how long has it been since the last deal? consumer promotion timing expectations and promotional...

How long has it been since the last deal? Consumerpromotion timing expectations and promotional response

Yan Liu & Subramanian Balachander

Received: 7 January 2013 /Accepted: 15 October 2013 /Published online: 16 November 2013# Springer Science+Business Media New York 2013

Abstract When modeling consumers’ forward-looking behavior using choice data onfrequently purchased products, the common approach assumes that consumers haverational expectations about future promotions. Previous studies modeled suchexpectations using a first-order Markov (FOM) process. However, empirical evidencefrom several categories suggest that inter-promotion intervals can last several weeksimplying that a FOM process that conditions future expectations of prices only oncurrent-period prices can be limiting. We utilize a Proportional Hazard model (PHM) tocharacterize consumers’ rational expectation of future price promotion. We first showthat estimating a dynamic structural model that uses a FOM specification for rationalexpectations can bias estimates of promotion effects with both simulation analysis andscanner panel data from four consumer packaged goods product categories. Secondly,we empirically show that a structural model employing a PHM specification forpromotion expectations fits the data better than ones that assume only a FOM priceor promotion expectation. Lastly, we show using an analysis of promotion policychanges that a structural model with a FOM expectation can lead to suboptimalmanagerial decisions.

Keywords Promotion . Promotion expectations . Dynamic discrete choicemodel .

Hazardmodel . Stockpiling

JEL Classification M3 . C23 . C25

Quant Mark Econ (2014) 12:85–126DOI 10.1007/s11129-013-9141-3

Y. Liu (*)Mays Business School, Texas A&M University, 4112 TAMU, College Station, TX 77843, USAe-mail: [email protected]

S. BalachanderKrannert Graduate School of Management, Purdue University, 403 W. State Street, West Lafayette,IN 47907, USAe-mail: [email protected]

1 Introduction

As a key marketing mix variable, consumer sales promotion accounts for approximate75 % or more of the marketing communications budget for many companies (Kotlerand Keller 2006). Although brands have used sales promotions to retain market share(e.g. diaper promotion wars, Wall Street Journal 2002), concerns exist that excessivesales promotion can lead to deal-to-deal buying and diminish brand equity. Partly as aresponse to such concerns, P&G reduced price promotions and switched to everydaypricing in the early 1990s (Ailawadi et al. 2001). A key issue that a manager faces whenmodifying a price promotion policy is estimating consumer response to changes in thepolicy. Past research has shown that in empirical models of consumer purchase ofpackaged goods that are frequently promoted and storable, forward-looking andstockpiling behavior of consumers can be important (Gönül and Srinivasan 1996;Erdem et al. 2003; Hendel and Nevo 2006; Osborne 2010). Specifically, the researchshows that a consumer’s purchase decision can depend on current prices andpromotions as well as her expectation of future price promotions.

The past literature on the modeling of consumer purchase decisions when consumersare forward-looking about future prices or promotions can be classified into twostreams. The first, early stream of research derives normative implications for consumerpurchase decisions when consumers have expectations for future prices (Assuncao andMeyer 1993; Meyer and Assuncao 1990; Krishna 1992). A second stream of research,which is more closely related to this paper, uses consumer purchase data to estimatedynamic structural models of purchase decisions by forward-looking consumers whohave expectations about future prices or promotions. In this stream, Gönül andSrinivasan (1996) estimate a structural dynamic programming model of consumers’decisions to purchase (purchase incidence) in a category when consumers haveexpectations about future price promotions. Sun et al. (2003) extend this analysis toalso consider consumer brand choice while Erdem et al. (2003) and Hendel and Nevo(2006) also consider consumer quantity choice. Sun (2005) and Chan et al. (2008)extend structural dynamic models to incorporate endogenous consumption decisions.Osborne (2010) studies the implications of changing promotional frequency andpromotional depth on long run demand.

When available data is limited to consumer’s choice and the marketing environment,dynamic discrete choice models suffer from an under-identification problem (Rust1994). In particular, consumer’s preference, discount rate and consumer’s belief onfuture market conditions cannot be jointly identified non-parametrically from choicedata (Magnac and Thesmar 2002). In practice, the analyst assumes a discount rate toreflect the current interest rate.1 As consumers’ beliefs are unobserved with purchasedata, a common practice is to assume that consumers form rational expectations onfuture market conditions, such as price and promotion (e.g. Erdem et al. 2003; Hendeland Nevo 2006).2 Accordingly, the evolution of market conditions is estimated from the

1 Rather than assume a discount rate, Geweke and Keane (2000), Yao et al. (2012) and Ching et al. (2013) arguethat consumer discount rate can be identified if the consumer’s current utility is observable. Magnac and Thesmar(2002) and Fang andWang (2013) suggest another way of identifying discount rate by using exclusion variableswhich affect the state transition probability but not the consumer’s static utility function.2 Survey data on consumer expectations can be used to relax assumptions about rational expectation. SeeManski 2004 for a review on this approach and Erdem et al. (2005) for an implementation.

86 Y. Liu, S. Balachander

data and consumer expectations are assumed to match this evolution process. Further,policy evaluations in these studies also assume that consumers form rationalexpectations of new changed policies.3

In this study, we follow this standard approach of assuming that consumers haverational expectations on future price promotions and in assuming a specific discountrate for consumers. However, our objective is to explore the consequence of using asimple, yet more flexible way of representing the promotion and price process in thedata that can be the basis of modeling consumers’ rational expectations. We use datafrom four product categories and show that the first-order Markov (FOM) process usedby previous studies to model consumers’ rational expectations of future prices orpromotions could be substituted for by a proportional hazard Model (PHM) processthat is more consistent with the observed price promotion process, and hence moreconsistent with the assumption of rational expectations. We then explore the bias thatwill result in estimates of consumer preferences if the less appropriate FOM process isused to model rational expectations of price promotion.

We begin with a review of how previous studies employing structural dynamicmodels have modeled consumers’ rational expectations on future prices or promotions.Gönül and Srinivasan (1996) and Sun et al. (2003) model consumers’ expectations onprice promotion incidence with a FOM process under which the probability ofpromotion of a brand in the period immediately following a given period depends onlyon the promotion status during the given period. Similarly, Erdem et al. (2003) modelconsumers’ expectation of future prices with a price process (referred to as the EIKprice process hereon) in which the probability and amount of price change expected fora brand in the period following a given period depend on the brand price and thecategory-average price during the given period. Further, while Sun et al. (2003) modelconsumers’ expectations on promotion depth with a normal distribution whose meanand variance are consistent with the data, Erdem et al. (2003) use an autoregressiveprocess to model the same. Hendel and Nevo (2006) reduce the state space by using asingle quality adjusted “price” (or inclusive value) for each size. The inclusive valueincludes price and advertising effects of all brands for each size. However, this “price”also follows an exogenous FOM transition process in their paper. Osborne (2010)builds on Erdem et al. (2003) and Sun et al. (2003) in the modeling of rationalconsumer price expectations. Thus, all of the above papers that estimate dynamicstructural models assume that consumer expectations of future prices or promotionsin the period following a given period are conditioned only on prices or promotions inthat given period, thus following a FOM process.

Although the assumption of a FOM process helps reduce the state space andcomputation cost of dynamic programming, empirical evidence from several categoriessuggest that inter-promotion intervals can last several weeks implying that a FOMprocess that conditions rational expectations of future promotions only on current-period promotions can be limiting. Figure 1(a) through (d) plot the time series of pricepromotion incidence and promotion discount depth at the retail level in a single retail

3 To the contrary, recent papers by Seiler (2013) and Ching et al. (2009) argue that consumers may notobserve prices every period, and therefore, it may not be reasonable to assume that consumers formrational expectations about future prices conditioned on past prices. This issue is an interesting ripe area forfurther research.

How long has it been since the last deal? 87

chain for brands in four frequently purchased product categories: tuna, mayo, peanutbutter, and tomato soup. The cyclical nature of price promotions in these plots suggeststhat rational expectations of price promotions may well depend on the time elapsedsince the previous promotion rather than simply on whether there was a promotion inthe previous period. Indeed, some theoretical models of price promotions dynamicssuggest that promotions occur in cycles such that a period of high prices by competitivefirms is followed by a price promotion by some firm or firms to be followed again by aperiod of high prices and so on (Sobel 1984; Conlisk et al. 1984).4 Thus, these modelsimply that the incidence of price promotion may depend stochastically on the period ofthe promotion cycle, which is the time elapsed since the previous price promotion.Next, as we can see from the plots, the inter-promotion interval can be quite irregularfor a brand, particularly if price promotions occasionally carry over from week to weekas occurs most prominently in the peanut butter category making the inter-promotioninterval zero in such weeks. Further, there is also considerable variation in the inter-promotion interval across brands and categories. Table 1 presents descriptive statisticsat the retail level for these four product categories. As can be seen from the data, theaverage inter-promotion interval for brands varies from a low of 1.3 weeks to as high as13 weeks across these four categories. In spite of such variation in inter-promotion

4 The idea in these papers is that competing firms optimally maintain high prices when there are few price-sensitive consumers in the market causing these consumers to accumulate in the market as they wait for a pricereduction. Eventually as the number of accumulated price-sensitive consumers becomes sufficiently high, oneor more of the firms find it profitable to reduce their prices clearing the market of price-sensitive consumers.After this clearing, firms find it optimal to price high again thus spawning a promotion cycle.

00.10.20.30.40

0.10.20.30.4

a b

c d

1 6 11 16 21 26 31 36 41 46

Starkist COS

Dis

coun

t%

Weeks

1 6 11 16 21 26 31 36 41 46 51

Hellmann's Kraft Private label

Weeks

0

0.4

0

0.4

0.8

0

0.4

0.8

1 6 11 16 21 26 31 36 41 46 51

Peter Pan Skippy J.M. Smucker

Weeks

Dis

coun

t %

1 6 11 16 21 26 31 36 41 46 51

Progress(1.19oz) Campbell(2.69oz)Campbell(0.67oz)

Weeks

Dis

coun

t %D

isco

unt %

0

0.4

0

0.4

0

0.4

0.8

0

0.4

0

0.4

0

0.4

0.8

Brand Promotion in Canned Tuna Category Brand Promotion in Mayo category

Brand Promotion in Peanut Butter Category Brand Promotion in Tomato Soup Category

Fig. 1 a Brand promotion in canned tuna category b Brand promotion in mayo category c Brand promotionin peanut butter category d Brand promotion in tomato soup category


intervals within a brand and across brands and categories, our analysis shows that thepromotion processes in the four categories are more consistent with cyclical promotionsthan with a FOM process.

If we assume that consumers have rational expectations on future promotions andprices when estimating a dynamic structural model, the above data suggests that a moreappropriate model of rational expectations than a FOM process would be one thatcomes closer to the cyclical promotion pattern observed in the data. In this study, wepropose a proportional hazard Model (PHM) to represent consumers’ rationalexpectations on promotion incidence while estimating a dynamic structural model toinvestigate consumers’ purchase incidence, brand choices and quantity decisions.Under the PHM, the expected probability of promotion is a function of the discretetime since the previous promotion consistent with the empirical evidence andtheoretical models. Thus, this model allows consumers’ promotion expectations todepend on promotion activity in the past several periods instead of simply the lastperiod, without increasing computational complexity. An alternative approach wouldbe to use a higher-order Markov process because such a process would allow forcorrelation in promotions across a number of periods. However, incorporating a higher-Markov model for promotion expectation complicates the model development andincreases computation burden dramatically. Note, however, that in comparison to theFOM process, the PHM model incorporates a different information set on which

Table 1 Descriptive statistics from data

Brand name Marketshare %

Regular priceper can ($)

Promotionfrequency %

Inter-promotiontime (weeks)a

Promotiondiscount (%)b

Mean S.D. Mean Max Min S.D. Mean S.D.

Canned tuna SKW 56.4 0.76 0.00 12.9 7.7 8 3 1.99 23.6 0.07

N=11035NH=836

COSW 20.9 0.75 0.00 17.0 5.9 11 0 3.25 25.2 0.07

SKO 11.3 0.76 0.00 12.9 7.7 8 3 1.99 23.6 0.07

COSO 3.4 0.75 0.00 17.0 5.9 11 0 3.25 25.2 0.07

Mayo Hellmann’s 62.3 3.03 0.13 21.2 4.7 15 0 3.61 31.5 0.12

N=2203 NH=308 Kraft 8.5 2.72 0.05 28.9 3.5 9 0 1.85 25.8 0.09

Private Label 13.2 1.69 0.02 73.2 1.4 8 0 1.36 35.3 0.12

Peanut butter Peter Pan 35.4 1.98 0.05 44.2 2.3 14 0 4.06 34.7 0.03

N=2324 NH=290 Skippy 38.5 1.93 0.00 46.1 2.2 10 0 4.06 34.0 0.05

J.M. Smucker 13.9 2.13 0.00 7.7 13.0 15 9 4.80 35.1 0.10

Tomato soup Campbell(0.67 oz.)

43.0 0.88 0.00 13.5 7.4 16 0 7.11 20.1 0.09

N=3561 NH=294 Campbell(2.69 oz.)

25.3 1.11 0.00 25.0 4.0 36 0 9.90 33.4 0.04

Progresso(1.69 oz.)

15.4 1.67 0.00 21.2 4.7 14 0 4.42 50.1 0.08

N number of observations; NH number of householdsa Inter-promotion time is the average number of weeks between two promotion events. We assume that inter-promotion time for promotions in consecutive weeks is zerob% of Regular Price


rational expectations are conditioned (Manski 2004). In doing so, the PHM processexpands the state space in comparison to a FOM process because the latter requiresonly a binary variable to indicate whether there was a promotion in the last period.Using covariates, the PHM also allows consumers’ expectations of promotions to bedependent across choice alternatives (cf. Erdem et al. 2003 and Hendel and Nevo2006). While we model promotion incidence using a PHM, we draw the promotiondepth, conditional on a promotion, from a stationary probability distribution. Thisdrawing of promotion depth from a distribution is consistent with the use of mixedstrategies in theoretical models to represent price promotions (Sobel 1984).

Our findings and contributions to the literature are as follows. First, we demonstratethat consumer rational expectations more consistent with the empirically observedpromotion processes can be embedded in a dynamic structural consumer model usinga PHM specification of promotion expectations. Such a specification allows for theprobability of promotion of a brand to be dependent on the time since the previouspromotion as well as competitive promotion activity. Second, using simulation data, weshow that using a FOM specification for rational expectations when the underlyingpromotion process follows a PHM specification causes an upward bias in estimatedpromotion effects, when estimating a dynamic structural model of consumers’ purchaseincidence, brand and quantity choices. Third, we compare the dynamic structuralconsumer models using three specifications for consumers’ rational expectations onprice promotion and prices, namely, the PHM, the FOM promotion process (cf. Sunet al. 2003) and the EIK price process (Erdem et al. 2003). We carry out thiscomparison in the four product categories discussed above that have very differentpromotion patterns. We show that specifying a simple FOM promotion process or anEIK price process for rational expectations can bias estimates of promotion effects incomparison to a PHM specification in all four product categories. However, this biasgets smaller with increase in the promotion frequency because of the shorter inter-promotion intervals. Therefore, in a heavily promoted product category, such as peanutbutter, this bias is at a minimum. Fourth, we show using policy simulations that the biasin estimation of promotion effects by the FOM and EIK specifications of rationalexpectations critically affect profit estimates from promotion, and can lead to incorrectdecisions on promotion frequency.

The intuition for why using a different or less appropriate specification for rationalexpectations can bias estimates of promotion effects is as follows. Consider Fig. 2which shows a brand that is promoted approximately every 5 weeks (it is promoted inWeeks 1 and 6) and whose promotion probability (hazard) is increasing in the number

00.10.20.30.40.50.60.70.80.9

1

1 2 3 4 5 6 7

Pro

mot

ion

Pro

babi

lity

Weeks

FOM True Hazard Promotion indicator

Fig. 2 Alternative models ofpromotion expectation


of weeks since the previous promotion, consistent with a PHM process. Now, supposewe model rational expectations of such a promotion process with a FOM process. Sincethe projected future probability of promotion with such a model depends only onwhether or not the brand was promoted in the current period, the expected promotionprobabilities stay the same from week 3 to week 6. Thus, for week 6, even though thebrand has not been promoted for four weeks, the consumer would not expect a higherprobability of promotion than in week 3, contrary to the true increasing hazard. Moreimportantly, with an FOM specification for rational expectations, the consumer assignsa higher probability of promotion than a PHM 2 weeks after the promotion (weeks 3and 4), as seen in Fig. 2. Therefore, under a model that assumes an FOM specificationfor rational expectations, consumers would optimally buy less during the promotion inweek 1 and become more likely to buy in the following non-promotion weeks as theyrun out of product, in comparison to a model that uses the more appropriate PHMspecification. In other words, if consumers’ rational expectations follow the moreappropriate PHM specification in the data, they would purchase more in the promotionweeks and less in non-promotion weeks than a dynamic choice model with an FOMspecification would predict. Thus, in order to explain the higher quantity purchasedduring a promotion in the data, a model with an FOM specification has to impute ahigher value to the estimated promotion sensitivity parameter. This intuition explainswhy promotion effects estimated using a dynamic choice model with an FOMspecification for rational expectations are biased upwards both in our simulation andempirical results.

We wish to reiterate that our focus is on choosing a specification for the pricepromotion process that would be more consistent with rational expectations in our dataand to shed light on the importance of appropriately specifying consumers’expectations in dynamic discrete choice models, when assuming rational expectations.We do not claim to identify consumers’ true beliefs or expectations about pricepromotions because, as noted earlier, these beliefs are neither observed, nor can theybe identified (non-parametrically) with choice data alone (Magnac and Thesmar 2002;Manski 2004). Identification of consumers’ true beliefs about future price promotionsmay be achieved with survey data (Manski 2004), but we do not have or assume accessto such data in our study. Next, as noted earlier, the information sets that conditionconsumer rational expectations differ for the FOM and PHM specifications. While theFOM specification conditions rational expectations on price promotions in the previousperiod, the PHM specification assumes a consumer information set that consists of thetime elapsed since the previous promotion for each brand. We argue that the latterinformation set is not too demanding as there is empirical evidence that consumers “liein wait” for price promotions (Mela et al. 1998). Nevertheless, a test of the moreappropriate specification of rational expectations is the degree to which thecorresponding structural dynamic model would predict consumer choices, both in-sample and out-of-sample. As noted above, estimates of consumer preferenceparameters adjust to rationalize the observed data under different specifications forrational expectations of price promotion. However, it is intuitive that a more appropriatespecification of rational expectation should result in greater predictive validity. On thismeasure, we find that a structural dynamic model with a PHM specification of the pricepromotion process generally leads to the best in-sample and out-of-sample fit in thefour product categories we study. Although the PHM specification fits best in the four


categories we study, it is conceivable that the competing PHM or EIK specificationsmay be better in other settings. Therefore, it is advisable for a researcher to test thesealternative specifications for the promotion process in their particular setting and usethe most appropriate specification.

Our approach compares in the following ways to extant dynamic structural modelingapproaches (Sun et al. 2003; Erdem et al. 2003; Hendel and Nevo 2006). As notedearlier, unlike all of these papers, we allow rational expectations of promotion incidenceto depend on promotion activity in periods prior to the previous period by using thePHM. In our modeling approach, we separately model consumer expectations onpromotion incidence and the prices, conditional on the incidence of a promotion. Inadopting this decomposition of prices into promotion incidence and prices conditionalon promotion incidence, we are similar to Sun et al. (2003) and unlike Erdem et al.(2003) and Hendel and Nevo (2006) who directly model prices. We also let promotionexpectations depend on competitive promotions in the previous period unlike Sun et al.(2003) and similar to Erdem et al. (2003) and Hendel and Nevo (2006). We decomposeprices into promotion incidence and prices conditional on promotion for the followingreasons. First, this decomposition is consistent with the tradition of allowing fordifferent effects of regular price and price promotions on consumer behavior inempirical studies (e.g. Guadagni and Little 1983; Mulhern and Leone 1991). Second,Krishna et al. (1991) find that consumer knowledge about promotion frequencyexceeds their knowledge of sale prices, suggesting that consumers are likely toprimarily forecast promotion incidence. Lastly, the theoretical analysis of promotiondynamics (Sobel 1984) suggests that the timing of price promotions is strategicallychosen by firms to take advantage of the buildup of price-sensitive customers, while theprices, conditional on a promotion, are chosen simply as a draw from a mixed strategydistribution. Consequently, we favor the decomposition approach.

2 Model and estimation

2.1 Household’s utility function and dynamic programming problem

We model a household’s utility as arising from consuming a focal category and acomposite of outside goods subject to a budget constraint to be met by the expenditureson the focal category and outside goods as well as inventory holding costs. Thus, wederive household i’s utility of purchasing quantityQijt of brand j, j=1, 2,…, J, in week t(see Appendix A for details):

UijQt ¼Xj¼1

J

ψij Cijt þ γi�Cijt

2� �þ α1irpjtdijtQijt þ α2iprjtDjtdijtQijt þ α3iInvit þ α4iyit ð1Þ

The above utility specification follows previous literature (Sun et al. 2003; Erdemet al. 2003 and Sun 2005). In the above equation, Cijt is household i’s consumption ofbrand j in week t, ψij is the consumption benefit per unit of brand j to that householdand γi is a consumption saturation parameter (cf. Sun 2005). Of the other variables, rpjtis the regular price of brand j in week t, prjt is an indicator variable which equals one if


brand j is on promotion in week t and zero otherwise, Djt is the promotion discount, dijt,is a choice indicator which equals one if household i chooses brand j in week t, Invit isthe simple average of category inventories at the beginning and end of week t, and yit ishousehold i’s income in week t. α1i, α2ι, and α3ι are the price, promotion and inventoryholding-cost parameters. We allow for differences in consumer’s responses to regularprice and promotional discounts, consistent with previous literature (Guadagni andLittle 1983; Mulhern and Leone 1991; Sun et al. 2003)

When household i does not purchase any of the J brands in week t, we assume itchooses brand j=0 designating an outside good. The utility of household i whenchoosing j=0 (di0t=1) is

Ui00t ¼Xj¼1

J


2� �þ α3iInvit þ α4iyit

We assume that in each week t, a household decides whether to buy (purchaseincidence), which brand to buy (brand choice) and how much to buy (quantity choice)conditional on the observed marketing activities, expected future promotions and thecurrent household inventory. Moreover, this household makes decisions on how muchto consume of each brand, depending on its purchase decision. The household’s choicespecific value function, vijQt, which is the discounted some of future utilities fromchoosing brand j and quantity Q at time t, satisfies the following Bellman equation:

vijQt Invit; Ztð ÞÞ ¼ MaxCijt

U ijQt þ εijQt þ δE V itþ1 Invitþ1; Ztþ1ð Þ½ �� ð2Þ

In the above equation, δ is the consumer’s discount factor, E(.) is the mathematicalexpectation operator, Invit is the vector of inventory of all brands at the beginning ofweek t, and Zt is a vector of values at the end of week t of state variables that determinethe probability of promotion and the price in week t+1. Note that in Eq. (2), theexpectation is taken over promotion and prices in week t+1. Further, for week t, Cit isthe vector of brand consumption quantities. Equation (2) says that when choosingbrand j and quantity Q at time t, the consumer chooses Cit that will maximize the sumof the utility in week t and the expected value, Vit+1, going forward from week t+1. Thelatter quantity depends on the values of the two state variables, Invit and Zt, at t=t+1.The value function, Vit, is given by the following:

V it Invit; Ztð Þ ¼ Maxdijt ;Qijt

vijQt Invit; Ztð ÞÞ þ εijQt� �

Consistent with previous marketing literature, we assume that error tem, εit(j, Q), inthe above equation follows an i.i.d. extreme value distribution. This error term capturesthe impact of unobserved factors that influence consumer’s choice in week t. We canthen calculate the probability that household i purchases brand j in quantity Q at time tby the multinomial logit formula:

Prob dijt ¼ 1;Qijt ¼ q Invit; Ztj� �

¼ exp vijqt Invit; Ztð Þ� Xk

XQ

exp vikQt Invit; Ztð Þ� ð3Þ


Note that in the denominator of Eq. (3), the summation is not only over brand choicesbut also over discrete quantity choices. Because yit enters vijQt linearly (see Eqs. (1) and(2)) and is constant across choices for household i, it does not affect the choiceprobability. Therefore, we set yit to zero without loss of generality (cf. Erdem et al.2003). The household i’s inventory of brand j at the beginning of week t, Invijt, evolvesaccording to the following equation:

Invijt ¼ Invijt−1 þ Qijt−1−Cijt−1 ð4Þ

2.2 Evolution of promotion and price state variables (Zjt)

As noted earlier, we follow the previous literature and assume that consumers haverational expectation on future price and promotions. To represent these rationalexpectations, we fit a model to the price and promotion process observed in our data.In particular, we compare the appropriateness of three alternative specifications, thePHM, the FOM processes and the EIK price process, for the price and promotionprocess. Each of these specifications differs in the consumer information set, and thusthe conceptualization of the state variable Zjt of brand j that determines rationalexpectations of promotion incidence and prices.

2.2.1 Zjt: Proportional Hazard Model (PHM) process

We noted in the introduction that the FOM model might be limiting in describingpromotion incidence based on the empirical evidence from several categories.Therefore, we make a brand’s likelihood of promotion in a given period depend onthe brand’s promotion history beyond just the previous period. We do so by using thetime since last promotion of a brand as the argument of the baseline hazard function in aPHM (see Seetharaman and Chintagunta 2003 for a review of the PHM and itsapplication in marketing). In addition to the baseline hazard, the PHM has a covariatefunction, which we use to capture the effects of competitive promotion activities. Usingthe PHM has the following two advantages. First, this specification effectively capturesthe notion that probability of a price promotion depends on the time elapsed since thelast promotion (cf. Sobel 1984). Second, the base line hazard with differentspecifications could capture various shape of the promotion probability, such asincreasing, decreasing and inverted U shape.

We use an exponential covariate function and derive the following discrete time PHMspecification for the promotion probability of brand j (see Appendix B for details):

Prob prjt ¼ 1 Tjt;X jt

� � ¼ h Tjt

� ��eX jtβe−∫Tjt0 h uð ÞeXjtβdu ð5Þ

In the above specification, Tjt is the number of weeks elapsed since the brand j’sprevious promotion as of the end of week t. Xjt is a vector of covariates of brand j inweek t. Thus, Zjt={Tjt, Xjt} for the PHM specification. In our empirical application, Xjtconsists of the time since last promotion of competing brands in week t. The function,


h(Tjt) is the baseline hazard and eX jtβ is the covariate function. Given Eq. (5), Tjtevolves as follows:

Tjt ¼ Tjt−1 þ 1 if prjt ¼ 00 if prjt ¼ 1

�

Thus, Tjt resets to zero when brand j has a promotion. Note that Eq. (5) introducesdependence of promotion probability on the promotion history beyond the previousperiod in a computationally simple way. For example, if the promotion schedule forbrand j in a 5-week sequence (represented by a vector of promotion indicators, prjt) is(0,1,0,0,0), Tjt equals 3 at the end of week 5. However, if the promotion schedule is(0,1,0,0,1), Tjt equals zero at the end of week 5. Note however that the promotionvectors (1,0,0,1,0) and (0,0,0,1,0) both result in Tjt=1 at the end of week 5, consistentwith our assumption that the time since the previous promotion is the information setthat drives expectation of a brand’s promotion probability.

Conditional on a price promotion for brand j in period t, we assume that thepromotion depth, Dscntjt, follows a normal distribution whose mean and variancecorrespond to those observed in the data. Conditional on a brand not being onpromotion, its anticipated price is its regular price. Similar to promotion depth, theregular price for each brand is drawn from a normal distribution whose mean andvariance are as observed in the data.5 Sun et al. (2003) make similar assumptions aboutconsumer expectations for promotion depth and regular prices.

2.2.2 Zjt: First-order Markov (FOM) process

Traditionally, a FOM process has been used to model the promotion incidence process(cf. Sun et al. 2003; Gönül and Srinivasan 1996; Assuncao and Meyer 1993). Thisprocess assumes that the probability of promotion of a brand in a given period dependsonly on whether there it was promoted in the previous period. Thus, Zjt=prjt under thisspecification. More formally,

Prob prjt ¼ 1 pr j;t−1 ¼ 1� � ¼ π j1 and Prob prjt ¼ 1 pr j;t−1 ¼ 0

� � ¼ π j0

For the process generating regular price and depth of promotion, we have the sameassumptions as in the PHM process.

2.2.3 Zjt: EIK price process

Erdem et al. (2003) use a first-order price process to model the generating process forthe extended price, which is the regular price minus promotional discounts, if any. Wedenote the extended price as epjt=rpjt−Djt. There are two components in the EIK priceprocess. The first component specifies the probability of the extended price of brand jstaying constant from 1 week to the next. It is a logit function of the difference between

5 Because the variance of regular prices and promotion discounts in the categories we study are small as seenfrom Table 1, when we draw from these distributions, we do not encounter the problem of drawing negativevalues. In other cases, a truncated normal or log-normal distribution may be appropriate to avoid drawingnegative values.


the extended price of brand j (epjt−1) and the average category extended price (ept−1 ) inthe previous period, where ept−1 ¼ 1

J ∑jepjt−1 . The logit function is as follows:

Prob epjt ¼ epjt−1

� �¼

exp δ0 j þ δ1 j epjt−1−ept−1� �

þ δ2 epjt−1−ept−1� �2�

1þ exp δ0 j þ δ1 j epjt−1−ept−1� �

þ δ2 epjt−1−ept−1� �2� :

Thus, similar to PHM process, the EIK price process also depends on competingbrands’ prices. The second component is an autoregressive process specifying theextended price when there is a price change with μjt~MVN(0,Σ):

ln epjt� �

¼ β0j þ β1jln epjt−1� �

þ β2j1

J

Xj

ln epjt−1� �( )

þ μjt

Therefore, the state variable vector, Zjt={epricejt}, j=1, 2,…, J. Note that in theframework of Erdem et al. (2003), consumer expectations about the extended pricealone matters because the extended price alone enters the utility function. Thus, to beconsistent with Erdem et al. (2003), when using the EIK process, we modify theconsumer utility function (Eq. 1) to be the following:

UitjQ ¼Xj¼1

J


2� �þ α1iepjtdijtQþ α3iInvit þ α4iyit

The above utility function used in conjunction with the EIK price processspecification only captures the extended price effects, while the utility function withthe PHM and the FOM specification (Eq. 1) captures the separate effects of the regularprice and promotion discount.

2.3 Heterogeneity, likelihood function and estimation

Following Erdem et al. (2003), we incorporate promotion and price expectations usinga two-stage estimation procedure. In the first stage, we estimate the promotionincidence parameters under the PHM or the FOM process or the EIK price process.We then consider the estimated parameters of the promotion and price process to beknown to consumers (thus forming the basis of their promotion or price expectations)in a second stage in which we model consumers’ dynamic choices. We estimate theparameters of the PHM, FOM and price process models using simulated maximum-likelihood methods. In our estimation of the PHM process, we consider threealternative specifications of the baseline hazard (see e.g. Seetharaman and Chintagunta2003): Erlang-2, Weibull, and Log-logistic(see Appendix B for further details of theestimation methods for the PHM specification).

We incorporate consumer heterogeneity in our second-stage model using a continuousdistribution. We assume the parameters in Eq. (1) ωi={ψij,γi,αi} are multivariate


normally distributed and ωi~N(ω0,Σω).6 Given the state space vector (Invit, Zt) and the

observed brand and quantity choices of households betweenweek t=1 andweek t=Γ, weform the likelihood of household i’s choice decision over time as follows:

Li ¼ log½∫ω ∏t

Xj

Xq dijt � Ind Qijt ¼ q

� �Prob dijt ¼ 1;Qijt ¼ q

Invit; Zt;ωi

� �dωi � ð6Þ

To compute Prob(dijt=1, Qijt=q) in the above likelihood function, we use Eq. (3),which requires us to compute the household value function in Eq. (2). Similar to Erdemet al. (2003), we approximate the solution to the infinite horizon dynamic programmingproblem in Eq. (2) by backsolving the finite horizon DP problem for a sufficiently largenumber of time periods, until the consumer’s value function ceases to changesignificantly (less than 10−3) as we backsolve one more period.7 The discount rate isassumed to be 0.99, consistent with previous literature. The state space for each modelconsists of the inventory vector, Invit, and the specific promotion and price process statevariable vector, Zt, for each model. Note that the inventory variable in our state spacevector is a discrete variable measured as the number of units of the product in question.For the FOM and PHM specifications, Zt also consists of discrete variables, with theprevious period’s promotion indicator serving as Zt for the FOM model. For the PHMspecification, Zt consists of a vector of Tjt, the discrete time (in weeks) since theprevious promotion for all brands (including competitor brands). However, for theEIK price process, Zt consists of brand prices from the previous period. Even thoughprice is a continuous variable, our approximation procedure for the value function usesonly a discrete number of price points as described below.

For all specifications, we approximate the value function by first calculating it atselected Chebychev nodes of our state space vector, (Invit, Zt).

8 We use five Chebychevnodes per dimension of our state space resulting in 15625 grid points (56=15625).9 Wethen use the polynomial interpolation method (Keane and Wolpin 1994) to approximatethe value at the other state-space points in order to perform the backward recursionsolution process. The polynomial interpolation significantly reduces the computationburden and allows us to solve for the endogenous consumption in a closed form (seeAppendix D for details on polynomial interpolation). For computational ease, wedivide households into groups based on observed quantity choices. For example, thequantity sets chosen for the five consumer groups in the canned tuna category are 1–3,1–4, 1–5, 1–6, and 1–7 cans respectively. We simulate the starting inventory ofhouseholds at the start of the data set using a procedure similar to Erdem et al. 2003.

6 Even though a continuous distribution for heterogeneity increases the computation burden, we use thisformulation for ease of comparison of the mean promotion coefficient across models. Sun et al. (2003) andSun (2005) are examples of papers in the literature that use a continuous distribution for heterogeneity. In ourapplication, we use 30 draws to simulate the likelihood function. Moreover, we use C++ to solve the dynamicprogramming problem to speed up the computation. Even with this improvement, model estimation may takeup to 2 weeks for each model on a modern computer.7 The number of periods we backsolve ranges from a little less than 100 to around 600, depending on themodel parameters.8 For Tjt, we assume the state space to consist of integers from 0 to T0, where T0 is the maximum number ofinter-promotion weeks observed in the data for brand j. We get similar results when T0 is 1.5 times themaximum inter-promotion interval observed in the data.9 We have done sensitivity analysis with 7, 6 and 5 nodes for each dimension with canned tuna data and mayodata. The results are robust. The sensitivity analysis results are reported in Appendix F.


Specifically we assume that households start with zero inventory 100 weeks prior to thestart of the data set. We then simulate the households’ purchases and consumption forthe 100 weeks to arrive at an expected starting inventory.

2.4 Discussion of the alternative specifications for the promotion and price process

We begin by commenting on some identification issues. As noted in the introduction,households’ expectations on future promotion are not observed in our choice data andcannot be identified non-parametrically with such data (Magnac and Thesmar 2002;Manski 2004). Thus, our emphasis is on comparing different specifications for rationalexpectations about promotions and prices. In particular, the PHM specification assumesa different information set for basing rational expectations, in the form of the time sincelast promotion for each brand, while the information set for the FOM and EIK priceprocess consists of the promotions and prices observed in the previous period. Ajustification for the PHM information set is provided by Mela et al. (1998) who findevidence suggesting that consumers lie in wait for promotions. Conditional on theinformation set assumptions, the first stage estimation in our study with the three priceexpectation specifications identifies the parameters that would generate expectationsclosest to the assumed rational expectations. Next, as noted in Section 1, a consumerwhose rational expectation corresponds to a FOM process (as with the FOM or the EIKprice process specification) rather than a PHM is likely to purchase less duringpromotion weeks as she anticipates a greater likelihood of another promotion in theweeks following the promotion. This difference in purchase behavior under differentexpectations will generally result in different values of the household utility parametersthat will rationalize the data. However, it is arguable that the most appropriatespecification of the first-stage consumer rational expectations model will lead to betterin-sample and out-of-sample fit. Thus, we use these fit measures to infer the mostappropriate specification for consumer expectations of price promotion.

Next, it is useful to summarize the pros and cons of the PHM specification, incomparison to the FOM and EIK specifications. As noted earlier, the PHMspecification, unlike the alternative specifications, allows for the incidence of pricepromotion to depend on promotion history extending beyond the previous period byemploying the time elapsed since the previous price promotion. This flexibility in thedependence of promotion incidence on promotion history comes with a few costs. First,the PHM model assumes an information set that extends further back in time incomparison to the other specifications as noted above. The benefit from this assumptionof a larger information set is best resolved by measuring how well the model performsin predicting consumer choices as noted above. A second cost of the flexible promotiondependence is computational as the state space for the PHM model is enlarged incomparison to the FOM model to incorporate the time since the last promotion.However, if the maximum inter-promotion interval observed in the data is not toohigh, the increase in state space may not represent a computational challenge.Nevertheless, in comparison to the EIK price process, the promotion-centered PHMand FOM specifications may offer a computational advantage because they do notinclude past prices in the state space. As price is a continuous state variable, we need todiscretize it in approximating a solution to the consumer’s dynamic programmingproblem under the EIK specification, and the granularity of the discretization is often


limited by computational considerations. In contrast, the state variable representing thetime since last promotion used by the PHM specification is a discrete variable,measured in our application as the number of weeks.

A second point of difference between the models is that the utility function with thePHM and FOM specifications (Eq. 1) incorporates the separate effects of the regularprice and promotion discount, while the utility function used in conjunction with theEIK price process specification only captures the extended (or net) price effects.Therefore, the PHM and FOM specification requires information on a regular price,which is often not explicitly specified in purchase data and needs to be inferred. Forexample, the data used in this study indicates if the price reported in a week is apromotion price or not. For a given promotional week, we assume that the price of theprevious no-promotion week closest to the promotion week as the regular price. Thus,the regular price needs to be inferred from the data which may be considered adrawback of the PHM and FOM specifications in comparison to the EIK specifications.

A third related point of difference between the specifications is that the PHM andFOM specifications assume that the promotion depth, conditional on a promotion,follows a normal distribution. In contrast, the EIK specification models the promotedprice through an autoregressive process.

As a final point of comparison, both the PHM and EIK specifications allow thepromotion or price process to depend on past competitive promotions, albeit in differentways. Specifically, the PHM specification includes the time since the last promotion ofcompetitors in a covariate function in the model, while the EIK specification models theincidence of price change and price as a function of the average of competitors’ pricesin the previous period.

3 Estimation results

3.1 Data

We estimate the dynamic structural choice models in four product categories, cannedtuna, mayo, peanut butter and tomato soup. We analyze the canned tuna category usingA.C. Nielsen’s household-level scanner panel data from Sioux Falls, SD. The dataspans 100 weeks from 1986 to 1987. We use the first 50 weeks of data for estimationand the following 50 weeks of data for validation. The store level data includes regularprice, discount depth and promotion indicator for the week. The regular price iscomputed as the weekly price before a promotion-related dip in the price. We analyzemayo, peanut butter and tomato soup categories using the IRI Academic Data Set(Bronnenberg et al. 2008).10 The data for our analysis spans 104 weeks over 2004–2005. We use the first 52 weeks data for estimation and the following 52 weeks data forvalidation. The IRI store level data include a promotion indicator for the week and a net(promoted) price for each product. For the weeks when there is a promotion, we assumethe price of the previous no-promotion week closest to this week as the regular price.

10 All estimates and analyses in this paper based on Information Resources, Inc. data are by the authors andnot by Information Resources, Inc.


In all four categories, we consider only the leading brand-sizes in order to reduce thecomputational burden. The brand-sizes included in our analysis account for 83.7 % to92.1 % of the market share in their respective categories. For all these categories, we onlyinclude households who make a minimum of four purchases and exclusively shop in asingle store or store chain. We estimate our models on all purchases of the selected brand-sizes by these households (cf. Sun et al. 2003; Erdem et al. 2003). Figure 1(a) through (d)plot the time series of price promotion incidence and promotion discount depth for thefour categories. Table 1 reports descriptive statistics for the four categories together withthe number of households and observations included in our analysis. As already noted, thefour categories included in this study are quite different in terms of the average intervalbetween promotions: tuna has some of the highest average inter-promotion times whilepeanut butter has some of the lowest. Further, the highest standard deviations in the inter-promotion intervals for particular brands are seen in the tomato soup category.

3.2 Analysis with simulated data

To illustrate the potential bias from mis-specifying the model for consumer’s promotionexpectations, we compare the estimation results of the three competing dynamicstructural models of consumer’s choice with simulated data. These models, namedChoice-PHM, Choice-FOM and Choice-EIK use respectively a PHM, an FOMspecification and an EIK price process specification for the promotion process in thefirst-stage model. We first generate promotion data for a product category with threebrands over a 150 week period using the PHM with a known set of parameters. Theaverage inter-promotion interval is 7.14 weeks for all three brands. We then generate thebrand and quantity choices (one or two units) for 1,000 households, assuming thatconsumers have rational expectations consistent with the true PHM promotion process.We call this data the PHM simulation data. We then generate another similar set ofpromotion, brand and quantity choice data using the FOM specification. We refer to thesecond simulated dataset as FOM simulation data. To facilitate model comparisons and tominimize other influences on the results, we assume that the consumers are homogeneousand have fixed consumption rate per period in the simulation. Moreover, we assume thatthe consumption saturation rate in Eq. (1) is zero for all consumers (γi=0).

Because the Choice-EIK model does not separate the regular price and promotioneffects as the Choice-PHM and the Choice-FOM models do, we cannot directlycompare the promotion and price coefficients across the three models. Therefore,for the three models, we compare promotion elasticities of brand 1 in a randomlychosen week. To calculate the promotion elasticities, we assume that there is a pricereduction in the randomly selected week equal to the average discount observed in thedata (see Appendix C for details of the elasticity calculation procedure).

Table 2 reports the estimation results for the simulated data. The results show that theChoice-FOMmodel estimated on the PHM simulation data overestimates the promotiondiscount parameter and the promotion elasticity. This indicates that significant bias isinduced by using the FOM specification for rational expectations if the consumer’srational expectations are in actuality consistent with the true PHM process. Thisoverestimation of promotion elasticity with the Choice-FOM is consistent with theintuitive explanation provided in the introduction. Specifically, if households’ rationalexpectations of promotions conform to the PHM rather than an FOM specification, they


Tab

le2

Estim

ationresults

with

simulationdata

Parameters

PHM

simulationdata

FOM

simulationdata

Truevalue

Choice-PH

Mmodel

Choice-FO

Mmodel

Choice-EIK

model

Truevalue

Choice-PH

Mmodel

Choice-FO

Mmodel

Choice-EIK

model

Brand

11.10

1.06(0.07)

1.15(0.07)

1.14(0.06)

1.10

1.07(0.07)

1.13(0.07)

1.11(0.06)

Brand

21.20

1.18(0.07)

1.17(0.07)

1.25(0.06)

1.20

1.16(0.07)

1.17(0.07)

1.23(0.06)

Brand

31.30

1.29(0.06)

1.39(0.07)

1.46(0.05)***

1.30

1.29(0.07)

1.29(0.07)

1.38(0.06)

Regular

Price

−3.00

−3.11(0.09)

−3.22(0.09)**

−3.00

−2.94(0.09)

−3.09(0.09)

Prom

.discount

8.00

7.92(0.10)

8.31(0.09)***

8.00

7.87(0.10)

8.04(0.11)

ExtendedPrice

−5.64(0.04)

−5.17(0.07)

Inventory

−0.50

−0.50(0.01)

−0.53(0.01)***

−0.43(0.04)**

−0.50

−0.47(0.05)

−0.52(0.05)

−0.44(0.04)

Totalprom

otionelasticity

a4.38

4.27(0.15)

4.81(0.14)***

2.99(0.07)***

4.47

4.33(0.13)

4.58(0.13)

2.80(0.06)***

Decom

posedelasticities

Purchase

incidence

0.39

0.36(0.03)

0.45(0.04)

0.35(0.02)*

0.41

0.40(0.04)

0.46(0.03)

0.35(0.02)***

8.90

%8.43

%9.34

%11.55%

9.17

%9.24

%10.62%

8.08

%

Conditio

nalbrandchoice

1.51

1.47(0.09)

1.60(0.10)

0.98(0.04)***

1.54

1.44(0.07)

1.49(0.07)

0.97(0.03)***

34.42%

34.43%

33.32%

32.80%

34.45%

33.26%

34.41%

22.40%

Conditio

nalp

urchasequantity

2.49

2.44(0.14)

2.76(0.08)***

1.66(0.04)***

2.52

2.49(0.08)

2.63(0.08)

1.48(0.03)***

56.71%

57.14%

57.40%

55.66%

56.38%

57.51%

60.74%

34.18%

-LL

67861.4

68643.5

69143.8

67979.4

67581.3

68924.8

Standard

errorsin

brackets

***significantly

differentfrom

true

valueatp=0.01

**significantly

differentfrom

true

valueatp=0.05

*significantly

differentfrom

true

valueatp=0.10

aProm

otionElasticity

ofbrand1in

Week7


would purchase higher quantities during a promotion than would be predicted by a FOMspecification. The rationale is that in the weeks following a price promotion, householdsexpect the promotion probability to be lower under a PHM specification for rationalexpectations than with an FOM specification. Consequently, a model estimated with anFOM specification for rational expectation has to impute higher values to the estimatedpromotion parameter in order to explain the greater number of purchases observedduring promotions, as would occur under a PHM model of rational expectations. Toconfirm this intuition, we compute the optimal purchase quantities for the Choice-PHMmodel and the Choice-FOM model for the PHM simulation data assuming identicalparameters for the second-stage consumer model. Table 3 shows that the Choice-FOMmodel predicts lower sales during promotion weeks and higher sales during non-promotion weeks consistent with our expectation.

Table 2 also shows that the promotion elasticities are underestimated with theChoice-EIK model when the consumer’s rational expectations are consistent with thetrue PHM promotion process. There are two factors explaining the possible downwardestimation bias of the Choice-EIK model. First, as we discussed earlier, the EIK priceprocess uses an FOM process to model price change so that the estimated probability ofa price change is very similar to that of the FOM specification (Fig. 3). Thus, theChoice-EIK model can lead to an overestimation of promotion elasticities for the samereasons that a Choice-FOM model results in such an overestimation. Second, recall thatthe Choice-EIK model does not separate the regular price effect and promotiondiscount effect in its specification. Previous research (Mulhern and Leone 1991) hasshown that when an empirical model does not separate regular price and promotioneffects, the estimated price coefficient represents a weighted combination of the regularprice effect, which is usually negative, and the promotion discount effect, which isusually positive. Thus, the lack of separation of regular price and promotion effects inthe Choice-EIK model can depress promotion elasticities in comparison to the Choice-PHM and the Choice-FOM specifications that do separate these effects. Apparently, inthe simulation, this downward shift in promotion elasticities is stronger for the Choice-EIK model than the potential overestimation caused by use of a FOM process,explaining the lower promotion elasticities obtained with this model.

Table 2 also shows that with the FOM simulation data, the promotion elasticitiesestimated by the Choice-PHM model are lower than those estimated by the Choice-FOM model, but the difference is not statistically significant. This is because the PHMis flexible and can closely capture the FOM rational expectation process. Therefore,even if the consumer’s rational promotion expectation follows an FOM process, thebias induced by using the Choice-PHM model is likely to be small.

Table 3 Optimal household purchase quantities under different expectations with simulated data

Choice-PHM model Choice-FOM model

Total Household Unit Purchases 26039 25568

Household Unit Purchases during Promotion Weeks 11014 10087

Household Unit Purchases during Non-promotion Weeks 15025 15481


3.3 First-stage results: promotion incidence models and the EIK price process

For the alternative models of PHM, FOM and the EIK price process, Tables 6 and 7 reportthe maximum-likelihood estimates (MLE) of the parameters in the canned tuna and mayocategories. The parameter estimates for the other categories are reported in the Appendix E.Table 4 presents the first-stagemodel fit for the promotion incidence process in terms of theBayesian Information Criterion (BIC) in the four categories. Note that the logit function ofthe EIK price process models the price change incidence, which is similar to promotionincidence. Therefore, we can directly compare themodel fit of the logit function in the EIKprice process with the PHM and FOM specifications. The PHM specification performssignificantly better than both the FOM specification and the EIK price process according tothe BIC in all four categories.11 Among the three PHMmodels, the Weibull hazard modelfits the promotion data the best according to the BIC in the canned tuna and mayocategories. The log-logistic hazard model fits the best according to the BIC in the peanutbutter and tomato soup categories. We use the most appropriate PHM specification foreach category in estimating the dynamic structural model.

We now compare the model fit of the FOM specification and the EIK price process.Table 4 indicates that the EIK price process fits the data better than the FOMspecification in the canned tuna and mayo categories. However, the reverse is true inthe other two categories as seen from Table 4. As discussed in the introduction, the EIKprice process is essentially a FOM process. The advantage of using the EIK priceprocess rather than the simple FOM specification to model promotions and prices isthat the former takes into account competitive prices. However, this advantage is notevident in two of the four categories according to the BIC.

Figure 3 plots the estimated hazards of the Weibull PHM model specification, theestimated FOM model probability of π0=Pr(promt+1=1|promt=0) and the estimatedprice change probability (changing from regular price to lower price only) of the EIKprice process for SKW/SKO in the canned tuna category. The Weibull PHM modelindicates a monotonically increasing shape for the baseline hazard. As discussed earlier,the FOM specification assumes that the promotion probabilities remain constantbetween promotions. The EIK price process specification has a very similar pattern

11 Even though PHM may be the model that fits the promotion process best, using the FOM to modelconsumer expectations of promotion would not be a misspecification if consumers’ information sets would bemore consistent with the assumptions of an FOM specification.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Prom

otio

n Pr

obab

ility

Weeks

Promotion Indicator

PHM(Weibull)

FOM

EIK Price Process

Fig. 3 Estimated promotion incidence process in tuna data for StarKist


to the FOM specification except for a few dips between promotions. These dips are dueto the inclusion of competitors’ price covariates in the EIK price process specification.

Having compared the fit of the PHM, FOM and EIK price process specifications to thepromotion incidence data, we now compare in Table 5 the mean square error (MSE) of theextended price (regular price minus promotional discounts) predicted by these modelsfrom the actual values observed in the data. We calculate the predicted extended price in a

Table 4 Comparative fit of competing models with promotion incidence data

PHM FOM EIK price process

Weilbull Log-logistic Erlang-2

Canned tuna -LL 44.46 47.57 47.39 148.36 101.47

BIC 100.48 106.70 100.56 308.28 220.94

Mayo -LL 55.80 65.23 122.09 173.16 170.86

BIC 135.82 154.67 262.33 358.43 355.83

Peanut butter -LL 8.43 8.42 13.25 64.08 64.15

BIC 37.45 37.43 41.96 138.45 140.31

Tomato soup -LL 17.73 14.46 24.36 63.57 65.64

BIC 56.09 49.51 64.17 137.43 143.30

Table 5 Comparative fit of competing models with extended price data

Mean Squared Error (MSE)a of extended price

PHM FOM EIK

Canned tuna SKO/SKW 0.224 0.269 0.314

COSO/COSW 0.342 0.464 0.362

Sum 0.566 0.733 0.676

Mayo Hellmann’s 26.066 26.417 35.919

Kraft 13.374 13.39 27.211

Private label 23.136 23.434 25.868

Sum 62.576 63.241 88.998

Peanut butter Peter Pan 4.915 4.805 5.014

Skippy 6.524 6.586 5.933

J.M. Smucker 1.578 2.749 2.256

Sum 13.017 14.14 13.203

Tomato soup Campbell(0.67 oz.)

0.093 0.195 0.121

Campbell(2.69 oz.)

12.058 16.608 11.741

Progress(1.69 oz.)

15.301 15.075 15.592

Sum 27.452 31.878 27.454

aMSE ¼ ∑ E epriceð Þ−epricedata½ �2Numberof Observations


period by combining the model’s predicted probability of promotion (or price change) inthe period with the expected prices with and without promotion. In all four categories, wefind that the PHM specification has the smallest sum ofMSE across the brands suggestinga better fit for the PHM than the other two specifications. However, the EIK price processpredicts extended price slightly better than the PHM for Skippy brand peanut butter andCampbell brand 2.69 oz. tomato soup. As a result, the PHM predicts extended price onlymarginally better than the EIK price process in the peanut butter category and tomato soupcategory. This difference in first-stage fit across models and categories is also reflected inthe second-stage estimation results as we see in the following section.

3.4 Second-stage results: dynamic choice models

3.4.1 Model fit

We estimate three competing dynamic structural models of consumer’s purchaseincidence, brand choice, and quantity choice in the four categories. Tables 6 and 7report the estimation results of the three competing models in the tuna and mayocategories. The parameter estimates for the other two categories are reported inAppendix E. The parameters have the expected sign in all cases suggesting face validity.

Table 8 presents the log-likelihood values, BIC and the hit rate (percentage of brand-quantity choices correctly predicted) in both the calibration and holdout samples for thefour categories. We also include the z-test statistics from testing whether hit rates (andpromotion elasticities) are significantly different among the competing models. Basedon these fit measures, we conclude that the Choice-PHM model fits the data better andpredicts better than both the Choice-FOM model and the Choice-EIK model in all fourcategories. The Choice-PHM model fits significantly better than the competing modelsin the canned tuna and mayo categories, while its fit is only marginally better(insignificantly) in the peanut butter and tomato soup categories. The difference in fitacross categories may be explained by the average inter-promotion intervals in thecategories. In both canned tuna and mayo categories, most brand inter-promotionintervals are in a moderate range of 3 to 11 weeks which appears to be the scenariowhere using a PHM to represent consumers’ rational expectations may be particularlyadvantageous. On the other hand, peanut butter is a heavily promoted category withpromotions in almost half of the weeks so that the inter-promotion interval is short.Therefore, the advantage of using the PHM to represent consumers’ rationalexpectations does not appear to be significant in such a category with shorter inter-promotion intervals. In contrast, the tomato soup category has some very long inter-promotion times (see Fig. 1(d) and Table 1) and yet the Choice-PHM does not performsignificantly better than the other models. We conjecture that this result may be becauseof the relatively irregular promotions for brands in this category as indicated by thehigh variances for inter-promotion intervals for these brands. Presumably, under theseconditions, the information set assumptions of the FOM specification or the EIK priceprocess may be equally appropriate for representing consumers’ rational expectations.

We now comment on the fit comparison between the Choice-FOM and Choice-EIKmodels in the four categories. While the Choice-EIK model fits worse than the othercompeting models in the canned tuna and mayo categories according to the BIC and hitrate, it performs marginally better than Choice-FOM in the other two categories. The


Tab

le6

Estim

ationresults

intuna

category

Firststage

PHM

FOM

EIK

priceprocess

Weilbull

Log-logistic

Erlang-2

LogitFu

nctio

nAutoregressive

SKW/SKOa

γ0.13(0.00)***

0.15(0.01)***

0.08(0.01)***

π1

0.06(0.03)*

δ 01

1.53(0.27)***

β01

−0.21(0.04)***

α1.87(0.29)***

5.04(0.10)***

π0

0.14(0.0

2)***

δ 11

43.19(13.40)***

β11

−1.13(0.32)***

T cosw/coso

−0.16(0.70)

−0.06(0.77)

−0.20(0.75)

β21

1.10(0.42)**

COSO

/COSO

aγ

0.19(0.01)***

0.25(0.02)***

0.11(0.01)***

π1

0.05(0.02)*

δ 02

1.06(0.26)***

β01

−0.20(0.03)***

α2.04(0.23)***

2.52(0.51)***

π0

0.20(0.04)***

δ 12

38.78(12.43)**

β11

−1.31(0.26)***

Tskw/sko

−0.55(0.69)

−0.45(0.72)

−0.15(0.72)

δ 3−3

32.60(100.98)**

β21

1.53(0.36)***

#of

parameters

66

44

6

-LL(BIC)

44.46(100.48)

47.57(106.70)

47.39(100.56)

148.36(308.28)

101.47(220.94)

Second

stage

Choice-PH

Mmodel

Choice-FOM

model

Choice-EIK

model

Mean

Std.

Deviatio

nMean

Std.

Deviatio

nMean

Std.

Deviatio

n

SKW

0.43(0.08)***

0.87(0.48)*

0.53(0.03)***

0.56(0.14)***

0.54(0.18)***

0.76(0.37)**

SKO

0.14(0.06)***

0.58(0.07)***

0.22(0.08)***

0.24(0.35)*

0.28(0.01)***

0.37(0.62)

COSW

0.27(0.04)***

0.95(0.23)***

0.35(0.15)**

0.27(0.03)***

0.44(0.24)*

0.65(0.13)***

COSO

0.01(0.00)**

0.31(0.05)***

0.17(0.10)*

0.23(0.10)**

0.24(0.08)***

0.43(0.01)***

Riskaversion

−0.07(0.04)**

0.96(0.51)*

−0.06(0.03)**

1.00(0.34)***

−0.02(0.01)**

1.65(0.15)***

Regular

price

−3.98(0.11)***

0.03(0.01)***

−4.13(0.24)***

0.02(0.00)***

Prom

otion

discount

7.88(0.17)***

0.04(0.02)*

8.04(0.15)***

0.04(0.00)***

Extendedprice

−6.33(0.96)***

0.06(0.03)**

Inventory

−0.03(0.00)***

0.002(0.00)***

−0.03(0.00)***

0.004(0.00)***

−0.04(0.01)***

0.004(0.00)***

Hitrate

calib

ratio

n(H

oldout)

0.81(0.79)

0.77(0.74)

0.73(0.69)

-LL(BIC)

23853.02

(47783.03)

24150.93

(48378.85)

24964.61

(49996.59)


Tab

le6

(contin

ued)

Decom

posed

elasticities

cTo

tal

Purchase

Incidence

Brand

Choice

Purchase

Quantity

Total

Purchase

Incidence

Brand

Choice

Purchase

Quantity

Total

Purchase

Incidence

Brand

Choice

Purchase

Quantity

6.65

(0.11)

1.80(0.05)

27.07%

0.59(0.01)

8.87

%4.26(0.10)

64.06%

7.11

(0.09)

1.76(0.03)

24.75%

0.74(0.01)

10.41%

4.61(0.09)

64.83%

4.15

(0.42)

1.07(0.17)

25.78%

0.55(0.08)

13.25%

2.53(0.38)

60.96%

Standard

errorsin

brackets***:significantatp=0.01

**:significantatp=0.05

*:significantatp=0.10

aSinceSK

WandSK

Ohave

thesameprom

otionpattern,theysharethesamesetsof

estim

ates.T

hesameistrue

fortheCOSW

/COSO

pair

bdecomposedelasticity

percentage

inbrackets

cProm

otionElasticity

ofSK

Win

Week3


Tab

le7

Estim

ationresults

inmayocategory

Firststage

PHM

FOM

EIK

priceprocess

Weilbull

Log-logistic

Erlang-2

LogitFu

nctio

nAutoregressive

Hellm

ann’s

γ1.74(0.31)***

0.81(0.21)***

0.10(0.02)***

π1

0.38(0.30)

δ 01

−1.46(0.87)*

β01

0.45(0.17)**

α1.03(3.68)

1.31(0.26)***

π0

0.19(0.03)***

δ 11

−0.18(0.44)

β11

0.31(1.09)

T kraft

0.67(0.10)***

2.00(0.05)***

0.07

(0.05)

β21

0.60(0.66)

T Private

−0.84(0.10)***

1.47(0.05)***

0.11(0.08)

Kraft

γ0.34(0.07)***

0.53(0.04)***

0.57(0.07)***

π1

0.25(0.10)**

δ 02

−1.16(1.16)

β02

0.55(0.10)***

α0.96(4.54)

4.16(0.16)***

π0

0.23(0.05)***

δ 12

7.42(2.27)***

β12

−0.89(0.14)

T Hellmann’s

0.02(0.06)

0.60(0.05)***

1.03(0.06)***

β22

0.52(0.29)

T Private

0.01(0.10)

0.48(0.05)***

0.44(0.06)***

Private

γ14.60(0.53)***

2.77(1.29)**

0.54(0.23)**

π1

0.69(0.16)***

δ 03

5.05(1.48)***

β03

−0.22(0.41)

α0.53(0.08)***

1.31(1.32)

π0

0.30(0.09)***

δ 13

−7.99(2.74)***

β13

−0.72(0.53)

T Hellmann’s

−0.09(0.10)

−0.04(0.05)

−0.01(0.04)

δ 3−5

.59(1.79)***

β23

1.57(1.30)

T kraft

0.63(0.10)***

−0.05(0.05)

−0.06(0.04)

-LL(BIC)

55.80(135.82)

65.23(154.67)

122.09(262.33)

173.16(358.43)

170.86(355.83)

Second

stage

Choice-PH

Mmodel

Choice-FO

Mmodel

Choice-EIK

model

Mean

Std.

Deviatio

nMean

Std.D

eviatio

nMean

Std.

Deviatio

n

Hellm

ann’s

5.56(1.30)***

0.62(0.26)**

4.28(0.73)***

0.63(0.16)***

3.86(0.94)***

1.35(0.63)**

Kraft

0.15(0.03)***

1.85(0.91)**

0.09(0.02)***

1.13(0.26)***

0.02(0.00)***

0.53(0.15)***

Private

0.31(0.12)***

1.29(0.56)**

1.00(0.24)***

1.53(0.31)***

0.89(0.13)***

0.61(0.05)***

Riskaversion

−0.08(0.03)**

0.93(0.18)***

−0.03(0.01)**

0.89(0.53)

−0.08(0.01)***

0.61(0.53)

Regular

price

−1.39(0.16)***

0.97(0.51)*

−2.33(0.73)***

0.96(0.59)

Prom

otion

discount

1.64(0.05)***

0.53(0.91)

1.83(0.08)***

1.53(0.61)***

Extendedprice

−2.44(0.09)***

1.12(0.12)***

Inventory

−0.20(0.04)***

0.78(0.28)**

−0.23(0.07)***

0.78(0.06)***

−0.02(0.01)*

0.25(0.29)***


Tab

le7

(contin

ued)

Hitrate

calib

ratio

n(H

oldout)

0.82(0.79)

0.78(0.75)

0.72(0.69)

-LL(BIC)

5781.67(11698.23)

5843.97(11822.82)

5913.54(11942.70)

Decom

posed

elasticities

aTo

tal

Purchase

Incidence

Brand

Choice

Purchase

Quantity

Total

Purchase

Incidence

Brand

Choice

Purchase

Quantity

Total

Purchase

Incidence

Brand

Choice

Purchase

Quantity

7.59

(0.17)

3.10(0.107)

40.84%

0.01(0.00)

0.13

%4.48(0.14)

59.03%

8.78

(0.28)

3.52(0.15)

40.09%

0.08(0.01)

0.91

%5.18(0.23)

(59.00

%)

5.66

(0.16)

2.42(0.10)

42.76%

0.12(0.01)

2.12

%3.12(0.12)

55.12%

Standard

errorsin

brackets****:significantatp=0.01

**:significantatp=0.05

*:significantatp=0.10

aProm

otionElasticity

ofHellm

ann’sin

Week12

anddecomposedelasticity

percentage

inbrackets


reasons for the lack of decisive advantage to either specification are three-fold. First,recall that both the FOM specification and the EIK price process use essentially a FOMprocess to predict promotion or price change and thus parity in model fit may beexpected. Second, we noted earlier that although the EIK price process considerslagged competitive prices in predicting prices it does not fit the promotion incidence(price change) data better than the FOM in any category except canned tuna. Third,previous research has shown that estimating regular price and promotion discounteffects, as the FOM specification does but the EIK process does not, can improvemodel fit (Mulhern and Leone 1991).

In sum, we may infer from the results on model fit that it is advantageous to allowpromotion expectations to be contingent on the time elapsed since the previouspromotion in categories with moderate lengths of average inter-promotion times andwhere inter-promotion times are not too irregular. Therefore, in the remainder of thissection, with a view to focusing on the interesting results, we only compare theestimation results of the three models in the canned tuna and mayo categories as theyhave moderately long and not too irregular inter-promotion times.

Table 8 Promotion elasticities and model fit comparisons of structural dynamic choice models

Z statistic

Choice-PHMmodel

Choice-FOMmodel

Choice-EIKmodel

Choice PHM vs.Choice FOM

Choice PHM vs.Choice EIK

Canned tuna -LL 23853.02 24150.93 24964.61

BIC 47783.03 48378.85 49996.59

Hit Rate (Calibration) 0.81(0.01) 0.77(0.01) 0.73(0.02) 4.01*** 4.71***

Hit Rate (Holdout) 0.79(0.01) 0.74(0.01) 0.69(0.02) 4.74*** 6.01***

Promotion Elasticity 6.65(0.11) 7.11(0.09) 4.15(0.42) 3.24*** 17.59***

Mayo -LL 5781.67 5843.97 5913.54

BIC 11698.23 11822.82 11942.70

Hit Rate (Calibration) 0.82(0.01) 0.78(0.01) 0.72(0.01) 2.67*** 8.30***

Hit Rate (Holdout) 0.79(0.01) 0.75(0.01) 0.69(0.01) 2.72*** 8.82***

Promotion Elasticity 7.59(0.17) 8.78(0.28) 5.66(0.16) 3.63*** 5.89***

Peanut butter -LL 2604.46 2607.18 2606.48

BIC 5232.94 5238.38 5233.55

Hit Rate (Calibration) 0.83(0.09) 0.83(0.01) 0.83(0.01) 0.11 0.01

Hit Rate (Holdout) 0.75(0.08) 0.74(0.01) 0.75(0.01) 0.07 0.01

Promotion Elasticity 6.95(1.08) 7.65(0.13) 6.88(0.56) 0.64 0.06

Tomato soup -LL 2881.11 2887.15 2883.30

BIC 5786.25 5798.32 5787.19

Hit Rate (Calibration) 0.81(0.13) 0.80(0.16) 0.81(0.31) 0.00 0.00

Hit Rate (Holdout) 0.73(0.12) 0.73(0.15) 0.73(0.27) 0.03 0.00

Promotion Elasticity 1.69(0.22) 1.79(0.23) 1.26(0.41) 0.31 3.02***

Standard errors in brackets. We compute the standard error for hit rate and promotion elasitciities withbootstrap (Efron and Tibshirani 1993)

***(Z value >2.33) suggests 99% confident level that the difference are statistically significant,**(Z value >1.65)suggests 95 % confident level that the difference are statistically significant. *(Z value >1.29) suggests 90 %confident level that the difference are statistically significant


3.4.2 Estimates of promotion elasticities

For the three competing structural models, we compare promotion elasticities of the leadingbrand in each category in response to a promotion (price-change) in a randomly chosenweek. Tables 6 and 7 report the estimated promotion elasticities of StarKist (canned tuna)andHellmann’s (mayo) under the threemodels. In both categories, the estimated promotionelasticities are highest for the Choice-FOM and lowest for the Choice-EIKmodel. With theChoice-FOM model, the estimated promotion elasticity in canned tuna (mayo) category is7.11(8.78), which is 6.9 % (15.7 %) greater than that predicted by the Choice-PHMmodel.Moreover, Table 8 shows the differences between the elasticities estimated from these twomodels are significant in both canned tuna andmayo categories. Thus, estimated promotionelasticities are higher in both canned tuna andmayo categories when using the FOMmodelfor consumer promotion expectations. This overestimation of promotion elasticity isconsistent with the simulation analysis in Section 3.2. The intuition is that, with a FOMspecification of rational expectations, consumers buy more during non-promotion weeksand less during promotion weeks, as discussed in the Section 3.2. As a result, the estimationassigns higher values to the promotion parameters in order to explain an observed purchasepattern that may be more consistent with a PHM specification of rational expectations.Moreover, Tables 6 and 7 shows that the elasticity estimated from the Choice-EIK modelfor the canned tuna (mayo) category is 4.15 (5.66), which is 37.6 % (25.4 %) lower thanthat predicted by the Choice-PHMmodel and consistent with the simulation analysis.12 Theintuition is that for the Choice-EIKmodel, the potential overestimation of promotion effectsfrom using an FOM process is smaller than the decrease in estimated elasticity due to notseparating the effects of regular price and promotion.

The difference in overall promotion elasticities also translate to differences across themodels when we decompose the promotion elasticities into those attributable to purchaseincidence, conditional brand choice, and purchase quantity (see Tables 6 and 7). Wefollow the decomposition approach in Gupta (1988). See Appendix C for details. Tobetter appreciate the managerial significance of the difference in the elasticity estimates,consider the brand choice (equivalently, brand-switching) elasticity in the canned tunacategory which is estimated at 0.59 with the Choice-PHM model, at 0.74 with theChoice-FOM model, and at 0.55 with the Choice-EIK model. For a brand like StarKistwith a market share of about 56 %, the difference in brand switching elasticity estimatedusing the Choice-PHM and Choice-FOM models implies an estimated difference of(0.74–0.59)*56 or 8.40 % share points in the impact of a 1 % change in promotionfrequency. The corresponding difference in brand switching elasticity estimates betweenthe Choice-PHM and Choice-EIK models is (0.55–0.59)*56 or 2.24 % share points forthe effect of a 1 % change in promotion frequency.

4 Policy simulation and profit analysis

The structural model distinguishes itself from reduced-form models especially in termsof policy change analysis (Lucas 1976). We now explore the differences in policy

12 Table 8 shows that the differences between the elasticities estimated from these two models arestatistically significant.


implications that might arise when using a Choice-PHM model instead of a Choice-FOM model or a Choice-EIK model. For this purpose, we conduct a policy simulationusing the holdout sample in which we assume that managers change the promotionpolicy for the Chicken of the Sea (COSW/COSO) brand by increasing its promotionfrequency by approximately 30 %. Specifically, given that COSW/COSO is onpromotion for 7 weeks during a 50 week time period, we assume that COSW/COSOoffers a promotion in two additional randomly-selected weeks. We then re-estimate thefirst-stage promotion process parameters under the new policy for the different modelsto reflect rational expectations of the new policy. We then calculate the change in salesand profit of COSW/COSO over the 50-week period. We formulate the profit functionfor a given promotion schedule as:

π ¼Xt

salest� pricet

1þMark up=100ð Þ−Dscntt�promt

Pass through=100−marginal cost

� ð7Þ

In the above equation, salest is the quantity sold in week t, which is computed bysumming over the household panel, the purchase probability from Eq. (3), evaluated at themodel parameter estimates. To accommodate the estimated heterogeneity in the parameters,we simulate the purchase probability using the heterogeneity parameter estimates.Mark_upand Pass_through are the assumed retailer’s markup and pass-through rate in percent, andmarginal_cost is the manufacturer’s marginal cost. Note that the profit is the sum of theprofit from promotion and non-promotion weeks. For the policy simulation, we assume a20 % markup and 80 % pass through rate (see Dhar and Hoch 1996; Jedidi et al. 1999 forsimilar assumptions). We assume the marginal cost to be $0.28 per unit.

Table 9 presents the policy simulation results and compares it to the baseline valuesrealized under the existing promotion schedule as observed in the data. We observe thatthe Choice-FOM model predicts a greater sales gain (compared to the baseline) than theChoice-PHMmodel while the Choice-EIKmodel predicts a smaller sales gain. This resultis understandable because the Choice-FOMmodel imputes a greater promotion effect thanthe Choice-PHM model and the Choice-EIK model imputes a smaller promotion effectthan the Choice-PHM model. Interestingly, the greater sales gain with the Choice-FOMmodel is because of higher sales during promotion weeks and lower sales during non-promotion weeks. Thus, the Choice-FOM predictions imply greater purchase accelerationand stockpiling that diminish sales at regular prices.13 In contrast, the smaller sales gainwith the Choice-EIK model is because of lower purchase acceleration and stockpiling.

Naturally, the predicted profitability of the policy change by the three models isdifferent because profit margins with and without promotions are different. Table 9shows that the Choice-FOM model suggests that the policy change negatively impactsprofit (−1.59 %) while the Choice-PHM model suggests a positive impact (1.23 %).Thus, the use of the Choice-FOM model instead of the Choice-PHM model can lead tosub-optimal decisions on promotion policy. Moreover, the Choice-EIK model suggestsa greater profit improvement (3.56 %) than the Choice-PHM model does, perhapspainting an overly rosy picture of the impact of the policy change.

13 Note, however, that this result is because of the larger promotion parameter estimate with the Choice-FOMmodel. When the parameters of the structural model are held constant, an FOM specification for consumerexpectations leads to lower sales during promotion weeks in comparison to a PHM specification as discussedin the introduction.


5 Conclusions and managerial implications

In this study, we compare competing dynamic structural models of consumer choicethat use alternatively a PHM model, a FOM model or an EIK price process to representrational expectations of promotion and price processes. Our key findings using scannerpanel data from four categories are the following. First, the PHM specification fits thepromotion and price process of brands better than the FOM and EIK price processspecifications in all four categories. Thus, the PHM specification would more closelyrepresent consumers’ rational expectations although the PHM specification does relyon a different consumer information set. Second, we find that the structural model thatassumes a PHM model for consumers’ rational promotion expectations fits the databetter and has better predictive performance in all four product categories than ones thatuse a FOM specification or an EIK price process for consumers’ rational promotion andprice expectations. In particular, the structural model with the PHM expectation fitssignificantly better than the competing models in the categories with moderately longinter-promotion times. On the other hand, it fits marginally better than the competingmodels in categories with short or irregular inter-promotion times. Third, the estimatedpromotion elasticities are biased upwards (by as much as 15.7 %) for the structuraldynamic model that uses a FOM specification for rational expectations aboutpromotion incidence and are biased downwards (by as much as 37.6 %) for thestructural dynamic model that assumes an EIK price process specification.

The implication for managers from the above findings is that the model used forspecification of rational expectations about promotion can significantly affect estimatesof promotion effects and that the PHM model may be a more appropriate specificationfor such expectations. Fourth, in evaluating the effect of a 30 % increase in promotionfrequency of the COSW/COSO brand as a policy simulation exercise, the dynamicstructural model using a FOM expectation specification predicts higher sales gainswhile the dynamic structural model using the EIK price process predicts lower salesthan a model that uses a PHM model of rational promotion expectations. Further, the

Table 9 Policy simulation with 30 % increase in COSW/COSO (Tuna) promotion frequency

Baseline(Data)

Effect of 30 % increase in promotion frequency

Choice-PHMmodel

Choice-FOMmodel

Choice-EIKModel

Promotion weeks Sales (cans) 1760 2644 2821 2326

Profit ($) 138 273 291 240

Non-promotion weeks Sales (cans) 1305 1116 1007 1261

Profit ($) 494 367 330 414

Total over 50 weeks Sales (cans) 3065 3761 3827 3587

% Change 22.69 % 24.87 % 17.04 %

Profit ($) 632 640 622 654

% Change 1.23 %** −1.59 %** 3.56 %*

**** significant at p=0.01**: significant at p=0.05*: significant at p=0.10

Approximate standard error is calculated using delta method


model with a FOM specification and the model with a PHM specification predictopposite impacts on profit from the policy change. This result emphasizes again that thespecification of rational expectations about future promotions can have a significanteffect on the profitability assessments from promotion policy changes.

Although the PHM specification does well in the four categories we study, it ispossible that the competing specifications are better in other settings. Therefore, it isbest that researchers study the most appropriate promotion and price process modelbefore incorporating it in a consumer dynamic choice model. Moreover, in categoriessuch as peanut butter and tomato soup, where there are extended periods of promotionor no promotion as seen in Fig. 1, it is important to recognize that even the PHMspecification may be far from a good model for the promotion process. The modelingof the promotion strategies in these categories could be an interesting area for furtherresearch. Finally, it might be useful to model the firms’ strategies that are a source of theobserved promotion process to provide a more structural explanation of the observedpromotion process. This area is also a worthwhile direction for future research.

Acknowledgment The authors acknowledge several useful comments from the Editor and anonymousreviewers as well as from Pradeep Chintagunta.

Appendix A: Details of derivation of utility function

For each week t, we define household i’s utility function Uit as a function of itsconsumption, Cijt, of each brand-size j, j=1, 2, J, and that of a composite of outsidegoods, Oit, as follows:

Uit ¼Xj¼1

J


2� �þ α1iOit ðA:1:Þ

In Eq. (A.1.), ψij denotes the household i’s consumption benefit associated with aunit of brand j, γi is the risk aversion parameter and α1i measures the consumptionbenefit associated with outside goods. Given household i’s income of yit in week t, werequire that weekly household expenditures satisfy the following budget constraint:

yit ¼Xj¼1

J �rpjt−prjtDjt

�dijt

�Qijt þ Oit þ βInvit ðA:2:Þ

The right-hand side of Eq. (A.2.) includes inventory holding costs as well asexpenditures on the focal category and outside goods with the price of outside goodsnormalized to one. rpjt is the regular price of brand j in week t, prjt is an indicatorvariable which equals one if brand j is on promotion in week t and zero otherwise,Djt isthe promotion discount,. The variable, dijt, is a choice indicator which equals one ifhousehold i chooses brand j at time t, Qijt is the quantity bought if households choosebrand j, Invit is the simple average of category inventories at the beginning and end ofweek t, and β denotes unit holding cost. For household i, if Invijt−1 denotes theinventory of brand j at the beginning of week t−1 at and if Cijt−1 is the consumption


of brand j in week t−1, the inventory of brand j at the beginning of week t (end of weekt−1) is given by

Invijt ¼ Invijt−1 þ Qijt−1−Cijt−1 ðA:3:Þ

Since Invit is the average of category inventories at the beginning of and at the endof week t, we obtain

Invit ¼Xj¼1

J

Invijt þ Qit−0:5Xj

Cijt: ðA:4:Þ

Combining Eq. (A.1.) and (A.2.), we obtain the following expression for ahousehold i’s utility function contingent on choosing brand j and quantity Q in week t.:

UijQt ¼Xj¼1

J


2� �

−α1irpjtdijtQijt þ α2iprjtDjtdijtQijt þ α3iInvit þ α4iyit ðA:5:Þ

In the above equation, α3i=-α1iβι, where α3i now represents the household utility frominventory. In order to allow different household responses to regular price and promotion,we modify the above equation so that the promotion response parameter is different fromthe price response parameter (cf. Guadagni and Little 1983; Sun et al. 2003). Morespecifically, we denote the price parameter to beα1i and the promotion response parameterto be α2i Finally, we absorb the negative signs before the coefficients into the coefficientsthemselves, where applicable. Thus, we have the following revised form of UitjQ:

UijQt ¼Xj¼1

J


2� �

þ α1irpjtdijtQijt þ α2iprjtDjtdijtQijt þ α3iInvit þ α4iyit ðA:6:Þ

Appendix B: Details of the PHM model

In this section, we discuss the PHMmodel specification and estimation details. We startby discussing four specifications of the baseline hazard, the Exponential, Erlang-2,Weibull, and log-logistic hazards that have been widely used in the marketing literature(see e.g. Seetharaman and Chintagunta 2003; Saha and Hilton 1997).

Erlang-2

h Tð Þ ¼ γ2T= 1þ γTð Þ

In the above equation, γ>0 and T is the time elapsed since the brand’s previouspromotion. The Erlang-2 hazard has a monotonically increasing shape.

Weibull

h Tð Þ ¼ γα�Tα−1

In the above equation, γ,α>0. The Weibull hazard is the most popularbaseline hazard specification in the marketing hazard model literature


because of its flexibility. It can be flat or monotonically increasing ordecreasing. .

Log-logistic

h Tð Þ ¼ γα γTð Þα−1= 1þ γTð Þα½ �

In the above equation, α, γ>0. The Log-logistic hazard can be monotonicallyincreasing or inverted U-shaped.

The covariate function of the PHM model is as follows:

ψ j X tj

� � ¼ eX tjβ j

This exponential covariate function ensures that the hazard function is non-negative.Thus, the standard PHM for our formulation is

h T ;X tj

� � ¼ h Tð ÞeX tjβ j

From the definition of a hazard model, we can also write the hazard function as

h T ;X tj

� � ¼ f T ;X tj

� �1−F T ;X tj

� � ðB:1:Þ

In the above equation, F(T,Xtj) stands for the cdf and f(T,Xtj) stands for the pdfcorresponding to the hazard function. Rearranging Eq. (B.1.) yields

f T ;X tj

� � ¼ h Tð Þ�eX tjβ j�S T ;X tj

� �where S(T,Xtj)=1−F(T,Xtj) is the survivor function which indicates the probability ofbrand j “surviving” a promotion in week t. We can rewrite Eq. (B.1.) as

dF T ;X tj

� �1−F T ;X tj

� � ¼ h Tð Þ�eX tjβ j�dT

The above first-order differential equation can be solved as

Z F

0

dF T ;X tj

� �1−F T ;X tj

� � ¼ Z T

0h uð Þ�eX tjβ j�du

Solving the above equation yields

S j T ;X tj

� � ¼ e∫T0 hj uð Þ�eX ujβ j�dT ðB:2:Þ

f T ;X tj

� � ¼ h Tð Þ�eX tjβe−∫T0 hj uð ÞeXβdu


To estimate the parameters of the hazard model at the individual brand level withright censoring, we specify the following likelihood function:

L ¼ ∏i

�f i T i−Ti−1;X j

� ��νi� S Ti−Ti−1;X j

� �� 1−νiwhere vi is the right censoring indicator. (T1,T2…Tn)_ stands for the calendar timeassociated with promotion of brand j.

Since promotions occur at discrete points in time, we model the household’sexpectation of promotion timing in a discrete fashion. We can rewrite Eq. (B.2.) as

S T ;X tj

� � ¼ e−XT

u¼1eXujβij

R uu−2hj wð Þdw

where u is discrete time, measured in weeks. The household’s expectation of promotionin discrete time t is then given by

Pr T ;X tj

� � ¼ S j T ;X tj

� �S j T−1;XT−1; j� � ¼ 1−e−e

X tjβij ∫TT−1hj uð Þdu ðB:3:Þ

In the above equation, Pr(T, Xtj) is the promotion probability (also called thediscrete-time hazard) in week t with covariates Xtj. We use maximum likelihoodmethod to estimate the discrete-time PHM parameters. The likelihood function is,

L ¼ ∏T¼1

ΓPr T ;XTj

� �δTj� 1−Pr T ;XTj

� �� 1−δTj ðB:4:Þ

where δtj is an indicator variable that equals to 1 if there is a price promotion for brand jin week t and 0 otherwise, Pr(T, Xtj) is the consumer’s anticipated promotionprobability of brand j in week t (Eq. B.3). Considering that promotion changes everyWednesday in the data set, we only consider discrete time PHM in this study.

Appendix C: Calculation and decomposition of promotion elasticity

We numerically calculate the promotion elasticity as the percentage change in expectedpurchase quantity change with and without promotion for household i for brand-size jin week t:

ηB j¼

E Qjti prijt ¼ 1� �

−E Qjti prijt ¼ 0� �

E Qjti prijt ¼ 0� �

where E Qjti prijt ¼ x

� �¼ ∑

Qq�Pr dijt ¼ 1;Qijt ¼ q prijt

¼ x� �

.

Following Gupta (1988), we decompose promotion elasticity into conditionalpurchase quantity elasticity, ηijt

QE conditional brand choice ηijtBE elasticity and


purchase incidence elasticity ηijtIE. Let j=0 refer to “no purchase” in the focal

category so that di0t=1 if household i makes no purchases in the category inweek t. The, we have

ηijt ¼ ηijtQE þ ηijt

BE þ ηijtIE

ηijtQE ¼

E Qjti dijt ¼ 1; prijt ¼ 1� �

−E Qjti dijt ¼ 1prijt ¼ 0� �

E Qjti dijt ¼ 1prijt ¼ 0� �

ηijtBE ¼ Pr dijt ¼ 1 di0tj ¼ 0; prijt ¼ 1

� �−Pr dijt ¼ 1 di0tj ¼ 0; prijt ¼ 0� �

Pr dijt ¼ 1 di0tj ¼ 0; prijt ¼ 0� �

ηijtIE ¼ Pr di0t ¼ 0 prijt ¼ 1

� �−Pr di0t ¼ 0 prijt ¼ 0

� �Pr di0t ¼ 0 prijt ¼ 0

� �Note that we derive the purchase incidence probability, conditional brand choice

probability and expected quantity purchased from the joint probability of brand andquantity choice (Eq. 3). In contrast, Gupta (1988) calculates these probabilities fromseparately modeled purchase incidence, conditional brand choice and quantitydecisions.

Appendix D: Polynomial interpolation and optimal consumption decision

To solve for the dynamic programming problem, we first select a finite numberof grid points in the state space and compute the value function Vit(Invit,Zt) atthe selected grid points. We then use polynomial interpolation to find theapproximate value of Vit(Invit,Zt) in any other grid point.

We specify the polynomial for Vit(Invit,Zt) as a function of Invit, a vector ofinventory of all brands at the end of week t and Zt, a vector across brands ofstate variables that determine the probability of promotion and the price inweek t+1. Recall that Zjt is the number of weeks elapsed since the brand j’sprevious promotion as of week t for the Choice-PHM model, the dummyvariable of whether there brand j’s was promoted in the previous period forthe Choice-FOM model and extended price of brand j in the previous periodfor the Choice-EIK model. We specify

V it Invit; Ztð Þ ¼ ∑k∑ j ∂1jk Inv

kijt þ ∂2jkZ

kjt

h iþ ∑ j∑ j0∑k∑k 0∂

3jj0kk 0 Inv

kijt � Zk 0

j0t ðD:1:Þ


In the above equation, ∂ are the parameters to be estimated with an OLS regression.In order to solve for a closed form consumption decision, we assume k=1,2 and k '=1,2.Given Eqs. 10(1), (2) and (A.4.) and (D.1.), we can rewrite the value function ofchoosing brand j with quantity Qijt as

vijQt dijt;Qijt Invit;j prt;Dt Ztjð Þ� � ¼Xj¼1

J


2� �þ α1irpjtdijtQijt

þ α2iprjtDjtdijtQþ α3i

Xj¼1

J

Invijt þ Qit−0:5Xj

Cijt

!

þ δXl¼0;1:

Prob prjtþ1 ¼ l; Ztþ1

� ��V itþ1 Invitþ1; prjtþ1;E Dtþ1ð Þ Ztþ1j� ��

To solve for the optimal consumption decision, we take the first order condition ofVijQt(dijt,Qijt|Invit,(prt,Dt|Zt)) with respect to Cijt. This gives us a closed-form solutionfor Cijt.

Appendix E: Complete estimation results for peanut butter and tomato soupcategories

Tables 10 and 11 report the first stage estimation results in the peanut butter and tomatosoup categories for the alternative models of PHM, FOM and the EIK price process.The PHM specification performs significantly better than both the FOM specificationand the EIK price process according to the BIC in both categories.

Tables 12 and 13 report the second stage dynamic structural modelestimation results in the peanut butter and tomato soup categories. As wediscussed in the paper, the advantage of using the PHM to represent consumerexpectations does not appear to be significant in the peanut butter categorywhich has shorter inter-promotion intervals and in the tomato soup categorywhich has some very long inter-promotion times.

Appendix F: Robustness check: number of grid points per state space dimension

Table 14 report the estimation results with 7, 6 and 5 nodes or grid points foreach dimension of the state variables. We compare the results of threecompeting structural models using canned tuna data. To reduce the computationburden, we assume consumers are homogenous. Moreover, since SKW andSKO have the same promotion pattern and regular prices, we treat these twobrand-sizes as one brand choice for consumers in the estimation. We do thesame for the COSW/COSO pair. The reason for these simplifications is toreduce the number of state space dimensions and make the computation withlarge number of nodes per dimension possible. We find that the estimatedparameters are robust to the number of nodes selected.


Tab

le10

Estim

ationresults

ofthePH

M,F

OM

andEIK

priceprocessmodelsin

peanut

buttercategory

Brand

Parameters

PHM

FOM

EIK

priceprocess

Weilbull

Log-logistic

Erlang-2

Logitfunctio

nAutoregressive

Peter

pan

γ0.67(0.24)***

1.23(0.82)

0.74(0.35)**

π1

0.78(0.28)***

δ 01

1.19(0.48)**

β01

0.44(0.17)**

α3.30(0.38)***

1.42(0.56)**

π0

0.18(0.06)***

δ 11

1.36(0.45)***

β11

−0.77(0.26)***

T Skippy

0.85(0.14)***

−0.34(0.14)**

−0.04(0.14)

β21

0.18(0.66)

T J.M

.Smucker

−1.24(0.14)***

0.80(0.14)***

0.40(0.14)***

Skippy

γ1.20(0.90)

0.66(0.22)**

3.09(0.37)***

π1

0.79(0.14)***

δ 02

1.17(0.51)**

β01

0.47(0.22)**

α0.71(0.28)***

4.49(0.41)***

π0

0.19(0.05)***

δ 12

−0.02(0.92)

β11

−0.76(0.38)**

T Peter

Pan

0.33(0.14)**

0.54(0.14)***

4.83(0.14)***

β21

−0.10(1.02)

T J.M

.Smucker

−0.34(0.14)**

−1.03(0.14)***

6.53(0.14)***

J.M.Smucker

γ0.06*0.01)***

0.06(0.01)***

0.02(0.00)***

π1

0.00(0.00)

δ 03

0.52(1.31)

β03

0.47(0.22)*

α5.29(1.31)***

6.11(1.55)***

π0

0.09(0.02)***

δ 13

9.79(2.96)***

β13

−0.76(0.38)*

T Skippy

0.25(0.16)

0.23(0.18)

0.18(0.12)

β23

−0.10(1.02)

T Peter

Pan

0.23(0.30)

0.21(0.30)

0.13(0.17)

δ 30.77(2.91)

nNum

berof

Parameters

1212

96

7

-LL

8.43

8.42

13.25

64.08

64.15

BIC

37.45

37.43

41.96

138.45

140.31

Standard

errorsin

brackets

****significantatp=0.01

**significantatp=0.05

*significantatp=0.10


Tab

le11

Estim

ationresults

ofthePH

M,F

OM

andEIK

priceprocessmodelsin

tomatosoup

category

Brand

size

Parameters

PHM

FOM

EIK

priceprocess

Weilbull

Log-logistic

Erlang-2

Logitfunctio

nAutoregressive

Cam

pbell

(0.67oz.)

γ0.75(0.37)**

0.62(0.18)***

0.60(0.17)***

π1

0.43(0.77)

δ 01

0.42(3.65)

β01

−0.54(0.07)***

α7.38(0.51)***

2.00(0.40)***

π0

0.09(0.02)***

δ 11

−2.54(2.36)

β11

0.82(0.26)***

T Cam

pbell2.69

0.33(0.14)**

1.40(0.14)***

0.65(0.14)***

β21

0.23(0.20)

T Progresso

−0.69(0.14)***

0.17(0.14)

1.36(0.14)***

Cam

pbell

(2.69oz.)

γ1.00(142.71)

1.57(3.16)

0.65(0.21)***

π1

0.67(0.36)*

δ 02

0.92(0.41)**

β02

0.48(0.06)***

α0.41(0.06)***

0.85(−2.68)

π0

0.10(0.02)***

δ 12

−4.91(6.42)

β12

−0.06(0.17)*

T Cam

pbell0.67

0.00(0.14)

0.17(0.08)**

−0.36(0.14)***

β22

−0.63(0.28)

T Progresso

−0.04(0.14)

−0.22(0.14)

1.13(0.14)***

Progresso

(1.69oz.)

γ0.52(0.11)***

0.18(0.01)***

0.21(0.08)***

π1

0.30(0.24)

δ 03

7.72(5.51)

β03

0.37(0.05)***

α0.20(0.02)***

0.68(0.25)

π0

0.17(0.04)***

δ 13

4.54(1.71)***

β13

0.47(0.26)*

T Cam

pbell0.67

−1.15(0.14)***

−0.62(0.14)***

0.00(0.07)

β23

−2.01(0.71)***

T Cam

pbell2.69

0.70(0.14)***

0.77(0.14)***

−0.04(0.03)

δ 3−3

.19(3.05)

nNum

berof

Parameters

1212

96

7

-LL

17.73

14.46

24.36

63.57

65.64

BIC

56.09

49.51

64.17

137.43

143.30

Standard

errorsin

brackets





Tab

le12

Estim

ates,p

romotionelasticities

andmodelfitcomparisons

ofstructuraldynamicchoice

modelsfrom

peanut

butterdata

Choice-PHM

model

Choice-FOM

model

Choice-EIK

model

Mean

Std.

Deviatio

nMean

Std.

Deviatio

nMean

Std.

Deviatio

n

Peter

pan

0.42(0.09)***

0.85(0.09)***

0.42(0.08)***

0.89(0.19)***

0.37(0.05)***

0.90(0.19)***

Skippy

1.60(0.51)***

1.88(0.76)**

1.60(0.90)***

1.91(0.85)**

0.46(0.05)***

1.98(0.81)**

J.M.Smucker

0.23(0.05)

0.50(0.73)

0.23(0.02)***

0.58(0.78)

0.20(0.04)***

0.50(0.83)

Riskaversion

−0.002(0.00)***

1.46(0.09)***

−0.002(0.00)***

1.38(0.15)***

−0.002(0.00)***

1.39(0.15)***

Regular

price

−3.48(0.12)***

1.04(0.35)***

−3.48(0.24)***

1.05(0.38)**

Promotiondiscount

1.99(0.46)***

0.61(0.31)*

2.01(0.45)***

0.62(0.35)*

Extendedprice

−1.95(0.23)***

1.09(0.41)**

Inventory

−0.09(0.04)**

0.18(0.31)

−0.09(0.05)*

0.27(0.35)

−0.09(0.05)*

0.30(0.43)

Hitrate(Calibration)

0.83

0.83

0.83

Hitrate(H

oldout)

0.75

0.74

0.75

-LL

2604.46

2607.18

2606.48

BIC

5232.94

5238.38

5233.55

Promotionelasticity

b6.95(1.08)

7.65(0.13)

6.88(0.49)

Decom

posedelasticities

bPu

rchase

Incidence

Brand

Choice

PurchaseQuantity

Purchase

Incidence

Brand

Choice

Purchase

Quantity

Purchase

Incidence

Brand

Choice

Purchase

Quantity

0.31(0.07)

4.46

%2.64(0.61)

37.99%

4.00(0.62)

57.55%

0.30(0.00)

3.92

%2.93(0.07)

38.30%

4.41(0.11)

57.65%

0.38(0.05)

5.52

%2.41(0.29)

35.03%

4.08(0.48)

59.30%

Standard

errorsin

brackets




adecomposedelasticity

percentage

inbrackets

bProm

otionElasticity

ofPeterPanin

Week3


Tab

le13

Estim

ates,p

romotionelasticities

andmodelfitcomparisons

ofstructuraldynamicchoice

modelsfrom

tomatosoup

data

Choice-PHM

model

Choice-FOM

model

Choice-EIK

model

Mean

Std.

Deviatio

nMean

Std.

Deviatio

nMean

Std.

Deviatio

n

Cam

pbell(0.67

oz.)

3.28(0.04)***

1.64(0.39)***

2.89(1.06)**

1.61(0.44)***

3.28(0.20)***

1.47(0.64)**

Cam

pbell(2.69

oz.)

0.87(0.07)***

3.88(0.77)***

0.93(0.07)***

4.61(0.85)***

0.87(0.22)***

4.63(0.56)***

Progresso(1.69oz.)

0.16(0.09)*

3.67(0.38)***

0.15(0.09)

3.71(0.51)***

0.16(0.08)*

3.56(0.47)***

Riskaversion

−0.003(0.00)***

0.30(0.21)

−0.003(0.00)***

0.37(0.15)**

−0.003(0.00)***

0.38(0.01)***

Regular

price

−1.85(0.88)**

0.76(0.38)**

−1.89(0.78)**

0.22(0.17)

Promotiondiscount

2.06(0.40)***

0.94(0.21)***

2.15(0.40)***

0.66(0.34)*

Extendedprice

−1.82(0.82)**

0.53(0.31)

Inventory

−0.29(0.03)***

0.08(0.01)***

−0.31(0.03)***

0.03(0.01)***

−0.27(0.03)***

0.05(0.01)***

Hitrate(Calibration)

0.81

0.80

0.81

Hitrate(H

oldout)

0.73

0.73

0.73

-LL

2881.11

2887.15

2883.30

BIC

5786.25

5798.32

5787.19

Promotionelasticity

b1.69(0.22)

1.79(0.23)

1.26(0.41)

Decom

posedelasticities

bPu

rchase

Incidence

Brand

Choice

PurchaseQuantity

Purchase

Incidence

Brand

Choice

Purchase

Quantity

Purchase

Incidence

Brand

Choice

Purchase

Quantity

0.80(0.15)

47.34%

0.02(0.00)

1.18

%0.87(0.16)

51.48%

0.78(0.15)

43.58%

0.02(0.01)

1.12

%0.99(0.18)

55.31%

0.55(0.26

43.65%

0.02(0.02)

1.59

%0.70(0.31)

55.56%

Standard

errorsin

brackets




adecomposedelasticity

percentage

inbrackets

bProm

otionElasticity

ofCam

pbell(0.67

oz)in

Week4


Tab

le14

Sensitivity

analysisof

differentgridspoints

5nodesperdimension

6nodesperdimension

7nodesperdimension

Choice-PH

MModel

Choice-FOM

Model

Choice-EIK

Model

Choice-PH

MModel

Choice-FO

MModel

Choice-EIK

Model

Choice-PHM

Model

Choice-FO

MModel

Choice-EIK

Model

StarK

ist

0.02*(0.01)

0.03***(0.01)

0.04***(0.01)

0.02**

(0.01)

0.04***(0.01)

0.04***(0.01)

0.02**

(0.01)

0.04***(0.01)

0.02**

(0.01)

COS

0.01**

(0.00)

0.01**

(0.00)

0.01**

(0.00)

0.01***(0.00)

0.01**

(0.00)

0.01***(0.00)

0.01**

(0.00)

0.01***(0.00)

0.01***(0.00)

Riskaversion

−0.01***

0.00

−0.01***

0.00

−0.01***

0.00

−0.01***

0.00

−0.01***

0.00

−0.01***

0.00

−0.01***

0.00

−0.01***

0.00

−0.01***

0.00

Regular

price

−3.97***

(0.32)

−3.44***

(0.412)

−4.03***

(0.47)

−3.36***

(0.28)

−3.88***

(0.78)

−3.52***

(0.69)

Promotion

discount

7.79

(0.27)***

7.92***(0.12)

7.74***(0.14)

7.96***(0.76)

7.73***(0.79)

7.98***(0.68)

Extendedprice

−6.29***

(0.68)

−6.34***

(0.28)

−6.36***

(0.54)

Inventory

−0.03***

(0.00)

−0.04***

(0.01)

−0.04**(0.02)

−0.03***

(0.01)

−0.04**(0.02)

−0.04***

(0.01)

−0.03***

(0.01)

−0.04***

(0.01)

−0.04**(0.02)

-LL(BIC)

24464.58

(48941.69)

24778.72

(49569.98)

25046.71

(50082.97)

24447.85

(48908.23)

24734.81

(49457.08)

24965.80

(49942.05)

24382.81

(48753.08)

24717.10

(49444.64)

24956.97

(49905.35)

Elasticities

a6.58(0.35)

7.12(0.24)

4.12(0.77)

6.31(0.71)

7.18(1.20)

4.13(0.69)

6.30(1.38)

7.19(1.24)

4.14(0.99)

Standard

errorsin

brackets




aProm

otionElasticity

ofStarKistin

Week3


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