1. The Geometric Distributed-Lag Model

Advertising expenditure is a risk-taking investment. The economic rationale is straightforward: a campaign launched in period t builds brand awareness and consumer loyalty that generate revenue and operating profit in the current and subsequent periods, with diminishing intensity. The canonical theoretical foundation is Nerlove and Arrow’s (1962) goodwill-stock model, in which advertising accumulates into an intangible asset that depreciates at a constant rate. The resulting revenue or profit process is a geometric distributed lag in current and past advertising expenditures.

Let R(t), X(t), and P(t) denote net revenue (Compustat REVT), advertising expenditure (XAD), and operating income before depreciation and amortization (OIBDP) in year t. The structural model for revenue is:

\begin{aligned}
R(t) &= \alpha + \gamma \sum_{k=0}^{\infty} \lambda^k X(t-k) + u(t) \\
     &= \alpha + \gamma \Big[ X(t) + \lambda X(t-1) + \lambda^2 X(t-2) \\
     &\quad + \cdots \Big] + u(t)
\end{aligned}

The parameter γ is the impact multiplier: dollars of revenue generated per dollar of current-period advertising. The parameter λ (0 < λ < 1) is the carryover rate: the fraction of advertising’s productive effect surviving to the next period. The two parameters are structurally distinct and must be estimated separately. The same model applies when P(t) replaces R(t) as the dependent variable, and to any investment flow—research and development (XRD), net property, plant, and equipment (PPENT), capitalized software, or any other expenditure that generates multi-period productive capacity.

2. The Koyck Transformation

The structural equation contains infinitely many regressors and is not directly estimable. Koyck (1954) showed that the infinite sum can be eliminated by a single algebraic operation. Write the structural equation at t and at t−1, multiply the latter by λ, and subtract:

\begin{aligned}
R(t) = \hat\delta_0 + \lambda R(t-1) + \hat\gamma X(t) + \varepsilon(t), \\ \quad \hat\delta_0 \equiv \hat\alpha(1-\lambda), \quad \varepsilon(t) \equiv u(t) - \hat\lambda u(t-1)
\end{aligned}

This three-parameter equation is the Koyck estimating form. The original structural parameters are recovered as α = δ₀ / (1−λ), and the following derived statistics carry economic content:

3. Generalization to Any Investment Flow

The Koyck transformation is not specific to advertising. It applies to any expenditure that generates a depreciating (or amortization) stock of productive capacity. The identifying condition is that the investment flow, denoted Flow(t), accumulates into a stock according to the formula:

\text{Stock}(t) = \hat\lambda \cdot \text{Stock}(t-1) + \hat\gamma \cdot \text{Flow}(t)

This perpetual-inventory recurrence—familiar from the capital-stock literature; see Griliches (1979)—connects the transformation to several Compustat investment flows, as shown in the following table. In each case the estimating form is identical; only the economic interpretation of γ and λ changes.

4. Econometric Considerations

The composite disturbance ε(t) = u(t) − λ u(t−1) is MA(1). Because R(t−1) is correlated with u(t−1), OLS applied to the Koyck form is inconsistent: the bias survives as N grows. The preferred estimator is the instrumental-variable procedure of Liviatan (1963), which instruments R(t−1) with X(t−1). The identifying assumption—that past advertising is correlated with past revenue but uncorrelated with the current composite error—holds when u(t) is serially uncorrelated and X(t) is predetermined.

Serial correlation in the OLS residuals is tested by Durbin’s (1970) h-statistic, which replaces the invalid Durbin-Watson statistic when a lagged dependent variable is present:

h = \hat{\rho} \cdot \sqrt{\frac{N}{1 - N \cdot \mathrm{Var}(\hat{\lambda})}} \sim \mathcal{N}(0,1)

Annual data introduce a frequency-mismatch problem. Meta-analytic evidence (Clarke 1976) places the advertising effect half-life between two and six months for most consumer categories. At annual aggregation, intra-year decay is collapsed into the intercept δ₀ and the contemporaneous coefficient γ̂. The lagged dependent variable absorbs macroeconomic momentum unrelated to advertising, biasing λ̂ toward zero or even negative values. Annual data can identify the impact multiplier γ reliably; the carryover rate λ should be treated with caution and tested for its sign and plausibility. A weak-instrument F-statistic from the first stage of the IV procedure (rule of thumb: F ≥ 10) is a prerequisite diagnostic when the sample is small.

5. Empirical Evidence: The Gap, Inc. (Annual, 2002–2023)

The Gap, Inc. (Compustat GVKEY 004990; CIK 0000039911: SIC Code 5651: Apparel Retail) is a suitable test case: advertising expenditure is persistent and material, averaging approximately $550 million per year over the period, with a coefficient of variation of roughly 15 percent. Annual Compustat data yield N = 22 observations. Both equations—with R(t) and P(t) as dependent variables—are estimated by OLS (Newey-West HAC, lag = 3) and by 2SLS with X(t−1) instrumenting R(t−1) or P(t−1). IV estimates are reported as preferred.

The impact multiplier γ̂ is statistically significant and empirically plausible in both equations. For the revenue equation, one dollar of advertising generates approximately $6.20 of contemporaneous revenue—a figure consistent with the apparel meta-analytic range reported by Sethuraman, Tellis, and Briesch (2011). For the profit equation, the multiplier of $4.06 per dollar of advertising is directly relevant to transfer pricing analysis under the TNMM, where the benchmark is an operating margin rather than a revenue multiplier. The P(t) specification is therefore the preferred equation for intangibles valuation purposes: advertising creates pricing power and margin, not merely volume.

The carryover rate λ̂ is negative in both equations—an economically inadmissible result, as a negative λ implies that prior advertising destroys current profits. Three sources of this pathology are identifiable: (i) the true carryover operates at monthly frequency and is not separately recoverable from annual data; (ii) the COVID-19 shock in 2020 (GPS revenue fell $2.6 billion) exerts disproportionate leverage on the first-order autocovariance of both R and P; and (iii) the Liviatan instrument X(t−1) is weak given the low year-to-year variation in GPS advertising. The appropriate conclusion is that λ is not credibly identified from annual data in this sample. The long-run multiplier, which depends on γ/(1−λ), is correspondingly unreliable. The contemporaneous impact multiplier γ̂, however, is robust to these limitations and constitutes the defensible empirical result.

References

Fisher, Irving. 1925. “Our Unstable Dollar and the So-Called Business Cycle.” Journal of the American Statistical Association 20 (150): 179–202.

▶ Theoretical and empirical. Fisher introduced geometrically weighted distributed lags to model the relationship between price level changes and business conditions, using U.S. price and output series. He is the proto-innovator of the geometric weighting scheme that Koyck later formalized into a general parameterization. Fisher did not derive the algebraic transformation that renders the model estimable. Citing such an early-reference is an academic caprice.

Koyck, Leendert M. 1954. Distributed Lags and Investment Analysis. Amsterdam: North-Holland.

▶ Theoretical, with empirical application to Dutch fixed capital investment data. Koyck’s specific and non-trivial innovation is the algebraic transformation that converts an equation with infinitely many regressors into a three-parameter autoregressive form. This distinguishes him from Fisher: Koyck did not merely use geometric weights but showed how to eliminate them. The model was developed for investment analysis, not advertising; its application to advertising came a decade later. Koyck is the innovator of the transformation; Fisher is the innovator of the geometric-weight concept.

Nerlove, Marc. 1958. Distributed Lags and Demand Analysis for Agricultural and Other Commodities. Agricultural Handbook No. 141. Washington, DC: USDA.

▶ Theoretical and empirical; agricultural commodity data from U.S. government sources. Nerlove derived the identical estimating equation—a first-order autoregression in the dependent variable and the current regressor—from a behaviorally grounded partial-adjustment model in which agents adjust toward a target level. Because the derivation is independent of Koyck’s and rests on stronger theoretical foundations (optimizing behavior rather than a statistical parameterization), Nerlove is a co-innovator of the estimating form.

Nerlove, Marc, and Kenneth J. Arrow. 1962. “Optimal Advertising Policy under Dynamic Conditions.” Economica 29 (114): 129–142.

▶ Theoretical. Arrow and Nerlove introduced the goodwill-stock model of advertising: advertising expenditure accumulates into an amortizable intangible asset G(t) = (1−δ)G(t−1) + γ·Flow(t). This is the theoretical foundation that justifies applying the Koyck estimating form to advertising data. Arrow and Nerlove are the innovators of the advertising-as-investment framework; Koyck provided the algebraic tool that implements it.

Liviatan, Nissan. 1963. “Consistent Estimation of Distributed Lags.” International Economic Review 4 (1): 44–52.

▶ Theoretical. Liviatan identified the inconsistency of OLS applied to the Koyck estimating form (due to the MA(1) composite error) and proposed the instrumental-variable remedy: instrument the lagged dependent variable with the lagged exogenous variable. This is a methodological follow-on to Koyck; Liviatan is an innovator in the estimation literature but a follower in the modeling literature.

Palda, Kristian S. 1964. The Measurement of Cumulative Advertising Effects. Englewood Cliffs, NJ: Prentice-Hall.

▶ Empirical. Palda was the first to apply the Koyck distributed-lag model specifically to advertising data, using annual sales and advertising expenditure for the Lydia E. Pinkham Medicine Company (1908–1960). He demonstrated statistically significant carryover and estimated a long-run advertising multiplier. Palda is the innovator of the empirical advertising-spillover literature; but he lacked statistical discipline, not estimating his dynamic specification.

Almon, Shirley. 1965. “The Distributed Lag Between Capital Appropriations and Expenditures.” Econometrica 33 (1): 178–196.

▶ Empirical. Almon proposed polynomial distributed lags as an alternative to geometric (Koyck) weighting, applied to U.S. quarterly capital appropriations and expenditures data. Her approach imposes a polynomial shape on the lag weights rather than geometric decay, allowing for non-monotone lag distributions. Almon is an independent innovator of a competing parameterization, not a follower of Koyck; the two approaches answer different prior beliefs about lag-weight shape.

Durbin, James. 1970. “Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors Are Lagged Dependent Variables.” Econometrica 38 (3): 410–421.

▶ Theoretical. Durbin derived the h-statistic as a large-sample test for first-order serial correlation when the equation contains lagged dependent variables, replacing the Durbin-Watson statistic, which is asymptotically invalid in that setting. This is a methodological contribution that resolves a diagnostic gap in estimating Koyck and Nerlove equations. Durbin is a methodological innovator; his paper has no predecessor for this specific problem.

Clarke, Darral G. 1976. “Econometric Measurement of the Duration of Advertising Effect on Sales.” Journal of Marketing Research 13 (4): 345–357.

▶ Empirical; meta-analysis of 69 published advertising econometric studies across consumer packaged goods, durables, and services. Clarke documented that the true advertising effect half-life is typically two to six months, and that annual data aggregate intra-year decay into the contemporaneous coefficient, biasing the estimated carryover rate λ toward higher values. This is the key empirical finding motivating caution when the Koyck model is estimated on annual Compustat data.

Griliches, Zvi. 1979. “Issues in Assessing the Contribution of Research and Development to Productivity Growth.” Bell Journal of Economics 10 (1): 92–116.

▶ Theoretical and empirical; industry-level and firm-level R&D and productivity data for U.S. manufacturing. Griliches established that R&D expenditure should be treated as a capital investment and that the perpetual-inventory method (equivalent to the Koyck recurrence Stock(t) = λ·Stock(t−1) + γ·Flow(t)) is the appropriate way to construct an R&D capital stock from flow data. His framework generalizes the Nerlove-Arrow goodwill model to all intangible investment flows—including XAD, XRD, and capitalized software. Griliches is the innovator of the R&D-capital-stock literature and a follower of Koyck and Nerlove-Arrow on the modeling structure.

Sethuraman, Raj, Gerard J. Tellis, and Richard A. Briesch. 2011. “How Well Does Advertising Work? Generalizations from Meta-Analysis of Brand Advertising Elasticities.” Journal of Marketing Research 48 (3): 457–471.

▶ Empirical; meta-analysis of 751 short-run and 402 long-run brand advertising elasticity estimates from 56 studies. Sethuraman et al. documented that short-run advertising elasticity averages approximately 0.12 across product categories, and that long-run elasticities (capturing carryover) are larger. The apparel and specialty retail range relevant to the Gap example is at the upper end of the cross-category distribution. This paper is the standard empirical reference for benchmarking estimated advertising multipliers. Meta-analysis is the quantitative synthesis of empirical results across multiple independent published studies that estimate the same underlying parameter — typically an effect size such as an elasticity, correlation, or regression coefficient.