Carryover Effects and Distributed Lags

Summary

Chapter 4 is the dynamic core of the book. Marketing effects typically extend beyond the current period through carryover (advertising goodwill, habit, word-of-mouth). This note covers the Koyck geometric lag, Almon polynomial distributed lag (PDL), geometric lag with purchase feedback (GLPF), the general ADL model, and the temporal aggregation bias with recovery procedures.

Why Carryover Matters

Advertising in period $t$ may influence sales in periods $t,t+1,t+2,\dots$ through:

• Memory: consumers recall brand messages
• Goodwill stock: cumulative brand awareness
• Purchase feedback: a new buyer is more likely to repurchase
• Word-of-mouth: buyers influence non-buyers

The retention rate $\lambda \in (0,1)$ governs how quickly these effects decay.

1. Geometric (Koyck) Distributed Lag

Koyck / Geometric Lag

Starting from an infinite distributed lag:

$$Q_t={\beta }_0+\sum _{k=0}^{\infty }{\beta }_{1,k}{X}_{t-k}+w_t$$

Imposing geometric decay ${\beta }_{1,k}={\beta }_1{\lambda }^k$ and applying the Koyck transformation yields:

$$Q_t=(1-\lambda ){\beta }_0+{\beta }_1(1-\lambda )X_t+\lambda Q_{t-1}+(w_t-\lambda w_{t-1})\text{\hspace{1em}(Eq 4.10)}$$

• $\lambda$: carryover (retention rate); $0<\lambda <1$
• ${\beta }_1(1-\lambda )$: short-run effect
• ${\beta }_1$: long-run effect (total cumulative effect of a permanent unit increase)
• The error $v_t=w_t-\lambda w_{t-1}$ is MA(1), requiring GLS for efficient estimation

Long-Run Multiplier

For the Koyck model, the long-run effect of a permanent unit increase in $X$ is:

$$\text{LRM}=\frac{{\beta }_1(1-\lambda )}{1-\lambda }={\beta }_1$$

The mean lag (average delay before the effect materializes) is $\lambda /(1-\lambda )$ periods.

2. Geometric Lag with Purchase Feedback (GLPF)

GLPF Model

Augments the Koyck model with purchase feedback: a new customer created by advertising in period $t$ generates repeat purchases in future periods. The model (Eq 4.21):

$$Q_t=\alpha +{\beta }_1{X}_t+\varphi Q_{t-1}+\text{Purchase feedback term}+u_t$$

where $\varphi$ captures both advertising carryover and the repeat-purchase rate. Identification requires separating $\lambda$ (advertising decay) from the repurchase probability.

3. Almon Polynomial Distributed Lag (PDL)

Almon PDL

Instead of geometric decay, the PDL places polynomial restrictions on the lag coefficients:

$${\beta }_k={\alpha }_0+{\alpha }_{1}k+{\alpha }_2{k}^2+\cdots +{\alpha }_p{k}^p,k=0,1,\dots ,K$$

This smoothness restriction reduces the number of free parameters from $K+1$ to $p+1$ while allowing flexible lag shapes (inverted-U, hump-shaped, etc.).

The model is estimated by OLS on transformed regressors $Z_j={\sum }_{k=0}^K{k}^j{X}_{t-k}$ for $j=0,\dots ,p$.

4. Autoregressive Distributed Lag (ADL)

ADL Model

The general ADL(r, s) model combines lagged dependent variable (autoregression) with distributed lags of the input:

$$Q_t=\alpha +\sum _{i=1}^r{\gamma }_i{Q}_{t-i}+\sum _{k=0}^s{\beta }_k{X}_{t-k}+w_t\text{\hspace{1em}(Eq 4.24)}$$

• $r$: autoregressive order (number of lagged sales terms)
• $s$: distributed lag order (number of lagged advertising terms)
• The Koyck model is the special case ADL(1,0) with ${\gamma }_1=\lambda$

Long-run effect: $\text{LRE}=\frac{{\sum }_{k=0}^s{\beta }_k}{1-{\sum }_{i=1}^r{\gamma }_i}$

5. Lead-Lag Taxonomy (Doyle & Saunders 1985)

Six cases for the lead/lag relationship between advertising $X_t$ and sales $Q_t$ (Eqs 4.25-4.31):

Case	Pattern	Interpretation
1	Only $X_t$ significant	Pure contemporaneous
2	$X_t$ + $X_{t-1}$	Short carryover
3	$X_t$ + $Q_{t-1}$	Koyck carryover
4	$X_{t+1}$ + $X_t$	Anticipatory (lead) effect
5	Only $Q_{t-1}$	Habitual purchase, no ad effect
6	$X_t$ leads $Q_t$ via sales momentum	Feedback loop

6. Time-Varying Parameters

Return-to-Normality Model

Parameters can evolve stochastically. The return-to-normality model (Eq 4.36):

$${\beta }_t=(1-\varphi )\bar{\beta }+\varphi {\beta }_{t-1}+{\nu }_t$$

where $\bar{\beta }$ is the long-run mean and $\varphi$ governs persistence. When $\varphi =0$, ${\beta }_t=\bar{\beta }+{\nu }_t$ (IID noise around mean). Estimated via Kalman filter.

The Cooley-Prescott nonstationary parameter model (Eqs 4.37-4.40) allows $\bar{\beta }$ itself to evolve with a random walk, capturing structural change.

7. Ratchet Models (Asymmetric Response)

Ratchet Model

Asymmetric response: sales react differently to increasing vs. decreasing advertising:

$$Q_t={\beta }_0+{\beta }_1{X}_t^I+{\beta }_2{X}_t^D,{\beta }_1>{\beta }_2\text{\hspace{1em}(Eq 4.45)}$$

where $X_t^I=X_t-X_{t-1}$ when $X_t>X_{t-1}$ (else 0) and $X_t^D=X_t-X_{t-1}$ when $X_t<X_{t-1}$ (else 0).

Alternative: historical maximum model

$$Q_t={\beta }_0+{\beta }_1{X}_t+{\beta }_2\underset{i\le t}{\max }(X_i)\text{\hspace{1em}(Eq 4.47)}$$

capturing hysteresis: once a high advertising level has been achieved, a reduction does not fully undo the brand-building effect (fast learning/slow forgetting, Figure 4-2 in book).

8. Temporal Aggregation Bias

Clarke (1976) Aggregation Bias

Estimating carryover $\lambda$ from annual data yields estimates 20–50× longer than from monthly data for the same underlying process. The reason: monthly carryover effects are summed during temporal aggregation, inflating the apparent retention rate.

Recovery procedures:

• Bass-Leone (Eqs 4.60-4.61): algebraic relationship between weekly and monthly $\lambda$
• Weiss-Weinberg-Windal (Eq 4.62): instrumental variable approach
• Direct aggregation (Eq 4.63): explicitly aggregate the weekly model to match observed monthly data

Recommendation: use the shortest available data interval. Aggregation-adjusted monthly $\lambda \approx 0.7$ (empirical generalization — see Advertising and Promotion Effects).

Lag Operator Notation (Appendix)

For reference, the lag operator notation used in the book:

$$L{X}_t\equiv X_{t-1}\text{\hspace{1em}(Eq 4.64)}$$ $$L^k{X}_t=X_{t-k}$$

The Koyck model in lag polynomial form:

$$(1-\lambda L)Q_t=(1-\lambda ){\beta }_0+{\beta }_1(1-\lambda )X_t+w_t-\lambda w_{t-1}$$

A general rational lag is $B(L)/C(L)$ (Eq 4.72), which nests ADL models.

Cross-Links

• Static baseline: Functional Forms in Marketing
• Reaction functions and competitive dynamics: Reaction Functions and Competitive Dynamics
• ARIMA extension: Single Marketing Time Series
• Transfer function extension: Transfer Function Model
• Empirical carryover estimate ($\lambda \approx 0.43$ monthly): Advertising and Promotion Effects
• Hysteresis in long-run analysis: Multivariate Persistence and Cointegration

See Also

• Linear-Gaussian State-Space Models — dynamic models in state-space form