Design of Static Response Models

Summary

A static response model estimates the contemporaneous (same-period) relationship between marketing instruments and sales. This note covers the design choices — variable selection, functional form, competition specification — that precede estimation. Static models are the foundation; Chapter 4 adds dynamic extensions.

When Static Models Are Appropriate

A static model is appropriate when:

1. Carryover effects are negligible (very fast decay, $\lambda \approx 0$)
2. The data are cross-sectional (many brands/markets at one time point)
3. The researcher seeks a reduced-form summary of long-run effects only

When dynamics are important, the static model will mis-attribute carryover to current period and bias elasticities upward. See Carryover Effects and Distributed Lags.

Variable Selection Principles

Model Completeness

A well-specified static response model must include:

• Focal instrument: the marketing variable of primary interest (e.g., advertising, price)
• Control variables: other marketing mix elements that affect sales (price, distribution, promotion)
• Competitive variables: rival brands’ price, advertising, and promotions
• Environmental variables: seasonality, economic conditions, category trend

Omitting relevant variables biases included coefficients — see Omitted Variables Bias.

Competitive Specification

Three main approaches to incorporate competition:

Approach	Model	Advantage
Absolute levels	$Q=f(X_{\text{own}},X_{\text{rival}})$	Direct interpretation
Share of voice	$Q=f(\text{SOV})$ where SOV $=X_{\text{own}}/\sum X_j$	Captures competitive intensity
Market share	$MS=g(X_{\text{own}},{\mathbf{X}}_{\text{rivals}})$	Bounded outcome, MCI/MNL structure

The market-share approach ensures logical consistency (shares sum to 1) — see Market Share Models.

Interaction Terms

Static models can capture moderation via interaction terms:

$$Q={\beta }_0+{\beta }_{1}A+{\beta }_{2}P+{\beta }_3(A\times P)+ϵ$$

where ${\beta }_3$ captures how advertising modifies price sensitivity (or vice versa). This is equivalent to letting $\partial Q/\partial P$ be a function of $A$:

$$\frac{\partial Q}{\partial P}={\beta }_2+{\beta }_{3}A$$

Related to Bayesian moderation analysis in the Bayesian statistics module.

Dummy Variables and Categorical Marketing Variables

Feature advertising and display are typically binary:

• $F_{it}=1$ if brand $i$ ran feature ad in week $t$, else 0
• $D_{it}=1$ if brand $i$ had in-store display in week $t$, else 0

In multiplicative models, dummies enter as:

$$Q={\beta }_0\cdot e^{{\gamma }_{F}F+{\gamma }_{D}D}\cdot (\text{other terms})$$

so $e^{{\gamma }_F}$ is the feature multiplier (ratio of sales with feature to sales without feature).

Specification Checklist

Before estimating, verify:

• All key marketing instruments included (avoid OVB)
• Functional form consistent with prior theory (concavity, saturation)
• Competitive variables included or argued to be orthogonal to focal instrument
• Seasonality and trend controlled (dummy variables or detrending)
• Sample period homogeneous (no structural breaks)
• Data level appropriate (store/market/national match advertising measurement)

Connection to Dynamic and Hierarchical Models

Static models are estimated equation-by-equation. When:

• Multiple time periods exist: extend to dynamic models (ADL, distributed lags)
• Multiple brands/markets exist: extend to panel models (SUR, random effects)
• Bayesian shrinkage is desired: extend to hierarchical Bayes

Cross-Links

• Functional forms catalogue: Functional Forms in Marketing
• Market share systems: Market Share Models
• Aggregation issues: Aggregation of Relations
• Dynamic extension: Design of Dynamic Response Models
• Estimation: Parameter Estimation in Market Response
• OVB: Omitted Variables Bias