Design of Dynamic Response Models

Summary

Dynamic models extend static response models by incorporating time: current sales depend on past advertising, past sales (carryover), and past competitive actions. This note covers the design decisions for dynamic models — order selection, identification of lag structures, and the distinction between short-run and long-run effects.

From Static to Dynamic

The static model $Q_t={\beta }_0+{\beta }_1{X}_t+{ϵ}_t$ assumes marketing effects are fully realized within the measurement period. In practice, advertising creates awareness that decays gradually, price promotions can cause inter-temporal substitution (stockpiling), and distribution gains persist.

Adding dynamics means asking:

1. How many lags? Determine maximum lag length $K$
2. What shape? Geometric (Koyck), polynomial (Almon), or unrestricted
3. What type of lagged dependent variable? Autoregressive, error correction, or pure MA

The General ADL Framework

Any linear dynamic model can be expressed as an ADL (Autoregressive Distributed Lag):

$$Q_t=\alpha +\sum _{i=1}^r{\gamma }_i{Q}_{t-i}+\sum _{k=0}^s{\beta }_k{X}_{t-k}+w_t$$

Key design choices:

• $r$ (AR order): test using AIC, BIC, or Ljung-Box test on residuals
• $s$ (lag order on $X$): test using pattern of OLS coefficients or AIC
• Restrictions: impose geometric decay (Koyck) or polynomial smoothness (Almon) to reduce parameters

See Carryover Effects and Distributed Lags for the full specification.

Identification Strategy

Direct-Lag Identification

The preferred identification strategy for lag structure is the direct-lag regression: estimate an OLS model with many lags of $X$ (and possibly of $Q$), then examine the pattern of estimated coefficients to determine where they cut off vs. die out (analogous to PACF/ACF in ARIMA identification — see Single Marketing Time Series).

If coefficients cut off sharply after lag $s$: use MA($s$) noise / no autoregression If coefficients die out gradually: impose geometric decay (Koyck) If coefficients form a smooth hump: use Almon PDL

Short-Run vs. Long-Run Effects

Distinction of Short and Long Run

For ADL(1,0): $Q_t=\alpha +\gamma Q_{t-1}+{\beta }_0{X}_t+w_t$

• Short-run effect (impact multiplier): ${\beta }_0$
• Long-run effect (total multiplier): ${\beta }_0/(1-\gamma )$

The long-run effect exceeds the short-run effect whenever $\gamma >0$ (positive carryover). A common error is to report only the short-run coefficient and interpret it as the “advertising elasticity” — this understates the true ROI by a factor of $1/(1-\gamma )$.

Model Order Selection

Criterion	Formula	Use
AIC	$-2\ln \hat{L}+2k$	Minimize; rewards fit, penalizes parameters
BIC	$-2\ln \hat{L}+k\ln T$	Stricter penalty, prefers parsimonious models
Adjusted R²	$1-(1-R^2)(T-1)/(T-k-1)$	Maximize
Ljung-Box Q	Tests residual autocorrelation	Diagnostic, not selection

Simultaneity and Endogeneity

A key design decision is whether marketing variables are endogenous (jointly determined with sales) or exogenous. The decision rule (spending = function of lagged sales) means advertising is predetermined but not strictly exogenous. Simultaneity biases OLS — see Parameter Estimation in Market Response for 2SLS.

Connecting to ARIMA

Dynamic response models can be written in the ADL form, which relates to the ARIMA framework:

• Noise component of ADL: if $w_t$ follows ARMA($p$,$q$), the model is an ARMAX
• Removing the $X$ terms yields pure ARIMA — see Single Marketing Time Series
• Adding multiple inputs yields the transfer function model — see Transfer Function Model

Cross-Links

• Koyck, PDL, ADL details: Carryover Effects and Distributed Lags
• Transfer function extension: Transfer Function Model
• Estimation issues: Parameter Estimation in Market Response
• Time series tools: Single Marketing Time Series
• Reaction dynamics: Reaction Functions and Competitive Dynamics