Adstock is the assumption, written into a statistical model as arithmetic, that advertising bought in one week keeps working in the weeks that follow. A television spot aired on a Monday does not stop influencing purchases on the Tuesday. Some fraction of its effect persists, decays and eventually falls to nothing. Adstock is the transformation converting a series of media spend or impression figures into a running stock of accumulated advertising pressure, so that a regression of sales on media is fitted against that stock rather than against raw weekly spend.

Textbook treatments give the term two dimensions: the decay or lagged effect, and saturation or diminishing returns. Practice has largely separated them. Adstock now refers to the carryover half, while diminishing returns are handled by a distinct saturation function applied before or after it. The separation matters, because the two transformations are estimated jointly and can absorb one another's signal.

How the transformation works

The standard formulation is geometric. Media execution at time t-s enters the stock at time t with a weight equal to a decay parameter raised to the power s. Google's Meridian documentation defines the function as a normalised weighted average across lags zero to L, where L is the maximum lag duration and the weights are divided by their own sum. Normalisation is not cosmetic: a fixed budget then produces a fixed total of adstock spread across weeks, rather than a total that grows as the decay parameter rises.

Meta's Robyn package illustrates the raw mechanics with a worked example in its reference manual. Spend of 100 on day one with a decay parameter, which Robyn calls theta, of 0.7 leaves 70 carried into day two and 49 into day three. The same arithmetic gives the half-life, the number of periods needed for the effect to fall to half its starting level.

Two further settings govern the shape. The maximum lag caps how far back the window reaches; Meridian recommends two to ten periods with geometric decay and four to twenty with its binomial alternative. Ordering relative to saturation is a separate flag: Meridian applies adstock first by default, with a parameter to reverse the sequence.

Decay curves and their parameters

Geometric decay is the one-parameter default across most implementations, forcing a constant proportional loss each period and putting the peak effect in the same period as the exposure.

Robyn adds two Weibull options. The cumulative density form lets the decay rate itself change over time, in either C or S shape, while keeping the peak in the first period. The probability density form permits the peak to arrive later, the only way in that package to model a genuinely delayed response. Robyn recommends a shape bound of 0.0001 to 2 and a deliberately conservative scale bound of 0 to 0.1, warning that scale increases the half-life sharply. The flexibility costs compute time: either Weibull option runs up to twice as long as geometric. Robyn's rule of thumb for theta ranges runs 0.3 to 0.8 for television, 0.1 to 0.4 for out-of-home, print and radio, and 0 to 0.3 for digital.

Google added a binomial curve to Meridian in the release of September 30, 2025, alongside non-media variables and channel-level contribution priors. Its weights fall to zero at lag L+1 and the curve is convex below a parameter value of 0.5, linear at 0.5 and concave above it. Google's own trade-off table is blunt about the consequence: geometric decay is vulnerable to underestimating long-term effects, binomial to underestimating short-term ones. Meridian's default prior on the decay parameter is uniform across zero to one, with Beta(1,3) suggested to push mass toward fast decay and Beta(3,1) toward slow decay.

PyMC-Marketing, the third widely used implementation, ships geometric, delayed, Weibull and binomial variants with a default maximum carryover of twelve periods.

From goodwill stock to open-source code

The economics came first. Marc Nerlove and Kenneth Arrow modelled advertising as an investment building a decaying stock of goodwill in Economica in 1962, and the distributed-lag machinery that makes such a stock estimable traces to Leendert Koyck's 1954 work on investment analysis.

Simon Broadbent, then at Leo Burnett in London, coined the word. His paper "One Way TV Advertisements Work" appeared in the Journal of the Market Research Society in 1979, followed by "Modelling with Adstock" in the same journal in 1984. Sources differ on the volume number of the 1979 paper, some citing volume 21 and others volume 23. The concept spread through consumer packaged goods econometrics over the following two decades, largely inside agency and consultancy practice rather than published literature.

Bayesian treatment arrived with a 2017 Google technical report by Yuxue Jin, Yueqing Wang, Yunting Sun, David Chan and Jim Koehler, which proposed flexible functional forms for carryover and shape effects and remains a cited reference in Meridian's documentation. Open-source packaging followed: Meta released Robyn, Google unveiled Meridian in March 2024 and opened it globally on January 29, 2025, and PyMC Labs shipped a Python equivalent. A walkthrough published on May 28, 2026 covered Meridian's Hill curves and adstock mathematics.

Who sets the number

Nobody observes adstock. It is estimated, or assumed, and the parties doing so sit almost entirely on the buy side. Advertiser analytics teams, agency econometricians and independent measurement vendors specify the decay function, choose priors and defend the resulting numbers to finance departments. IAB Australia's vendor landscape, published in September 2025, profiled twelve active providers ranging from classical econometric shops to Bayesian and machine-learning approaches.

Sell-side involvement is the contested part, since both dominant open-source frameworks are built by companies that sell media. Prescient AI, launching a proprietary model in July 2025, positioned independence from media selling operations as its principal argument. Joint-industry bodies have started supplying inputs instead of models: Barb's Ads Hub, which passed 600 users by June 2026, planned a data pack for mix-model input, addressing the routine underestimation of linear television in models built on digital signals.

Why the decay rate decides the media plan

A decay parameter is not a diagnostic curiosity. It determines whether a channel can be flighted or must run continuously. Research reported in August 2026 made the point: a standard mix model estimated weekly carryover at 0.50 against a known true value of 0.20. A channel retaining half its effect week to week can be pulsed and rested; one retaining a fifth cannot. Those are different media plans and different annual budgets.

The parameter also decides how much credit upper-funnel activity receives. Google described its binomial curve explicitly as a way to capture brand recall driving purchases weeks after exposure. The stakes are visible in outcome research: TransUnion and MMA Global work published in October 2025 found conventional measurement capturing 56 percent of marketing impact at Ally Bank, 40 percent at Campbell's and just 17 percent at Kroger. A model with an aggressively short lag window reproduces that undercounting by construction.

Mix modelling has drawn renewed attention as deterministic tracking weakened, a shift documented as early as 2024 and which a French trade body restated in February 2026 in recommending hybrid measurement.

Limitations and disputes

Practitioners and academics do not agree on plausible values. A 2008 working paper in the Munich Personal RePEc Archive reports academic studies suggesting half-lives of roughly seven to twelve weeks against industry practice of two to five weeks, with fast-moving consumer goods brands averaging 2.5 weeks. The gap is a factor of three or more on the same underlying quantity.

The deeper objection concerns identification. Ryan Dew of Wharton, Nicolas Padilla of London Business School and Anya Shchetkina of Wharton posted an identification paper to arXiv on August 14, 2024, running 2,187 simulation settings with 100 datasets each. Nonlinear and time-varying effects proved frequently inseparable from standard mix-model data, and conflation rates rose with carryover: at a stock coefficient of 0.8, the time-varying case reached 99 percent. The two model families imply different optimal budgets, so the conflation is not academic.

Endogeneity compounds it, because budgets are not randomly assigned. Governance is a live concern too: CIMM warned on July 31, 2026 that default priors in widely adopted open-source frameworks can reconfigure data asymmetry rather than remove it, an argument previously tested against Meta's Robyn.

What adstock is not

Attribution lookback windows count conversions occurring within a fixed period after a click or impression. They are counting rules applied to observed events. Adstock is a modelled decay applied to spend, estimated from aggregate time series, with no requirement that any individual be tracked.

Saturation describes diminishing returns to a given level of pressure, typically through a Hill or logistic function. Adstock describes persistence over time. The two are complementary transformations and, as the identification literature shows, mutually confusable.

Frequency measures how often a person is exposed. Adstock says nothing about individuals; it operates on aggregated channel-level series.

Awareness tracking measures recall through surveys. Broadbent's original model was built partly to explain such data, but adstock today is a variable inside a sales equation, not a survey metric.

Recent developments

Meridian gained a code-free Scenario Planner in February 2026, moving into open beta with per-channel spend constraints by August 2026. Geographic experimentation arrived alongside it: Meridian GeoX and Meridian Studio were announced on May 5, 2026, with experiment results designed to calibrate the model rather than sit beside it.

Calibration is where adstock research is now concentrated. Structural estimation from four geo-experiments recovered decay of 0.19 against a true 0.20, compared with 0.50 from the observational specification. Retail media measurement has taken up the same theme, with the IAB arguing that legacy methods understate the channel's value. Measurement confidence, meanwhile, has not moved much.

Timeline

  • 1954: Leendert Koyck publishes distributed lag methods for investment analysis
  • 1962: Marc Nerlove and Kenneth Arrow model advertising as a decaying stock of goodwill in Economica
  • 1979: Simon Broadbent coins adstock in the Journal of the Market Research Society
  • 1984: Broadbent publishes "Modelling with Adstock" in the same journal
  • 2017: Google researchers publish Bayesian methods for carryover and shape effects
  • March 2024: Google unveils Meridian on a limited basis
  • August 14, 2024: An arXiv identification paper documents conflation between carryover and nonlinear effects
  • January 29, 2025: Meridian opens globally after testing with hundreds of brands
  • July 15, 2025: Prescient AI launches a proprietary alternative to open-source foundations
  • September 2025: IAB Australia profiles twelve mix-modelling vendors
  • September 30, 2025: Google adds binomial adstock decay to Meridian
  • February 19, 2026: Meridian Scenario Planner launches
  • May 5, 2026: Meridian GeoX and Meridian Studio announced
  • May 28, 2026: A developer episode details Meridian's adstock and Hill mathematics
  • July 31, 2026: CIMM warns on default priors in open-source frameworks
  • August 21, 2026: Structural estimation paper on recovering adstock from geo-experiments posted to arXiv

Summary

Who: Simon Broadbent coined the term at Leo Burnett in 1979, building on goodwill-stock economics from Marc Nerlove and Kenneth Arrow and distributed-lag methods from Leendert Koyck. It is now specified by advertiser analytics teams, agency econometricians and measurement vendors, and implemented in code by Google, Meta and PyMC Labs. Critics include Ryan Dew, Nicolas Padilla and Anya Shchetkina on identification, and CIMM on framework defaults.

What: A transformation converting raw media spend or impressions into a decaying stock of accumulated advertising pressure, governed by a decay parameter, a maximum lag window and a choice of decay curve, usually geometric, binomial or Weibull.

When: The underlying economics dates to 1954 and 1962, the term to 1979, Bayesian formalisation to 2017, and open-source implementation to the Robyn and Meridian releases of the 2020s. Binomial decay entered Meridian on September 30, 2025.

Where: Inside marketing mix models rather than in ad servers, demand-side platforms or attribution tools. The parameter is estimated from aggregated weekly or daily time series, increasingly calibrated against geographic experiments.

Why: Advertising effects are not contemporaneous, and a model that ignores persistence attributes long-run sales to the wrong period or to the baseline. The decay rate directly determines whether budgets are flighted or run continuously, and how much credit brand activity receives, which is why a parameter that cannot be observed and is disputed by a factor of three still governs large allocation decisions.