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Marketing Letters 16:3/4, 267–278, 2005

c ? 2005 Springer Science + Business Media, Inc. Manufactured in the Netherlands.

Spatial Models in Marketing∗

ERIC T. BRADLOW

University of Pennsylvania

BART BRONNENBERG

UCLA

GARY J. RUSSELL

University of Iowa

gary-j-russell@uiowa.edu

NEERAJ ARORA

University of Wisconsin

DAVID R. BELL

University of Pennsylvania

SRI DEVI DUVVURI

University of Iowa

FRANKEL TER HOFSTEDE

University of Texas, Austin

CATARINA SISMEIRO

Imperial College, London

RAPHAEL THOMADSEN

Columbia University

SHA YANG

New York University

Abstract

Marketing science models typically assume that responses of one entity (firm or consumer) are unrelated to

responses of other entities. In contrast, models constructed using tools from spatial statistics allow for cross-

sectional and longitudinal correlations among responses to be explicitly modeled by locating entities on some

type of map. By generalizing the notion of a map to include demographic and psychometric representations,

spatial models can capture a variety of effects (spatial lags, spatial autocorrelation, and spatial drift) that impact

firm or consumer decision behavior. Marketing science applications of spatial models and important research

opportunities are discussed.

∗This paper is based upon the discussions of the Spatial Models in Marketing seminar at the Sixth Invitational

Choice Symposium, June 2004. Eric T. Bradlow, Bart Bronnenberg and Gary J. Russell served as co-chairs of the

session.

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Introduction

A consumer’s decision to adopt a new Internet service is affected by interactions with

other consumers who live in the same postal code area (Bell and Song, 2004). The utility

weights used by consumers to determine satisfaction ratings vary geographically due to

the impact of demographics and lifestyle on choice behavior (Mittal et al., 2004). Retailers

develop promotional policies based on the policies of other retailers in the same trading

area (Bronnenberg and Mahajan, 2001). Each of these is an example of a marketing context

in which the spatial location of a decision-maker plays a key role in the choice process. In

each instance, the spatial component creates a process in which the choice outcomes of one

individual are related to the choice outcomes of other individuals.

The basic tool for constructing models of choice interdependence is the stochastic theory

of spatial statistics (Anselin, 1988; Ripley, 1988; Cressie, 1993; Haining, 1997). Simply

put, spatial models assume that individuals (or, more generally, units of analysis, such as

postal codes) can be located in a space. Typically, responses by individuals are assumed

to be correlated in such a manner that individuals near one another in the space gen-

erate similar outcomes. (In a competitive context, individuals might generate dissimilar

(negatively correlated) outcomes.) The methodology can integrate complex spatial cor-

relations between entities into a model in a parsimonious and flexible manner. Because

spatial statistics was originally developed as a modeling tool in the physical and biolog-

ical sciences, much of the older literature in spatial statistics emphasizes the use of a

geographical map. In marketing, however, it is more appropriate to regard the space as

any type of map—geographic, demographic or psychometric—that describes the relation-

ship among individuals (or units). By generalizing the notion of a map, we can define a

spatial model as a stochastic model which uses known or unknown (latent) relationships

amongindividuals(consumers,managers,retailersetc.)topredicttheoutcomeofadecision

process.

The goal of this paper is to present a brief overview of spatial models in marketing

science. We begin by defining the elements of a spatial model: a map, a distance metric,

and model of spatial effects. We emphasize that the researcher need not use geography in

developing an interesting and useful model. We then consider issues of model specification

and calibration. We conclude with suggestions for new research in spatial models.

Constructing Spatial Models

The key assumption in the traditional marketing science literature is that the behavior of

one individual is conditionally independent of the behavior of another individual. Although

researchers in marketing science now routinely pool information across individuals to al-

low heterogeneity in parameters (Allenby and Rossi, 1998), the underlying model is still

constructed by assuming that each individual acts in isolation while making a decision.

In contrast, spatial models posit that a richer understanding of behavior can be obtained

by assuming the actions of different individuals are correlated. The key questions are why

these relationships exist and how they may be modeled.

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Typology of Spatial Models

All spatial models are constructed using a number a key components. In addition to an

outcome variable y, we assume that the researcher has available a set of covariates X and a

set of spatial relationships Z. Examples of X include product attributes, demographics and

marketing mix elements. In some cases, X can include lagged values of y, both over time

and across individuals. The identity of the variables in Z largely depends on the applica-

tion. However, Z can be viewed as the location of each individual on some type of map.

Unlike X, the location information in Z is typically assumed to be exogenous. (However, in

some applications, map positions Z are treated as parameters and estimated in the course of

the analysis (see, e.g., DeSarbo and Wu, 2001).) Finally, a spatial model includes a vector

of parameters ? that determines the relationships among y, X and Z. Formally, the task

of the researcher is to study the decision process by computing a reasonable estimate of ?

from the available information.

Using this notation, we can define two general types of spatial models of interest to

researchers in marketing. Type I models, denoted by the notation f (y|X, Z, ?), predict

the choice outcome y, conditional on the X variables and the map locations. Type I mod-

els constitute the vast majority of models considered in regional sciences (Cressie, 1993,

Haining, 1997) and in spatial econometrics (LeSage, 1999; Anselin, 2001, 2002). The

simplest models in this area are spatial regression models with the general specification

y = Xβ + e,

e ∼ N(0,?(Z,θ))(1)

where ?(Z, θ) denotes a properly specified covariance matrix in which the correlations be-

tween the responses of two individuals is monotonically decreasing in the distance between

the individuals on the map. Because the errors in (1) have a spatial correlation pattern, the

model can be used to predict the outcome variable of one individual at a specified location

by using the known responses and locations of all other individuals. This approach, known

as kriging, has been used in a marketing context to develop more accurate market-level

estimates of brand sales (Bronnenberg and Sismeiro, 2002). We consider more complex

Type I models later in this article.

TypeIImodels,denotedbythenotation f (Z | X, y,?),reversethelogicofthemodeling

process. Instead of predicting outcome variables y, we predict the locations Z at which

certain outcomes occurred. Models in this form are not generally discussed in the spatial

statisticsliterature.However,TypeIImodelsarelikelytobecomemoresalienttoresearchers

inmarketinggivenongoingtechnologicaladvancesinradiofrequencyidentification(RFID)

technology.

For example, consider the Path Tracker system developed by Sorensen Associates for

category management applications (Murphy, 2004; Sorensen Associates, 2004). Using

an RFID system, the locations of a consumer’s grocery cart in the store are recorded

over time, Zt, providing information on the relationship between store layout and pur-

chasing activity. Applied to these data, a Type II model would predict the consumer’s

path through the store, given information on purchases y, consumer characteristics X1

(Larson et al., 2005), and possibly, most importantly, store layout X2. That is, as marketing

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managers experiment with store layouts, a more detailed analysis can be done to un-

derstand its impact not solely on end outcomes (y = sales), but on the traffic effect

of that design. In this way, manufacturers can begin to tease apart whether sales re-

sults may be due to poor traffic (awareness and consideration) or due to the product

itself.

TypeIImodelsarealsopotentiallyusefulinthepredictionofthesequenceinwhichinfor-

mation is used in consumer behavior experiments (Wedel and Pieters, 2000). For instance,

thesequenceinwhichrespondentssearchforinformationcouldbecriticaltounderstanding

the fundamental behavioral process. One additional area in which the outcome Z may be

of interest is in market basket analysis (Manchanda et al., 1999) when the order in which

customerspickupitemsmayprovideimportantinsightsforcross-selling,andasmentioned

before, store layout. In contrast to Type I models, Type II models do not generally make

use of the tools of spatial statistics. For this reason, we restrict attention to Type I models

in the remainder of this article.

Maps and Distance Metrics

Clearly, the most distinctive element of spatial models is the existence of a map. In regional

sciences and spatial econometrics, the map is typically geographical in nature, indicating

wheretheentitiesofinterest(firms,consumersetc.)arelocated.Theroleofthemapissimilar

to the role of time in time series models: spatial models typically assume that proximity

on the map implies high correlation in the response variables. However, in contrast to time

series models, the map is multidimensional—two or more dimensions—and can imply a

rich variety of spatial relationships. For example, in ecological studies, spatial correlations

are stronger in some directions (east-west) than others (north-south) due to prevailing wind

patterns (Cressie, 1993). Similarly, spatial models in marketing often seek to differentially

weightinformationinmodelingthecorrelationstructureofresponsevariables(Bronnenberg

and Mahajan, 2001; Yang and Allenby, 2003).

The selection of an appropriate map is of singular importance in spatial modeling. In

particular,itshouldbeemphasizedthatageographicalmapisnotnecessarilythebestchoice

for marketing applications. The space in a spatial model represents additional variables

(such as social networks, lifestyles or trading areas) that are not directly observed by the

researcher, but which are likely to determine the response variable. For example, Yang and

Allenby (2003) use both postal codes and demographics in defining the social network

of consumers. Moon and Russell (2004) base their analysis solely on a pick-any map of

consumer ideal points (a latent map), ignoring geographical location entirely. Clearly, the

selection of a map implies an assumption about the variables that determine the relative

similarities of individuals.

Once a map has been selected, a distance metric must be defined. Again, the researcher

facesanumberofchoices.Onecommonapproachistoassumethatthecorrelationstructure

isisotropic:itdependsonlyupondistancesonthemap,notuponthedirection.Forexample,

inthemodelofequation(1),itcouldbeassumedthatthecovariancebetweentwoconsumers

(aandb)isproportionaltoexp(−d(a,b)/θ)whered(a,b)istheEuclidiandistancebetween

the consumers on the map and θ > 0 is a parameter to be estimated. Cressie (1993) and

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Figure 1. Definition of Neighborhood.

Haining (1997) provide extensive discussions on the specification of covariance structures

using both Euclidian and non-Euclidian (spherical) geometry.

In many applications, however, a continuous measure of distance is not appropriate. This

typically occurs when the unit of analysis is a collection of many individuals such as postal

code area, county or state. For example, Bell and Song (2004) model the probability that at

least one individual in a postal code area has purchased groceries from netgrocer.com at a

giventimepoint(duringathree-yearperiodinwhichtheInternetfirmwasbeingestablished

inthemarketplace).InFigure1(adaptedfromBellandSong,2004),weillustratetheuseof

acontinguitymatrixC todetermineaneighborhood.Becausetheseauthorsareinterestedin

modelingtheimpactofsocialinteractionsintheadoptionprocess,theydefinetheneighbors

of a target postal code as all other postal codes that share boundaries with the target. For

example,postalcode1isaneighborofpostalcodes2and3only.ThecontiguitymatrixC is

row conditional: the pattern of ones and zeros identifies all postal codes that are neighbors

of the postal code in the given row. (For reasons of model identification, the main diagonal

of C is set to zero.) Prior to model specification, the contiguity matrix is usually converted

into a row standardized spatial lag matrix W by rescaling the rows of C to sum to unity.

There are two reasons to prefer a neighborhood structure over a continuous distance

metric, when scientifically appropriate. First, as in Bell and Song (2004), the logic of the

particular application may dictate the use of a contiguity matrix. For example, a neighbor-

hood structure is the appropriate choice to represent a social network. A hybrid approach

is provided by Anselin (2002). He suggests that distance between economic agents be de-

termined by counting the number of nodes separating the agents on a graph representing

the social network In this context, the elements of the contiguity matrix C are integers and

the W matrix is a function of the inverse of these integers. Yang (2004) argues for a gen-

eralized contiguity matrix C that permits the possibility that influence between individuals

is asymmetric (e.g., opinion leaders impact opinion followers more than followers impact

leaders).

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Second, in any spatial analysis, there will always be some individuals located near the

edges of the map. When a continuous distance metric is adopted, these individuals have

less surrounding points than those in the interior of the map. This can lead to biases in

model parameters and poor forecasts at the edges of the map. A compromise is to define a

neighborhoodasthe K nearestindividualsusingaEuclidiandistancemetric.Thisdefinition

also has the useful property that individuals located in a sparse region of the space will have

the same number of neighbors as those in a dense region of the space. (However, given the

larger distances, the impact of these individuals may be smaller in magnitude.) In effect,

the Euclidian measure of unit distance is allowed to be larger when in sparse regions of the

map. Haining (1997) provides an extensive discussion of the statistical problems of edge

effects and possible solutions.

Modeling Spatial Effects

Spatial models can represent three different types of spatial patterns. Two of these effects

have already been briefly discussed. First, using the W matrix noted above, spatial models

can capture spatial lags, the idea that the individuals are directly affected by the known

decisions of other individuals (Yang and Allenby, 2003; Bell and Song, 2004). Models of

this sort are of particular interest in applications, such as spatial econometrics, in which

economic agents are known to interact during the choice process. Second, as shown in

equation (1), spatial models can capture spatially correlated errors, the idea that important

latent variables that drive purchase behavior can be inferred from consumer proximity on

the map (Russell and Petersen, 2000; Bronnenberg and Sismeiro, 2002; Yang and Allenby,

2003). (Similar work by Chintagunta et al. (2004) shows how omitted variables induce

a form of time and brand spatial dependence.) Models of this sort can be regarded as a

statistical adjustment for missing variables that determine the response variable, but are not

available to the researcher.

Third, spatial models can capture spatial drift, the idea that model parameters are a

function of an individual’s location on the map (Brunsdon et al. 1998; Fotheringham et al.

2002). Models of this sort can be regarded as a representation of unobserved heterogeneity

in which the parameters (as opposed to the response variables per se) follow a spatial

process. The theoretical justification for this type of model strongly depends upon the

application. For example, Mittal et al. (2004) argue that geography dictates the parameters

ofasatisfactionratingregressionmodelduetodifferencesinlifestyleandclimate.Jankand

Kannan (2003) argue that spatial patterning of utility model parameters can be expected

because geographical location is a surrogate for the demographics that determine consumer

tastes.

The book by Fotheringham et al. (2002) provides an extensive discussion of a class of

spatial drift models known as Geographically Weighted Regression (GWR). A Bayesian

treatmentofGWRmodels,designedtoimprovethestatisticalpropertiesofGWRparameter

estimates, is discussed by LeSage (2003). By exploiting a relationship between GWR and

weighted maximum likelihood estimation, Duvvuri et al. (2004) develop a logit model with

spatial drift. Intuitively, a GWR estimator can be regarded as an Empirical Bayes estimator

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where the prior for a particular individual is based upon the neighborhood structure of the

contiguity matrix.

Formally, we can write a model specification which includes all three types of spatial

effects by generalizing equation (1) as

y = ρWy + X β[Z] + e,

where β[Z] is a continuous function of the map coordinates Z and ρ > 0 is a scalar

parameter. In this structure, ρWy represents spatial lag effects, ?(Z, θ) represents spatial

correlation effects, and β[Z] represents spatial drift effects. The response variable y in

equation (2) is typically assumed to be an observable outcome such as brand sales. Note

thattheoutcomevariablecanalsobeordinalinnature(suchaspreferencescores)orconstant

sumdatacollectedaspartofallocationtasks;seeMarshallandBradlow(2002)foraunified

computational approach to these types of models.

AsnotedbyAnselin(2002),equation(2)canbeadaptedforchoicemodelingbyreplacing

y with a continuous latent utility variable u

e ∼ N(0,?(Z,θ))(2)

u = ρWu + X β[Z] + e,

and by linking u to observed choice using a random utility theory argument. Although this

generalization is simple in principle, it may not be appropriate for all marketing science

applications.Forexample,thespatiallagtermρ Wu impliesthattheutilityofoneindividual

is influenced by the utilities of other consumers. This is clearly not the same as assuming

that a given consumer’s choice is influenced by the observed choices of other consumers

(Anselin, 2002). Although ρ Wu can be replaced by ρ Wy (thus, inducing a form of state-

space dependence), model calibration must be approached carefully because the u values

are correlated and u determines y.

Equation (2) can also be generalized to deal with cross-sectional time series data by

allowing time to impact the model components as

e ∼ N(0,?(Z,θ))(3)

y(t) = ρWy(t) + X β[Z,t] + e(t),

where the errors e(t) are correlated over time according to some stationary time series

process. These models, known as spatio-temporal models, have been extensively studied

in the biostatistics literature (see, e.g., Waller et al., 1997). Because spatio-temporal mod-

els provide considerable flexibility in capturing different types of dependence, they offer

researchers in marketing science a promising direction for new work.

e(t) ∼ N(0,?(Z,θ))(4)

Statistical Issues

Collectively,thethreetypesofspatialeffectsimplythattheresearchermusttakeintoaccount

interdependence in calibrating spatial models. Note that equation (2) can be rewritten as

y = [(I − ρW)−1X]β[Z] + v,v ∼ N(0,(I − ρW)−1?(Z,θ)(I − ρW?)−1)

(5)

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where v = ρWv + e can be interpreted as a spatially-lagged error structure. Even when

the original errors e are not spatially correlated (i.e., ?(Z, θ) is a diagonal matrix), all

outcome variables y will be correlated due to the spatially-lagged error v. An analogous

propertyholdsforchoicemodelsconstructedbyreplacing y bylatentutilitiesu.Inpractical

terms, interdependence means that calibration of a spatial model is considerably more

complex than calibration of a traditional marketing science model because the standard

assumptionsofstatisticalconditionalindependenceareinappropriate.Clearly,simultaneous

equationestimationstrategiesmustbeusedformodelcalibration.Inthiscontext,simulation

technologies such as simulated maximum likelihood and Markov Chain Monte Carlo are

apt to be attractive choices for the applied researcher (Tanner, 1996; Train, 2003).

In calibrating a spatial model, the researcher needs to be aware that the general model

in equation (2) cannot be aggregated analytically without changing the model structure.

Aggregation in this context refers to grouping of individuals (e.g., analyzing segments in-

stead of consumers) or aggregating geographical areas (e.g., analyzing counties instead

of postal codes). This general issue, known in the spatial statistics literature as ecolog-

ical fallacy or the modifiable areal unit problem, implies that spatial effects present at

one unit of analysis may not be observed at another unit of analysis (Anselein, 2001,

2002). Sismerio (2004) addresses this problem by developing a generalized spatial model

which simultaneously incorporates the different spatial effects for different levels of anal-

ysis. Using simulated data, she shows that both local and large-scale effects can be re-

covered if the model is appropriately specified. In general, the researcher should select

a scale for the model specification which coincides with the intended use of the model

(Anselin, 2002).

Research Opportunities

Spatial models are interesting new tools for analyzing interdependence in behavioral out-

comes.Here,webrieflydiscusstopicsthatareofparticularinteresttoresearchersinmarket-

ing science. Our intent is to highlight aspects of spatial modeling that present opportunities

for future research.

Dimensionality

Spatial data present a number of challenges for the researcher. The most obvious character-

istic of spatial data is the sheer amount of information that must be stored. Specialized data

storage formats, data retrieval tools and data presentation software now exist in the form of

geographical information systems (Rigaux et al., 2002). GIS software tools emphasize the

use of efficient strategies for representing spatial information. For example, consider again

thecontiguitymatrixC inFigure1.Althoughthistypicallyisalarge N by N matrix(where

N is the number of individuals in the analysis), the number of ones (denoting neighbors) in

C isoftenverysmallrelativetothenumberofzeroes.Byrecordingonlythelocationsofthe

ones in C, the amount of data storage required is greatly reduced. Statistical software (such

as the Matlab spatial statistics toolbox) which allows for sparse matrix representations can

significantly reduce the time needed to calibrate a spatial model (LeSage, 1999).

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Several approaches exist that address the dimensionality problem of spatial models. An

alternativetoexpressingthespatialdependencethroughjointdistributions(presentedearlier

inequation(2))istoassumeconditionalindependenceofneighboringlocationsandtocreate

a model based upon a Markov Random Field (Besag, 1974, 1975). The main difficulty of

this approach is the need for “severe restrictions on the available functional forms of the

conditional probability distributions in order to achieve a mathematically consistent joint

probabilitystructure”(Besag,1974,p.196).Forexample,MoonandRussell(2004)estimate

a conditional autologistic model and constrain the pairwise relationship between locations

to be symmetric. In order to avoid the computation of a mathematically intractable joint

likelihood function, parameters are estimated using a pseudo-likelihood algorithm based

upon the conditional probability distributions.

Recent developments in the estimation and inference of joint autoregressive models with

the large data sets also limit the number of direct relationships among locations. The goal is

tosimplifycomputationsandreducememoryusageoflikelihood-basedapproaches(includ-

ing maximum-likelihood and Bayesian estimation). For example, Pace and Barry (1997,

1999) provide algorithms to quickly compute maximum-likelihood estimates when the de-

pendent variables (or its errors) follow a general spatial autoregressive process with few

direct relationships. LeSage and Pace (2000) introduce the matrix exponential spatial spec-

ification (MESS) that relies on a specific spatial transformation of the dependent variable.

Pace and Zou (2000) provide closed-form maximum-likelihood estimates for the particular

case of nearest-neighbor spatial dependence (allowing only one other location, the nearest-

neighbor,todirectlyaffecteachsampledlocation).Similarly,LeSage(2000)derivesexpres-

sions for the conditional distributions central in Bayesian estimation of nearest-neighbor

models that avoid complex matrix computations.

Analysis of Marketing Policies

Spatial data can be used to understand the geographical patterns of marketing variables.

In this context, the X variables may be endogenous to the model. For example, Anderson

and de Palma (1988) study regional patterns of price discrimination, taking into account

delivery costs and manufacturer locations. Using information on the location of consumer

residences, Thomadsen (2004) calibrates a choice model for banking services and develops

recommendations for optimal placement of automatic teller machines. Bronnenberg et al.

(2005)showthatcurrentbrandsharesforaconsumerpackagedgoodsproducthaveaspatial

distribution dependent on order of entry into the region and the (endogenously determined)

regional levels of advertising expenditure.

A promising use of spatial models in marketing science is the correction of endogeneity

in marketing mix response models (Bronnenberg and Mahajan, 2001; Bronnenberg, 2004).

They propose to jointly model the marketing mix and the response variables by modifying

the spatial regression model of equation (1) to models of the type

y = τ + X(τ)β + e1,

X(τ) = μ + λτ + e2,

e1∼ N(0,σ2

e2∼ N(0,σ2

1I)

(6)

2I)

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where τ are error components that follows a spatial lag pattern and X(τ) is (partially)

dependent on τ. In words, this model asserts that the base level of the response y exhibits a

spatial pattern. Moreover, this base level is used by marketing managers to set the observed

marketing mix expenditures (such as advertising budgets) found among the X(τ) variables.

A representation of this sort both corrects biases in the estimation of β and provides a

structural view of managerial decision rules with respect to marketing mix variables. An

analysis based upon equation (6) is particularly important if the goal of the research is to

develop optimal policy recommendations for marketing managers.

Interpretation of Spatial Effects

Most researchers in marketing are interested in spatial models primarily as a means of

understanding choice behavior. Yang and Allenby (2003) and Bell and Song (2004) use

spatial models to measure the impact of social influence on choice behavior. Ter Hofstede

et al. (2002) use spatial priors (in a hierarchical Bayes analysis) to understand geographical

dispersion of preference segments in Europe. Ter Hofstede (2004) extends this work by

linking spatial segmentation to the means-end chain framework (in which abstract values

lead to desired benefits which lead, in turn, to desired product features). For each of these

examples, the structure of the spatial model suggests that specific behavioral mechanisms

determine choice outcomes.

In this context, it is important to understand the spatial models—like most statistical

models—may not be informative about the constructs underlying behavior. However, a

researcher can persuasively argue for one type of spatial mechanism versus another by

collectingadditionaldatathatstronglyfavorsonetypeofmodelspecification.Forexample,

Arora (2004) argues that spatial models could be an effective tools in studies of group

decision-making (see, e.g., Arora and Allenby, 1999; Aribarg et al., 2002). Because the

data collection process would include direct observation of dyadic interactions (such as

discussions between parent and child), the researcher would be justified in using a spatial

lagmodeltomodeljointdecisionbehavior.Again,thestrongimplicationisthatsubstantive

knowledge of the application area must guide the use of spatial models. For this reason,

joint work between behavioral and quantitative researchers should be encouraged.

Conclusion

Spatial models are a new research area in marketing science. Because spatial models allow

for correlations in response variables across individuals, they represent an entirely new way

of understanding decision processes. Major opportunities exist for researchers who under-

stand the methods of spatial statistics and can craft specialized models to study substantive

marketing issues.

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