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JOURNAL OF APPLIED ECONOMETRICS J. Appl. Econ. 17: 49–59 (2002) DOI: 10.1002/jae.653 A SEGMENT-LEVEL HAZARD APPROACH TO STUDYING HOUSEHOLD PURCHASE TIMING DECISIONS DEMETRIOS VAKRATSAS 1 * AND FRANK M. BASS 2 1 McGill University, Montreal, Canada 2 University of Texas at Dallas, Dallas, Texas, USA SUMMARY The increasing availability of customer-level data and the willingness of marketers to customize the timing of their offers to consumers makes the accurate segment-level description of household purchase timing decisions a compelling issue. In this paper we employ a finite mixture accelerated failure time model to identify and characterize household segments in terms of their purchasing rates and their propensity to accelerate purchases due to marketing mix activities. Such an approach also promises to alleviate possible aggregation problems arising from the use of a common hazard rate for all households. An application to household panel data suggested that infrequent buyers show higher propensity to accelerate than frequent buyers do, and that positive duration effects are underestimated when not accounting for segment-specific hazard rates. Copyright 2002 John Wiley & Sons, Ltd. 1. INTRODUCTION The importance of purchase timing decisions as an element of the consumer buying process has been well documented in the marketing literature (Bell, Chiang and Padmannabhan, 1999; Chintagunta and Haldar, 1998; Gupta, 1988; Jain and Vilcassim, 1991; Jeuland, Bass and Wright, 1980; Neslin, Henderson and Quelch, 1985). Retailers, facilitated by the growth of scanner data, the use of loyalty cards and most recently the advent of ‘e-tailing’ can now routinely collect purchase timing data at the customer level (The Economist, 1998, 1999). Such customer-level data enable them to customize the timing and nature of their offers based on the purchase history of their customers (Rossi, McCulloch and Allenby, 1996), especially since price reductions and promotions have been frequently shown to induce consumers to accelerate their purchases (Blattberg and Neslin, 1990). These advances in database marketing have made the accurate description of purchase timing patterns at the customer or segment level a compelling issue. The need for a segment-level approach to the study of household purchase timing decisions can be further illustrated by the following example. A direct marketing company, based on an analysis of its database, classifies its customers into two segments, ‘A’ and ‘B’. Forty per cent of the company’s customers belong to segment ‘A’ and 60 per cent belong to segment ‘B’. Segment A customers buy the product on average once every 20 days (i.e. they are ‘heavy’ buyers) while segment B customers buy the product on average once every 60 days (i.e. they are ‘light’ buyers). The hazard rate describing the purchase timing pattern for each segment is assumed to be strictly increasing and following the Weibull specification. Figure 1 describes the aggregate hazard rate as well as the hazard rates for each segment. The figure suggests that aggregating the hazard Ł Correspondence to: D. Vakratsas, Faculty of Management, McGill University, 1001 Sherbrooke St. West, Montreal, Quebec, Canada H3A 1G5. E-mail: [email protected] Copyright 2002 John Wiley & Sons, Ltd. Received 20 February 1998 Accepted 26 February 2001

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JOURNAL OF APPLIED ECONOMETRICSJ. Appl. Econ. 17: 49–59 (2002)DOI: 10.1002/jae.653

A SEGMENT-LEVEL HAZARD APPROACH TO STUDYINGHOUSEHOLD PURCHASE TIMING DECISIONS

DEMETRIOS VAKRATSAS1* AND FRANK M. BASS2

1 McGill University, Montreal, Canada2 University of Texas at Dallas, Dallas, Texas, USA

SUMMARYThe increasing availability of customer-level data and the willingness of marketers to customize the timingof their offers to consumers makes the accurate segment-level description of household purchase timingdecisions a compelling issue. In this paper we employ a finite mixture accelerated failure time model toidentify and characterize household segments in terms of their purchasing rates and their propensity toaccelerate purchases due to marketing mix activities. Such an approach also promises to alleviate possibleaggregation problems arising from the use of a common hazard rate for all households. An application tohousehold panel data suggested that infrequent buyers show higher propensity to accelerate than frequentbuyers do, and that positive duration effects are underestimated when not accounting for segment-specifichazard rates. Copyright 2002 John Wiley & Sons, Ltd.

1. INTRODUCTION

The importance of purchase timing decisions as an element of the consumer buying processhas been well documented in the marketing literature (Bell, Chiang and Padmannabhan, 1999;Chintagunta and Haldar, 1998; Gupta, 1988; Jain and Vilcassim, 1991; Jeuland, Bass and Wright,1980; Neslin, Henderson and Quelch, 1985). Retailers, facilitated by the growth of scanner data,the use of loyalty cards and most recently the advent of ‘e-tailing’ can now routinely collectpurchase timing data at the customer level (The Economist, 1998, 1999). Such customer-level dataenable them to customize the timing and nature of their offers based on the purchase history of theircustomers (Rossi, McCulloch and Allenby, 1996), especially since price reductions and promotionshave been frequently shown to induce consumers to accelerate their purchases (Blattberg andNeslin, 1990). These advances in database marketing have made the accurate description ofpurchase timing patterns at the customer or segment level a compelling issue.

The need for a segment-level approach to the study of household purchase timing decisionscan be further illustrated by the following example. A direct marketing company, based on ananalysis of its database, classifies its customers into two segments, ‘A’ and ‘B’. Forty per cent ofthe company’s customers belong to segment ‘A’ and 60 per cent belong to segment ‘B’. SegmentA customers buy the product on average once every 20 days (i.e. they are ‘heavy’ buyers) whilesegment B customers buy the product on average once every 60 days (i.e. they are ‘light’ buyers).The hazard rate describing the purchase timing pattern for each segment is assumed to be strictlyincreasing and following the Weibull specification. Figure 1 describes the aggregate hazard rateas well as the hazard rates for each segment. The figure suggests that aggregating the hazard

Ł Correspondence to: D. Vakratsas, Faculty of Management, McGill University, 1001 Sherbrooke St. West, Montreal,Quebec, Canada H3A 1G5. E-mail: [email protected]

Copyright 2002 John Wiley & Sons, Ltd. Received 20 February 1998Accepted 26 February 2001

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50 D. VAKRATSAS AND F. M. BASS

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

1 11 21 31 41 51 61 71 81 91

Time (days)

Haz

ard

Rat

e

Segment ASegment BAggregate

Figure 1. Illustration of aggregation effects on hazard rates

rates of the two customer segments results in a non-monotonic hazard function, even though eachsegment’s hazard rate is increasing.

The company’s goal is to target its customers via customized offers at a time when they aremost likely to buy. The timing of the company’s offer, however, would be different, depending onwhether it uses the aggregate or segment-level hazard rates. The aggregate non-monotonic hazardrate suggests that the offer should be timed around the twentieth day or so without discriminatingbetween high and low intensity customers. The segment-level hazards, on the other hand, suggestthat both segments exhibit positive duration effects but differ in their intensity of buying.Furthermore, the two customer segments may differ in their response to the company’s marketing-mix activities, another element that the company may want to utilize when customizing its offers.

The segment-level modelling of household purchase rates and their response to retailermarketing-mix activities, the importance of which were highlighted in the previous discussionand example, is precisely the objective of this paper. More specifically, we employ finite mixturesof hazard rate models to describe the timing of household purchases and relate it to marketing-mix activities. Such mixtures enable us to identify and characterize household segments in termsof both their purchase rates and their response to marketing variables. The choice of a differenthazard rate to model each segment’s purchase timing decisions can also ease potential aggregationproblems. As suggested in our example, the non-monotonicity of hazard rates, frequently observedin studies using a common hazard rate for all households, may be the result of aggregating hazardrates of various shapes corresponding to different segments (Vakratsas, 1998).

Finally our choice of a finite mixture to capture household heterogeneity in purchase ratesand propensity to accelerate is based on its ability to provide a segment-level interpretation ofhousehold purchase timing decisions. Continuous mixtures of hazard rates, on the other hand,although sometimes preferable for providing a ‘finer’ heterogeneity approach (Allenby, Leone,and Jen, 1999; Rossi et al., 1996), cannot provide a segment-level behavioural interpretation.

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PURCHASE TIMING SEGMENTATION 51

2. MODEL FORMULATION

We assume that there exist S different household segments characterized by their purchase timingdecisions. We choose the Accelerated Failure Time (AFT) formulation to model the purchasetiming behaviour in each segment s (s D 1, 2, . . . , S).1 Given that a household belongs to segments, AFT describes the effects of a vector of explanatory variables x, on its interpurchase time T, inthe following manner:

log T D ˇ0sx C �s log T0 1

Model (1) is essentially a log-linear regression model with log T0, the standard baseline distribu-tion, assuming the role of the error term. The difference between the AFT and a typical regressionmodel is that in the former the standard baseline distribution is not necessarily normal but mayassume other distributional forms and can also handle censored observations. Our specificationallows the S segments to differ in terms of both their purchase intensity (purchase rates), and theirintensity of response to price and promotional activities (purchase acceleration). Such specificationcan become more flexible if each segment s is allowed to follow a different standard baseline dis-tribution. This can be achieved by using a standard baseline distribution that includes a variety ofdistributions as special cases allowing thus for specification testing (Bergstrom and Edin, 1992).Such is the generalized gamma distribution (Stacy, 1962; and for a marketing application Allenbyet al., 1999) which implies the following density function for log T:

flog Tjslog tjˇs D j sj �2

s

( �2

s

) �2s exp

[ �2

s

( s

(log t � ˇ

0sx

�s

))

� exp

( s

(log t � ˇ

0sx

�s

))] s 6D 0 2

With s and �s being respectively the shape and scale parameters. The generalized gamma nestssome of the distributions most commonly used to describe household purchase timing decisions,notably the exponential distribution ( s D 1, �s D 1), the Weibull ( s D 1), the gamma (�s D 1),and the lognormal ( s ! 0). The distribution’s ability to yield increasing, decreasing and non-monotonic hazard functions depending on the values of its scale and shape parameters is a furtherdemonstration of its flexibility, an especially useful feature in the context of our study. Theconditional likelihood given that household i with a purchase history xi belongs to segment s hasthe following expression:

Lijs Dni∏

jD1

flog Tjslog tijjˇs, xijυij 1 � Flog Tjslog tijjˇs, xij1�υij 3

1 The AFT formulation was chosen because it encompasses the generalized gamma distribution which is a very flexibledistribution for describing purchase timing behavior (Allenby et al., 1999), and can directly relate interpurchase times tomarketing mix variables as suggested by (1).

The proportional hazards formulation (PH) can also eventually lead to a relationship between marketing mix variablesand interpurchase time via the representation (we thank an anonymous reviewer for this suggestion): ln t D ˇ0x C � C ˛,where exp(˛) is a standard exponential random variable, � is the minus log of a positive random variable, and t is theintegral of the baseline hazard (Ridder, 1990). However, such relationship is less direct than in the AFT case. Nevertheless,mixtures of PH models were also estimated by the authors (results available upon request) and yielded results similar tothose of AFT models in terms of segment membership implications suggesting that the finite mixture approach is robustwith respect to the choice of AFT or PH formulation. For an application of a PH mixture model in marketing see Siddarthand Chattopadhay (1998).

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52 D. VAKRATSAS AND F. M. BASS

where:

tij = jth interpurchase time (spell) for household ixij = vector of explanatory (marketing) variables for household i on the jth purchaseni = total number of purchases for household i during the observation periodυij = 1 if the jth observation for the ith household is complete

= 0 if the jth observation for the ith household is right-censored

and Flog Tjs is the cumulative distribution function of log T.The unconditional likelihood for all households is then provided by the following formula:

L DN∏

iD1

S∑sD1

PisLijs 4

where N is the total number of households in the sample and pis denotes the probability thathousehold i belongs to segment s. Economic and marketing theory suggest that purchasing ratesand propensity to respond to marketing activities may depend on household demographics (Becker,1965; Blattberg et al., 1978). We therefore express pis as a function of household demographicvariables by employing the logit formulation (see Gupta and Chintagunta, 1994):

pis D exp˛s C �sdis∑

kD1

exp˛k C �kdk

s D 1, . . . , S 5

To ensure that the above probabilities add to 1, we standardize (5) with respect to the probabilityof belonging to segment S:

pis D exp˛s C �sdi

1 Cs�1∑kD1

exp˛k C �kdk

s D 1, . . . , S � 1 6

where di is a vector of demographic variables for household i, with piS D 1 �∑S�1kD1 pis and ˛s D

˛s � ˛S, �s D �s � �S. Segment S therefore becomes the ‘baseline’ segment, with �s reflecting thedifference in the effect of the demographic variables on the probability of membership in segments from the effect of the demographic variables on the probability of membership in segment S(Gupta and Chintagunta, 1994). The choice and the expected effects of such demographic variableson segment membership will be discussed in detail in the empirical application section. Themaximization of L in (4) is performed through the use an iterative algorithm (Broyden, Fletcher,Goldfarb, and Shanno) in GAUSS.

Posterior Household Classification

The probabilities pis of assigning a given household to one of the S segments can be updated bymeans of an empirical Bayes procedure based on the information on household purchase histories(Bucklin and Gupta, 1992; Kamakura and Russell, 1989). The posterior probability of classifying

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PURCHASE TIMING SEGMENTATION 53

household i to segment s given its vector of purchase history xi and the estimated vector ofparameters O is provided by the following formula:

Pisj O ,xiD pisLijs

S∑mD1

pimLijm

7

Each household is assigned to the segment with the highest posterior probability (Bucklin andGupta, 1992; Gupta and Chintagunta, 1994).

Selection of Number of Segments S

To determine the optimal number of segments for our mixture model we use the BayesianInformation Criterion BIC D LL � p/2 lnn (Allenby, 1990), where LL is the Log-likelihoodof the model, n is the total number of observations, and p is the number of model parameters.We select the multi-segment model that maximizes BIC.

3. EMPIRICAL APPLICATION

3.1. Data and Variables

The estimation of the proposed model was performed using AC Nielsen scanner data on catsuppurchases. Our sample consisted of 268 regularly purchasing households (each household madeat least 5 purchases during an 80-week period), that accounted for a total of 2790 interpurchasespells. The last spell for each household is censored at the time data collection ended. All householdpurchases made in the participating stores in the areas of data collection (Sioux Falls, South Dakotaand Springfield, Missouri) were recorded in the database. Price per ounce, absence or presenceof feature (dummy variable), and absence or presence of display (dummy variable) were themain marketing variables in our analysis. Since there were few purchases of multiple units andmost of the purchases were of one particular size (32 oz) we did not include a number of unitsor volume variable. Based on exploratory research, economic and marketing- theoretic arguments(Becker, 1965; Blattberg et al., 1978; Jeuland and Narasimhan, 1985; Murthi and Srinivasan, 1999;Narasimhan, 1984) we used three variables to capture the effects of demographics on segmentmembership:

Household size = number of household membersHousehold income = 1 if household income is above median

0 otherwiseFemale Employment Status = 1 if female head of household is full-time employed

0 otherwise.

Household size is used here as a proxy for a household’s holding and time costs. Largerhouseholds typically have higher consumption rates and have to stock larger quantities of theproduct leading to higher inventory (holding) costs (Jeuland and Narasimhan, 1985). Furthermore,larger households are more likely to have children (information that is not included in the data)a factor that would likely increase their costs of time (Blattberg et al., 1978). Since purchase

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54 D. VAKRATSAS AND F. M. BASS

acceleration leads to stockpiling (Blattberg and Neslin, 1990), we expect smaller households toexhibit higher propensity to accelerate because of their inventory flexibility and their lower costsof time. Household income can be considered a signal for a household’s willingness to respond toa retailer’s marketing activities. Lower income households, being more price-sensitive and havinglower costs of time, should exhibit higher propensity to respond to prices and promotions thanhigher income households (Becker, 1965; Blattberg et al., 1978; Murthi and Srinivasan, 1999).On the other hand, higher-income households have typically lower holding costs since they tendto live in larger residences and generally have more resources than lower-income households thatwould allow them to take advantage of price and promotional activities (Blattberg et al., 1978).Given these two opposing views regarding the effects of income on deal propensity and the mixedempirical results obtained in previous marketing studies (Blattberg and Neslin, 1990), we donot form any expectations on the effects of income on segment membership. Finally, female heademployment status is used as a proxy for a household’s opportunity costs of time. Households witha non-working housewife are expected to have more time to evaluate and benefit from promotionsthan households with both heads being fully employed (Narasimhan, 1984). We therefore expectthem to have a higher tendency to respond to marketing-mix activities.

3.2. Results

We estimated the generalized gamma mixture model with 1, 2, 3, and 4 segments. Table I reportsthe BIC values for the four mixture models suggesting that the two-segment model provided thebest overall fit (higher BIC). The results of the two-segment mixture estimation along with someposterior household classification statistics are reported in Table II.

Based on the size of the intercepts for the two segments and the mean interpurchase times impliedby the posterior classification results, one can easily infer that the two segments differ substantiallywith respect to their purchasing intensity. Segment 1 households appear to be infrequent (‘light’)buyers whereas segment 2 buyers appear to be frequent (‘heavy’) buyers. The 83/17 split betweenheavy and light buyers appears to be close to that suggested by exogenous classification rules (e.g.80/20—Schmittlein, Cooper, and Morrison, 1993). What exogenous rules fail to capture, however,when compared to the finite mixture model are the differences in the propensity to accelerateand the demographic profile of the two segments. Segment 1 buyers are both price and featuresensitive, whereas the purchase timing decisions of Segment 2 do not appear to be influenced bymarketing-mix activities. This coupled with the fact that the only significant demographic variabledetermining segment membership is household size, suggests that segment 2 households whichare larger and have higher purchase rates have holding costs constraints that do not allow them toresort to purchase acceleration.

Table I. BIC results for mixture models with various numbers of segments

Number ofsegments

No. ofparameters

(p)

Log-likelihood (LL) BIC(LL-(p/2 ð lnN)

1 6 �3830.8 �3854.62 16 �3597.6 �3661.13 26 �3561.9 �36654 36 �3550.5 �3693.3

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PURCHASE TIMING SEGMENTATION 55

Table II. Parameter estimates for the two-segmentgeneralized gamma mixture (t-ratios in parentheses)

Variable Segment 1 Segment 2

Intercept 3.96 3.11(41.68) (24.41)

Scale 0.80 0.76(48.84) (33.38)

Shape 0.71 0.64(12.3) (7.97)

Price 0.04 0.03(1.99) (1.07)

Feature �0.12 �0.08(�2.72) (�1.27)

Display �0.05 �0.06(�0.72) (�0.78)

Effects on membership probability for segment 1

Intercept 2.64(4.55)

Household income 0.55(1.39)

Female head �0.17Employment status (�0.46)Household size �0.35

(�2.44)

Posterior classification results

Segment size 0.83 0.17Ave. interpurchase time 67.2 26.9in days (20.2) (5.5)(std. deviation)% of total purchases 65 35

The suggestion that infrequent buyers are influenced by marketing-mix activities more thanfrequent buyers when making purchase timing decisions we believe is interesting, especiallywhen compared to reported differences in brand choice decisions between frequent and infrequentbuyers. Brand choice studies have suggested that in brand choice decisions it is the frequentbuyers that exhibit a higher propensity to respond to marketing-mix activities (Allenby and Lenk,1995; Kim and Rossi, 1994). This reveals a noteworthy reciprocity in the purchasing strategiesof frequent and infrequent buying households. Infrequent buyers, being smaller households, havemore stockpiling flexibility and can more easily accelerate their purchases in taking advantage ofprice cuts and promotions. Alternatively one may suggest that infrequently buying households,regardless of their size, have low utility and therefore a lower willingness to pay for the productcategory (‘low demanders’—Diamond, 1987). In such a case, price cuts and promotions ‘lure’low-demanding buyers into the category and induce them to accelerate their purchases. Thehigher propensity to accelerate exhibited by the infrequent buyers can therefore be supportedby either interpretation: that of inventory-flexible households or low-demanding households.Frequent buyers, on the other hand, have higher utility for the product category but largerholding constraints and find more beneficial to use price cuts and promotions to make their brand

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56 D. VAKRATSAS AND F. M. BASS

0

0.01

0.02

0.03

0.04

0.05

1 11 21 31 41 51 61 71 81 91 101 111

Time (days)

Haz

ard

Rat

e Segment 1

Segment 2

Aggregate

Figure 2. Segment-level and aggregate hazard Rates: Generalized Gamma specification

choice decisions using their experience and information about the product category, rather thanstockpiling.

Another issue we sought to explore in this study is that of dealing with the consequences ofaggregating segment-level hazard rates. The effects of aggregation on the shape of the hazardrates are demonstrated in Figure 2 which portrays the estimated baseline hazard functions for theaggregate generalized gamma model (based on the estimated parameters of the single-segmentmodel) and for each segment separately. The aggregate generalized gamma baseline hazard rate isnon-monotonic whereas the baseline hazard rate for each segment shows strong positive durationeffects. The aggregate hazard rate converges to that of segment 1 since the latter mainly consists ofhouseholds that exhibit low propensity to purchase and therefore have longer purchase cycles. Thenon-monotonicity therefore of the aggregate hazard rate, which has been observed even in one-parameter unobserved heterogeneity hazard rate models (e.g. Jain and Vilcassim, 1991), appearsin our application to be the result of aggregation.

Figure 3 describes the effect a feature promotion on the hazard rate of a household belonging tosegment 1. The feature promotion causes an increase in the hazard rate of a household belongingto segment 1, translating in an acceleration of its purchase. Segment 2 customers do not respond tomarketing-mix activities leaving therefore their baseline hazard rate unchanged. Thus if one lookedonly at the baseline hazard rates (Figure 2) s/he could have concluded that segment 2 customersshould be targeted more often because they have higher hazard rates. However, it is segment1 households who respond to promotions (Figure 3) and should thus be the focus of a targetedcampaign. The ‘jump’ in the hazard rate of segment 1 due to a promotion may provide marketerswith some input as to how early they should expect their customers to buy because of a promotion.

3.3. Specification Tests

One of the main reasons we chose the generalized gamma distribution to describe purchase timingdecisions is that it facilitates specification testing against nested distributions based on certain

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PURCHASE TIMING SEGMENTATION 57

0

0.01

0.02

0.03

1 11 21 31 41 51 61 71 81 91 101 111 121

Time

Haz

ard

Rat

e

No Feature

Feature

Figure 3. Effect of feature promotions on segment 1 hazard rate

Table III. Tests of alternative specifications against the generalized gammamixture (two-segment models)

Specification(hypotheses)

Log-likelihood LR (df) Nullhypothesis

result

Weibull mixture( 1 D 2 D 1)

�3616.8 38.4 (2) Reject

Exponential mixture(�1 D �2 D 1 D 2 D 1)

�3795.2 395.2 (4) Reject

Gamma mixture(�1 D �2 D 1)

�3690.9 186.6 (2) Reject

Log-normal mixture( 1 D 2 ! 0)

�3704.3 213.4 (2) Reject

Segment 1 Weibull( 1 D 1)Segment 2 generalizedgamma

�3608.6 22 (1) Reject

hypotheses about its scale and the shape parameters. In addition, specification combinations withsegment 1 specification being different from the one in segment 2 may also be tested. The resultsof such specification tests based on the Likelihood Ratio statistic (LR) are reported in Table III.The mixture of Weibull for segment 1 and Generalized Gamma for segment 2 provided thebest fit among all specification combinations and is the only combination reported here. All thenull hypotheses are rejected at the 1% level, suggesting that the generalized gamma mixture ispreferred over all alternative specifications. These tests suggest that although the segment-levelhazard rates appear relatively ‘simple’, household purchase timing decisions require complexprobability distributions for their description.

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58 D. VAKRATSAS AND F. M. BASS

4. CONCLUSIONS

In this paper we employed a finite mixture approach to modelling household purchase timingdecisions. Such framework offers a segment-level interpretation by allowing differences inpurchase rates and propensity to accelerate across groups of households, something that cannotbe achieved through the use of continuous mixture distributions. Furthermore, the use of differenthazard rates for each segment promises to alleviate aggregation bias problems arising in singlehazard-rate models.

The application of our model to scanner data on catsup purchases suggested that two householdsegments exist with different purchase rates and acceleration propensities. A comparison of thehazard rates estimated for each of the two segments and a single-segment aggregate hazard raterevealed that indeed aggregate hazard rates underestimate positive duration effects, just as ourdiscussion on aggregation effects suggested. Infrequent buyers appear to be influenced in theirpurchase timing decisions by marketing-mix activities more than frequent buyers. We interpretedthis result using household costs, intensity of demand, and information-theoretic arguments.Infrequent buyers, who are also smaller households, have lower utility and information accessbut higher inventory flexibility than frequent buyers and therefore are lured into the category bymarketing mix activities. Frequent-buying households, on the other hand, appear to be constrainedby holding costs. This may translate into strategic opportunities for the retailer. The retailer cancustomize his or her marketing activities by promoting the price attractiveness of the categoryas a whole to infrequent buyers, and by promoting the variety of promotional activities fordifferent brands in the category to frequent buyers. Given the ability and willingness of today’sretailers to customize their offers, such a strategy is feasible. The optimal design of retailerpromotional strategies taking into account the differential response of frequent and infrequentbuyers to marketing-mix activities in their purchase timing and brand choice decisions may be afruitful area for future research. Such research would help us to formalize the previous argumentsmade regarding the brand choice and category focus of promotional strategies for frequent andinfrequent buyers.

Finally, assuming independence of purchase timing and choice decisions (Bucklin and Gupta,1992; Gupta, 1988; Jeuland et al., 1980), brand choice can be incorporated into our modellingframework in a multiplicative fashion. Given, however, this independence and the fact that themarketing literature on brand choice decisions is fairly comprehensive, we opted to focus onpurchase timing decisions and utilize some of the extant results on brand choice in our discussion.

In summary, the finite mixture framework presented here produced a segmentation schemethat revealed significant differences in purchasing rates and response to marketing-mix activitiesbetween segments, and suggested that a common hazard rate cannot adequately describe positiveduration effects of purchase timing decisions.

ACKNOWLEDGEMENTS

This paper has benefited from presentations at the London Business School and Groupe ESSEC(Paris, France). The authors would like to thank in particular Albert Bemmaor and Bruce Hardiefor their helpful comments and suggestions.

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