response to spend flattens within a period, the volume counterpart to adstock’s timing.
- Diminishing returns - why each extra pound of spend tends to buy less response than the last.
- Marginal ROAS - the return on the next pound of spend, the metric that should drive scaling decisions.
- iROAS - the incremental revenue advertising actually creates, net of baseline demand.
Frequently asked questions
What is adstock?
Adstock is the carryover effect of advertising - the way the impact of an ad persists after exposure rather than vanishing immediately. In media modelling, this period’s response depends partly on current spend and partly on a decaying memory of previous spend, which is why advertising can keep driving conversions well after the impression was served.
What is the carryover advertising effect?
The carryover advertising effect is the same idea as adstock: a portion of the demand created by advertising arrives later than the exposure itself. Each period retains a fraction of the prior period’s adstock, so a single burst of spend produces a tail of delayed conversions that decays over subsequent days or weeks.
What is the adstock decay rate and half-life?
The adstock decay rate (often written as lambda, between 0 and 1) is the share of the previous period’s effect carried forward into the next. A higher lambda means impact lingers longer. The half-life is the number of periods it takes for the carried-over effect to fall to 50 percent, and it follows from lambda as half-life = ln(0.5) ÷ ln(lambda). For example, a weekly lambda of 0.5 gives a half-life of one week, while 0.8 gives roughly three weeks.
How is adstock used in media mix modelling?
In media mix modelling, raw spend or impressions are first passed through an adstock transformation so each period’s value includes a decaying contribution from earlier spend. The transformed series is then fed into a response model - typically a saturation curve such as the Hill function - so the model captures both the delay in advertising impact and its diminishing returns. The decay rate is usually estimated from the data, channel by channel.
What is the difference between adstock and saturation?
Adstock and saturation describe two different distortions. Adstock is about time: it spreads the effect of spend across later periods through a decay process. Saturation is about volume: it caps how much extra response each additional pound of spend can buy within a period. Media mix models apply both - adstock first to shift impact across time, then a saturation curve to model diminishing returns - because they are complementary, not competing, effects.
Why do short reporting windows undervalue demand creation?
Because adstock spreads impact across time, conversions that an ad genuinely caused can land outside a short attribution or reporting window. Channels that create demand - rather than simply harvest it at the point of intent - look weaker than they are, so budgets drift toward last-click harvesting channels and away from the demand creation that feeds them.
How does ElenIQ model adstock?
ElenIQ applies an adstock transformation to historic spend before fitting its response models, so each period’s effect includes a decaying contribution from earlier spend. Combined with saturation curves, this lets ElenIQ forecast the fuller, time-shifted impact of a budget change rather than only the conversions that fall inside a narrow window.
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