ASIMANSARI, CARL F.MELA, and SCOTT A.NESLIN*The authors develop a mode - PDF document

1 ASIMANSARI, CARL F.MELA, and SCOTT A.NES
ASIMANSARI, CARL F.MELA, and SCOTT A.NESLIN*The authors develop a model of customer channel migration and applyit to a retailer that markets over the Web and through catalogs.Themodel identifies the key phenomena required to analyze customermigration, shows how these phenomena can be modeled, and developsan approach for estimating the model.The methodology is unique in its Asmultichannel distribution becomes increasinglyprevalent, customers face an expanding array of purchaseand communication options. For example, online sales areexpected to increase 20% in 2006 to $211.4 billion, dou-bling the total revenue in 2003 (TheWall Street Journal2006). As such, multichannel customer management isbecoming a pivotal component in firmsÕmarketing strategy. Customer Channel Migration CONCEPTUAL FRAMEWORK Channel migration can also affect revenues when prices differ acrossIn Figure 1, we consider two forms of communicationÑcatalogs ande-mails (because these are the instruments used by the firm in our data).develop Òchannel loyaltyÓ according to their channel usageexperience?¥What is the role of marketing communications in channelmigration? Does marketing affect channel selection, demand,¥Do customer differences affect the channel migration process,and if so, how?We develop and estimate a model of customer channelmigration to investigate these substantive questions. Ourcontribution is twofold: First, we (1) propose a set of keyphenomena that are related to channel migration behavior,(2) show how these phenomena can be modeled, and (3)develop an estimation approach for such a model. Themigration model captures the effects of large numbers ofmarketing communications in the face of dynamics andcustomer heterogeneity. Second, we contribute to the sub-stantive knowledge base on c

2 ustomer channel migration.One key findin
ustomer channel migration.One key finding is that Web use, when we control for mar-keting and other factors, is associated with a permanentdecrease in the likelihood of buying from a firm, perhapsbecause the Internet can expand consideration sets andlower customer service levels.We proceed as follows: We first describe the modelingframework and use it to identify key phenomena to beincorporated into the model. Next, we describe our model.Subsequently, we describe our data and report the results.Finally, we summarize key findings and conclude by offer-CHANNEL MIGRATION FRAMEWORKChannel migration affects firm profit through its influ-ence on cost and revenue. For example, it has been claimedthat the Internet is more cost efficient than traditional chan-nels. Although this might suggest that companies shouldmigrate customers to the Web, the efficacy of this strategydepends on how migration affects overall demand. Thus,understanding how marketing actions are associated withdemand is crucial in grasping how customer channel migra-tion affects firm profitability.In Figure 1, we provide anoverview of the demand-side characteristics of channelWe assume that the customer jointly decides how muchto purchase from the firm and what channel to use. Bothbehaviors entail experience or learning effects, wherebyprevious purchases and channel selections can affect subse-quent behavior. In addition, purchase volume and channelselection may be linked contemporaneously. For example,heavy purchasers may prefer certain channels. Finally, mar-keting communications can affect purchase volumes andTo illustrate the ramifications of this framework, we con-sider its implications for research findings reinforced in theChain Store AgeInter@ctive Week2000; Kumar andVenkatesan 2005; Kuswaha an

3 d Shankar 2005; Thomas andSullivan 2005;
d Shankar 2005; Thomas andSullivan 2005; TheWall Street Journal2004; Yulinsky2000). As a result, there is speculation that Òmultichannelcustomers are the best customers for a retailer, because theybuy more and provide retailers with incremental gains overInter@active Week2000, p. 50). It might beconjectured that firms should cultivate multichannel buy-ing. However, the framework suggests several other possi-bilities regarding why the multichannel and Internet-loyalcustomers have high sales levels:¥Heavy users naturally migrate to the Internet (purchase vol-channel selection). Heavy usage might be correlatedwith various demographic factors.¥The Internet cultivates heavy buying; that is, customers buymore from a firm in the long run when they buy on the Internetpurchase volume).¥Customers respond differently to marketing (communica-purchase volume and channel share). For example,marketing.Each explanation has a different implication for the prof-gle them. For example, Internet buyers might not be proneto buy more; rather, they might receive more marketing,leading to the appearance of a greater proclivity to buy.Should a switch to the Internet be countervailed by lesserWeb sites or by lowering service levels), the prescription tomigrate people to the Web could be counterproductive.Thus, it is desirable to take a more systematic view of chan-nel migration. In the next section, we formalize the modelWe model purchase volume and channel selection as sug-gested by our framework. We model the purchase incidenceand order-size components of purchase volume using aType II Tobit specification and channel selection using a Customer Channel Migration We thank an anonymous reviewer for this suggestion.We derive this variable by differencing the latent utilities for Internetch

4 oice and catalog choice; therefore, it h
oice and catalog choice; therefore, it has a value of ±1. We add It is impossible to include a corresponding variable for catalog pur-purchases before the data. In contrast, the Web channel is new, so theamount of purchasing before our data is negligible for all households.Although we suppress the subscripts for the equation (b, q, or w) tosimplify the presentation, all parameters are equation specific.ceding month through either the catalog or the Web. In theorder-size equation (Experience), we define Lcat andLweb as the order size of the previous purchase on the cata-log or on the Web. The use of lagged incidence in the inci-dence equation and lagged order size in the order-size equa-tion mimics the definitions of the dependent variables. Thelagged volume and lagged channel selection effects.equation, we define Lcat and Lweb as the previous monthÕspurchase volume from the catalog or the Web, respectively.We also include Diff, which represents state dependence inchannel selection. We set Diff to 1 if the previous purchaseWe define Since as the number ofmonths elapsed since the previous purchase. We includethis recency measure in all three experience equations.The function Wuse, defined as Log(1+ Web purchasesto date), captures the permanent effect due to Web usage.We use Wusein all three equations. By definition, thisvariable is independent of the duration between Web pur-chases because we attempt to capture forgetting and othertransient effects due to previous channel usage through theLweb variables. We specify this to have diminishing mar-(Roberts and Urban 1988), we expect the first usage to havea greater effect on behavior than the last usage. Finally, weallow for individual-specific slopes for all the experienceeffects.Communications EffectsWe define comm

5 unication c as a particular communica-ti
unication c as a particular communica-tion sent by the firm at a particular time (these can bee-mails or catalogs). Therefore, two different catalogsmailed at the same time are considered different communi-cations, and the same catalog sent at two different times isconsidered two different communications. For this reason,totaling C= 723. Although each customer could havereceived 723 communications, in practice, no customerreceived this many, and the number received varies acrosscustomers. Rather than model the effect of each communi-cation separately, we decompose these effects (Campbell et(e.g., communications of like kind have the same effect)and (2) the time since the communication was sent (i.e., theeffect of the communication decays over time). In addition,we allow for the direct effect of a communication and itslikely to be decreasing marginal returns to these communi-cations, reflected in a negative interaction.Direct communication effects. We define the direct effect We use this functional form to ensure 0 It is possible to consider additional attributes (e.g., menÕs catalogs ver-sus womenÕs), but the catalog versus e-mail distinction is fundamental andenables us to investigate the propositions we stated previously (i.e., ourspecification is theoretically driven).The variable dindicates whether customer i has receivedcustomer receives communication c, and it equals 1 eachperiod thereafter. This ensures that the communication doesnot begin to have an impact until the customer receives it.The variable rtomer i received communication c. The parameter and reflects dynamics. We expect that between 0 and 1; a large exerts an impact well into the future. The parameter the magnitude of the direct effect of communication c andis household specific. Communication

6 s that are more effec-tive have higher v
s that are more effec-tive have higher values of We can sum Equation 7 across all communications tocompute the total direct effect of communications receivedcommunication effect parameters, 2169 parameters for each customer. To model these effectsparsimoniously, we describe each communication by a setof M attributes. We define acommunication c has attribute m and 0 when otherwise(m= 1, ƒ, M). Then, we can express the communicationeffect for communication c asWe can write the decay parameter asBecause M is small compared with C, we achieve greatparsimony while allowing different communication types tohave different effects. In our application, we use M= 2 andAccordingly,we have the following:We use the label Catalogs(the set of all communicationsthat are catalogs), and this equals exp()/[1+ exp(all catalogs. Similarly, we use the label expexp a ().icimcm (8)Total_Direct_Effect=icc (7)Direct_Effect= icc Customer Channel Migration With monthly aggregation, multiple purchases are negligible, amount-ing to .29% of total observations and 1.61% of choice occasions. Whenlation distribution for modeling unobserved heterogeneity.The t is a robust alternative to the normal because it has fat-ter tails. We use Bayesian methods for inference regardingthe parameters. Because the posterior distribution is notcompletely known, we use Markov chain Monte Carlo(MCMC) techniques to obtain draws from the posterior dis-tribution of the unknowns. We describe the priors and thefull conditional distributions for the unknown parameters inDATAData were provided by a retailer that sells consumerdurable and apparel products in mature categories over theInternet and through a catalog. The data span four years,from February 1998 to February 2002. We restrict attentionto active cu

7 stomers who bought at least three times
stomers who bought at least three times in atleast one of the years during this period. This restrictionallows for changes in behavior over time. In our data, 37%chases, 1% exclusively used the Internet, and 62% usedonly catalogs. The entire data set consists of 40,000 cus-48 months= 24,000observations are available for estimation; however, the ini-tialization period necessary to create lagged variablesreduces our estimation sample to 19,064 observations.The data consist of several files. A catalog purchase fileincludes information on how much was spent by whom andwhen, and an Internet purchase file provides the same infor-mation for Internet purchases. Catalog and e-mail data filesindicate who received which communications when. Inaddition, a demographics file includes the age, income, andwere purchased by the firm from companies that use eitherpublicly available data sources or surveys. We aggregatedata to the monthly level because the median purchase fre-quency is approximately 1.7 purchases per year. Finer gra-dations yield an excess of observations with zero sales, andsingle interval.sponds largely to the decision processes we model.Table 1presents the means of some of the key variablesin the raw data. Collinearity in the data is modest becausethe condition indexes for the regressor sets in each of ourthree equations are all below 30 (Belsley, Kuh, and Welsch communications variables and their interactions; correlations betweenthese variables range up to .94. Because collinearity increases standarderrors, our data afford a conservative test of our hypotheses.An item of note in Table 1 is the high levelSubstantial channel migration is evident in our data, andmultichannel and Internet buyers purchase more. In 1998, alarge proportion (96% of customers) made m

8 ore than 95%had fallen to 77%. Moreover,
ore than 95%had fallen to 77%. Moreover, people who purchasedthrough the Internet and the catalog tended to buy more; themean purchase level of those who made more than 95% oftheir purchases on the catalog was $267, in contrast to $444RESULTSWe estimated four models. The first, M1, is the fullmodel specified in Equation 4; it incorporates heterogeneityusing a multivariate t population distribution. The secondmodel, M2, is also the full model, but it assumes that therandom effects are distributed multivariate normal. Thismodel is useful in assessing the relative merit of using thet-distribution for the random effects. The third model, M3, assumes no marketing effects. The fourth, M4, doesthe communications have only an immediate impact and donot have a delayed effect). Both M3 and M4 uset-heterogeneity. M3 enables us to ascertain whether market-ing contributes to model fit, whereas M4 enables us to testthe role of marketing dynamics. Table 2displays the log-marginal likelihoods based on the MCMC draws.well as marketing and experience dynamics. The superior-ity of M1 over M2 suggests that the t-distribution, owing toits fatter tails, is better able to capture heterogeneity thanthe normal distribution. The superiority of M1 over M3Table 1 DESCRIPTIVE STATISTICS VariableMMdnSDMinimumMaximumAverage purchase ($)1311117526564Purchases (per year)2.21.71.308.3Income ($)84,48781,50040,7651,000150,000Age (years)5048122497.040.0503.4Catalogs (per month)3.43.21.6.37.9Catalog share (of orders)89%100%.2222%100% Table 2 MODEL COMPARISON ModelDescriptionLog-Marginal LikelihoodM1Full modelÐ16,896.6M2Normal heterogeneityÐ17,095.5M3No marketingÐ18,965.3M4No communications dynamicsÐ17,077.4 Customer Channel Migration PREDICTION RESULTS A: Purchase Volume Notes: In-sample from Perio

9 ds 2Ð45, and out-of-sample from Periods
ds 2Ð45, and out-of-sample from Periods 46Ð48. The solid dark line is actual data. The gray line is for the full model (M1),the dashed line is for the no-marketing model (M3), and the dotted line is for the no-dynamics model (M4). B: Channel Share The total effect is calculated as 1+ .125+ .125+ ...= 1/(1Ð.125)= 8/7.Finally, the decay parameter estimates are small butnonzero. The average (.14+ .11)/2= .125 (or 1/8), whichimplies an infinite horizon effect for the communicationson choice of 1/(1Ð 1/8), or 8/7.nications effect occurs in periods after the communicationis received. Although this appears small, this firm sends 45catalogs a year. As such, the communication decay is tanta-mount to increasing the effect of these catalogs by 45/7=across people, the revenue effects of these lagged factorsare considerable. The leads to the question whether thesedecay effects would be slower if fewer communications Customer Channel MigrationTable 5 PARAMETERS:YEAR 1 VERSUS 4 No-Migration GroupMigration Group MSEMSEChange in Marketing Catalogs/month1.02.091.54.16E-mails/month.95.081.76.21Response ParameterCatalog.10.01Ð.16.03E-mailÐ.43.01Ð.52.05 These two groups do not sum to 500 households because (1) theychanges from Year 1 to Year 4) and (2) they do not include those that didnot have sufficient data in Year 1 for initialization of experience variables.Internet MigrationTo ascertain more precisely why customers migrated tothe Internet, we calculate changes in the experience effects,marketing effects, and time effects in the channel selectionand 2001. The strategy is to observe how these factors2001 (n= 69) versus those who did not (n= 312) and totion can be written as follows:Because only one survey record exists for each customer,between 1998 and 2001. Thus, these factor

10 s cannot explainan increase in channel l
s cannot explainan increase in channel latent utility over time. However,experience, communications (marketing), and time contri-butions to latent utility do change. Thus, we calculate theaverages of these utilities for both years, difference them,and then compare these differences between customers whomigrated from the catalog to the Internet between Year 1(1998) and Year 4 (2001) and those who did not. Table 4Negative signs in Table 4 suggest that the correspondingfactor facilitates a migration to the Internet. First, experi-ence effects are equal for both groups. Therefore, thechange in experience utility is the same for both groups.Ergo, experience effects were not associated with migrationbehavior for the two groups. Second, the time factor isnegative for both segments, capturing a trend toward theInternet, and it is greater for customers who migrated. It istempting to argue that this larger effect arises from our defi-in later periods. However, note that we measured the trendeffect while controlling for experience and marketingeffects, and it is possible that these effects alone would havepredicted the migration toward the Internet. So, the findingthat those who migrated had a stronger trend toward theInternet is not an artifact of the model. Third, the change in EwCustomercharacteristicsExperiencitwiCommunicationsTimeeffectswitwitwit Again, a negative sign means that channel selection utility decreases,which, given our coding of w*, means that the factor is associated withcustomers using the Web.Note that additional e-mail by itself is not sufficient to migrate cus-tomers to the Web. They must also respond to that e-mail by moving to theWeb. Of the 36.2% of nonmigrating customers who received e-mail, onaverage, they received 2.6 additional e-mails per mo

11 nth, and their channelselection response
nth, and their channelselection response parameter averaged Ð.30. Of the 76.7% of migratingcustomers who received e-mail, they received on average 2.3 additionale-mails, and their average channel selection response parameter was Ð.54.marketing utility is positive for the no-migration group butnegative for the migration group. This suggests that market-ing both enhanced the likelihood that some customerswould migrate and inhibited the likelihood that other cus-tomers would migrate.Further inspection reveals that these differences in mar-keting utility arise from both changes in the levels of mar-keting and differences in marketing response across groups.In Table 5, we consider these factors. Customers whomigrated were exposed to more marketing and switchedBecause the migration groupÕsresponse parameter for e-mail is more negative than that forcatalogs and because the absolute level of change is higher,Changes in Purchase VolumesPreviously, we offered three explanations regarding why1. Migrating households were heavier users to begin with,2. A positive experience on the Web encouraged higher pur-chase volumes, and3. Migrating households simply reacted to marketing.To disentangle these explanations, we decomposed the inci-dence model latent utility (b*) into experience, marketing,customer characteristics, and time factors, similar to theprevious analysis. (We focus on incidence, rather than ordersize conditioned on incidence, because most of the variancein purchase volume is explained by the incidence model.)We report changes in these factors in Table 6. A positiveentry means that changes in this factor contributed to highersales in Year 4 than in Year 1. We also show the averagecustomer characteristic utility (which does not change overtime) and the average sales

12 response to catalog and e-mail.Table 6
response to catalog and e-mail.Table 6 eliminates the Òheavier-usersÓ explanation forincreased use because the average customer characteristicTable 4CHANGES IN CONTRIBUTION TO LATENT UTILITY:YEAR 1 VERSUS 4 No-Migration GroupMigration Group Mean ChangeSEMean ChangeSEExperienceÐ.15.01Ð.12.09Marketing.51.06Ð.55.17TimeÐ.56.04Ð2.05.12 Customer Channel Migrationor in the store. The attendant reduction in personal servicecould lead to lower loyalty (Ariely, Lynch, and Moon 2002;Kacen, Hess, and Chiang 2003). Thus, the notion thatmigration is unqualifiedly positive because it lowers coststion that it can be negatively associated with long-term pur-Another novel result is our finding of decreasing returnsfor communications in the purchase volume model.to think that the optimal e-mail strategy is to e-mail cus-tomers daily. However, decreasing returns imply that apulsing strategy might be more effective. That is, totaland diminishing their effectiveness (see Blattberg, Kim, andNeslin 2008). Note also that customers react differently tothe same marketing stimuli. For example, firms need tolearn which types of customers will be unreceptive to theInternet and to the company if they are channeled to theAlthough the firm we analyze can be characterized as aÒtypicalÓ retailer, channel migration can be affected byindustry, product line, marketing policy, customer base, andtime. For example, Zhang and Wedel (2004) find high Inter-net loyalty for grocery goods, suggesting positive statedependence in that industry. This can be explained by thelist feature offered by online grocers, in which consumersinvest considerable effort setting up a shopping list to facili-from 1998 to 2001. It is possible that firms have sinceaddressed the lack of a human interface and improvedtrolled exp

13 eriment. Accordingly, it is not possible
eriment. Accordingly, it is not possible to makestrong causal claims or rule out all alternative explanations.An alternative explanation is selection bias. For example,selectivity could become a problem in our channel choicemodel if unobserved variables governing the receipt ofe-mails are correlated with unobserved variables determin-2004). For example, an unobserved factor could be cus-tomer propensity to favor the Internet. This would makecustomers more likely to reveal to the company their e-mailaddresses and more likely to use the Internet to make pur-chases. This would induce a spurious correlation betweenreceipt of e-mails and channel selection. Although we can-not definitively rule out this possibility, we believe thatselectivity is not a severe problem in our case for threeFirst, we modeled selectivity with observed variables byincorporating surrogates for Internet propensity explicitlyin the choice model. Foremost among these is anindividual-specific trend term in Equation 16). Thisterm controls for changes in customersÕInternet propensityover time. We also include demographics, such as age andincome, that could be correlated with Internet propensity.Indeed, we find that younger people are more likely tochoose the Internet. Although the company in this case did ( not explicitly target e-mails on the basis of previous behav-ior, previous RFM factors, including lagged volume andchase, might be related to Internet propensity. For example,people who recently bought on the Internet may evidence aheterogeneous trend, demographics, and prior behaviors,we still find a significant e-mail coefficient with regard tochannel choice (Table 3).Second, several additional factors mitigate the potentialcorrelation between unobserved factors affecting channelchoice and

14 the receipt of e-mails. For example, man
the receipt of e-mails. For example, many cus-tomers who received e-mails did not migrate to the Internet(66%), and many customers who migrated to the Internetdid not receive e-mails (23%). If there were a strong corre-lation between unobserved factors, these patterns would bedifficult to obtain. In addition, there is considerable varia-tion in the number of e-mails received from week to week(some weeks the customer receives no e-mails, some weeksthe customer receives one, some weeks the customerreceives two, and so on). This pattern of variation differsfrom what might be expected in terms of changes in Inter-net propensity over time. This also suggests a weak linkbetween unobserved factors governing e-mail receipt andInternet choice. Furthermore, e-mails were not sent to any-one until well after some people commenced their Web pur-lation between unobserved factors governing receipt andchoice is likely not high. Finally, the firm indicated that itdid not target its e-mails. Rather, e-mail addresses are col-lected from prior purchases (both Web and catalog), andTogether, these factors suggest a low correlation betweenunobserved factors affecting e-mail receipt and InternetThird, to the extent these arguments indicate that we haveincluded effective surrogates for Internet propensity in themodel and that the factors we do not observe that influencereceipt of e-mails and channel selection behaviors are nothighly correlated, selectivity effects will be minimal. Per-haps for similar reasons, Gönül and Shi (1998) find selec-tivity (albeit in a catalog context) to be insignificant.We also note that our finding that e-mails could influencechoice of the Internet has both face and convergent validity.In terms of face validity, e-mails typically contain links tothe company

15 Web site, so it is easy for the custome
Web site, so it is easy for the customer totransfer to the Web site and make a purchase there. In termsof convergent validity, other researchers have found asso-It is theoretically possible to control for selectivity biasby appending two selection equations (one for e-mails andselection, purchase volume, and incidence. However, thismodel is already heavily parameterized. In light of theseconsiderations and the discussion in the foregoing para-graphs, the cost of selectivity controls (poor reliability andconvergence) exceeds the value of additional insights thatmight accrue. However, although our previous discussionwould suggest that selectivity is not problematic in our con- Customer Channel MigrationThis corresponds to Equations 13 and 14 in the text. Thefirst term in Equation B4 is the catalog effect (Cat_CatAPPENDIX C: DERIVATION OF STOCK VARIABLESThe computation of the direct effects in Equations 11aand 11b and the interaction effects in Equations 14aÐ14c isdifficult because of the large number of catalogs ande-mails in our data. The computational complexity arisesbecause we need to compute aggregates involving a largenumber of communication dummies for each observation inthe data and for each sampled value of the discount termswithin the MCMC iterations. However, considering that allcommunications of a given type (i.e., catalogs or e-mails) aperson receives within a month are exchangeable (i.e., theyhave the same effect on all subsequent observations), wecan use a much simpler representation that involves a recur-sive definition of the direct and interaction effects.For the recursive definitions, we do not need the commu-nications dummies. Because of the exchangeability of com-munications received in the same period, we need to knowonly two variables, Cca

16 talogs and the number of e-mails, respec
talogs and the number of e-mails, respectively, receivedby customer i in month t. We now show how these variablescan be used to compute recursively the direct and inter-actions effects.Direct EffectsAccording to Equation 11a, the direct effect of the cata-mies as follows:is a coefficient that is common to all termswithin the summation, we can ignore this coefficient anddefine RCatto be the raw catalog effect:Because of the exchangeability of catalogs received in agiven month, all catalogs in a month will have the same ().logCRCatditcatcCatas ().CCatditicatCatalogs ccc4Total_Interaction_Effecticcttictictictccboictictictictcatalogscbothe-mailse-mailicticictictcisacatalogisane-maile-mailictictict variable. Thus, the direct effect of catalogs in Month 1 canalternatively be written asThis is the number of catalogs received in Month 1 by cus-= 0 for all catalogs in Period 1. Thus, wehave the following:The direct effect of the catalogs in Month 2 is given byThus, we can compute the direct effect recursively by dis-counting the previous periodÕs direct effect and by addingthe direct effect due to all catalogs received in the currentmonth. Generically, the recursive scheme results in the fol-lowing representation:The computational complexity is reduced considerablybecause on any given observation, only a single term isadded to an already computed value obtained from the pre-The direct effect for e-mails can analogously be writtenrefers to the raw e-mail direct effect. Wecan compute the total e-mail direct effect, E-mailwith the coefficient Interaction EffectsWe begin by focusing on the Cat_Catinteraction effect.coefficient, we defineGiven the exchangeability of catalogs received within thesame period, considerable simplifications result. For exam-ple, for the first per

17 iod, we can writeto represent the intera
iod, we can writeto represent the interaction effects between all catalog pairsreceived in Month 1. Similarly, for Month 2, we can writethe interaction effect as (C11)RCatCaticati 222Š ()_CRCatCat ()_CRCatCatdditcatictictictictcCatalogs ()_CCatCatdditcatictictictict cCatalogs (),ititlitRE-mailRE-maile-mai ().CRCatCRCatititcatit ().CRCatCCCRCatiicatiicati22121=+=+ ().CRCatC ().CRCatdiicCatalogs Customer Channel Migrationtion is VFourth, we use the Metropolis method to make independ-ent draws for the elements in #, the vector of the trans-formed decay parameters. The likelihood isGiven the normal prior for p(likelihood, we use a random-walk Metropolis algorithm todraw each element independently. For generating candidatedraws, we use a normal proposal distribution centered onthe previous draw and with a variance of .02.dard deviation, . Given the likelihood in the previous step, we use a random-walk Metropo-lis step with the proposal distribution centered on the previ-ous draw and with a proposal variance of .1 that is tuned tobecause of positive definiteness requirements. The full con-ditional for any correlation is not completely knownbecause of the positive definiteness constraint. Therefore,we use the guided-walk Metropolis algorithm (Gustafson1998) to generate each correlation separately. In generatingthe candidate correlation from a normal proposal distribu-tion, we ensured that each correlation was obtained from aninterval that kept the correlation matrix positive definiteSeventh, the full conditional for is Wishart and isgiven by is the prior degrees ofAbraham, Magid M. and Leonard M. Lodish (1993), ÒAn Imple-mented System for Improving Promotion Productivity UsingStore Scanner Data,Ó Marketing ScienceAriely, Dan, John Lynch, and Youngme Moon (2002), ÒTa

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