Spatial Models in Marketing

Bradlow et al (2005)Marketing Letters

Introduction

• Interdependent entities– Consumers’ satisfaction ratings (Mittal et al 2004)

– Retailers’ promotional policies (Bronnenberg and Mahajan 2001)

• Objectives of this review paper– Define elements of a spatial model– Introduce various types of spatial models– Model spatial effects– Suggest new research directions

Elements of a spatial model

• A map describes the relationship among individuals– Geographic, demographic, or psychometric

• Distance metrics determine the strength of a relationship– Discrete or continuous– Isotropic: relationship depends on the distance, not on the direction– Individuals of shorter distance have a stronger relationship

• Spatial distance results in a spatial effect

Types of spatial models

• Type I models:– Predict the choice outcome y, conditional on the X variables and the

map locations Z– Simplest specification:

• Kriging: Predict the outcome variable of one individual at a specified location by using the known responses and locations of all other individuals.

• Type II models:– Predict the locations Z at which certain outcomes occurred– Not generally discussed* Opportunity for ABM?

,Z,X|yf

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,,X| yZf

Modeling spatial effects

• General specification

– Spatial lags• Outcomes are spatially interdependent

– Spatially correlated errors• Error terms are spatially interdependent

– Spatial drift• Parameters are a function of an individual’s location on the map

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Spatial lags

Spatial drift

Spatially correlated errors

Modeling spatial effects

• Variations– Replace choice outcome y with continuous latent utility u:

– Spatio-temporal models: incorporate cross-sectional time series data:

• Statistical Issues

– Outcome variables y are (1) spatially correlated and (2) spatially-lagged dependent

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)),(,0(~)( ),(],[)()( ZNtetetZXtWyty

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Research opportunities

• Dimensionality– Sheer amount of information that must be stored

• GIS software, Matlab spatial statistics toolbox, Markov random field– Estimation

• Simplify computations and reduce memory usage of likelihood-based approaches

• Analysis of marketing policies– Endogeneity between marketing mix and response variables

• Spatial distribution depends on order of entry into the region and regional levels of advertising expenditure (Bronnenberg et al 2005).

– Correction

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Research opportunities

I)N(0, ~ ,)( 2111 eeXy

Error component that follows a spatial lag pattern

Marketing mix variables are a function of the error term

Marketing mix variables

Research opportunities

• Interpretation of spatial effects– Impact of social influence on choice behavior (Yang and Allenby 2003; Bell and

Song 2004)

– Spatial priors in a hierarchical Bayes analysis to understand geographic dispersion of preference segments (Ter Hofstede et al 2002; Ter Hofstede 2004)

– Group decision making (Arora 2004)