Aggregation of Relations

Summary

Market response models are almost always estimated on aggregate data (stores, markets, or national), while the underlying behavioral processes occur at the individual level. This note covers the conditions under which aggregate models correctly recover individual parameters, and when aggregation bias arises.

The Aggregation Problem

Aggregation Bias

Aggregation bias occurs when the functional form of an individual-level response function does not carry over to the aggregate level. If individual $i$ has response $Q_i=f(X_i;{\beta }_i)$, the aggregate $Q={\sum }_i{Q}_i$ generally cannot be written as $f(\bar{X};\bar{\beta })$ unless $f$ is linear or strong distributional assumptions hold.

Conditions for Exact Aggregation

Exact Aggregation (Linear Case)

For the linear model $Q_i={\beta }_0+{\beta }_1{X}_i+{ϵ}_i$, summing over $N$ individuals:

$$Q=N{\beta }_0+{\beta }_1\sum _i{X}_i+\sum _i{ϵ}_i=N{\beta }_0+{\beta }_{1}N\bar{X}+N\overline{ϵ}$$

The aggregate OLS on $\bar{Q}$ and $\bar{X}$ recovers the same ${\beta }_1$. Exact aggregation holds for linear models.

For nonlinear forms (log-log, logistic, etc.), exact aggregation fails except in special cases. The aggregate elasticity is a weighted average of individual elasticities, and the weights depend on the distribution of $X_i$ across the population.

Approximate Aggregation (Second-Order Correction)

For a nonlinear function $f(X)$, a second-order Taylor expansion around the mean $\bar{X}$ gives:

$$E[f(X)]\approx f(\bar{X})+\frac{1}{2}{f}^{\prime \prime }(\bar{X})\cdot \text{Var}(X)$$

The correction term $\frac{1}{2}{f}^{\prime \prime }(\bar{X})\cdot \text{Var}(X)$ is:

• Zero for linear $f$ (exact aggregation)
• Negative for concave $f$ (aggregate overestimates mean individual response)
• Positive for convex $f$ (aggregate underestimates mean individual response)

Marketing Implications

Aggregation and Advertising Response

If individual advertising response is concave (diminishing returns at the individual level), the aggregate response function is also concave but may appear more linear. The slope at the aggregate mean understates diminishing returns — i.e., the aggregate model overestimates the return to further advertising at the margin.

This matters for budget optimization: optima derived from aggregate models may differ from those that maximize individual-level expected utility.

Cross-Sectional Heterogeneity and Random Coefficients

When ${\beta }_i$ varies across individuals (heterogeneous response), the aggregate model picks up an average effect. The random coefficients model (see Functional Forms in Marketing) explicitly models this heterogeneity:

$$Q_{it}={\beta }_{0i}+{\beta }_{1i}{X}_{it}+{ϵ}_{it}$$

where ${\beta }_{ji}\sim ({\bar{\beta }}_j,{\sigma }_{\beta }^2)$. The aggregate-level regression recovers ${\bar{\beta }}_j$ but conceals the distribution of individual effects, which matters for targeted marketing decisions.

Temporal vs. Cross-Sectional Aggregation

Type	Mechanism	Key Bias
Temporal (weekly → monthly)	Carryover effects compressed	Retention rate $\lambda$ inflated (Clarke 1976)
Cross-sectional (store → market)	Heterogeneous responses averaged	Elasticity biased toward mean
Functional-form	Nonlinear $f$ aggregated	Second-order correction term

See Carryover Effects and Distributed Lags for temporal aggregation corrections.

Cross-Links

• Functional forms: Functional Forms in Marketing
• Temporal aggregation bias: Carryover Effects and Distributed Lags
• Hierarchical models for heterogeneity: Hierarchical Linear Models
• Random coefficients: Functional Forms in Marketing