Abrand choice model with heterogeneous price-threshold parameters is used to investigate a three-regime

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1 Vol. 25, No. 4, July August 2006, pp issn eissn X informs doi /mksc INFORMS Researc Note Estimating Heterogeneous Price Tresolds Nobuiko Terui Graduate Scool of Economics and Management, Tooku University, Sendai , Japan, Wirawan Dony Daana Graduate Scool of Economics, Osaka University, Osaka , Japan, Abrand coice model wit eterogeneous price-tresold parameters is used to investigate a tree-regime piecewise-linear stocastic utility function. Te model is used to explore te relationsips between aspects of consumer price sensitivity and price tresolds using ierarcical Bayes modeling wit te Markov cain Monte Carlo (MCMC) metod. Tis study contributes to te modeling literature on discontinuous likelioods in coice models. Te empirical application using our scanner panel data set sows tat te reference effect and loss aversion are more marked after price tresolds are taken into eterogeneous price response models. Furtermore, loss aversion is attenuated by using price tresolds tan by an aggregate (omogeneity) model witout price tresolds. Key words: discontinuous likelioods; reference effect; price tresold; latitude of price acceptance; brand coice; Bayesian MCMC; eterogeneity; scanner panel data History: Tis paper was received November 11, 2004, and was wit te autors 5 monts for 2 revisions; processed by Greg M. Allenby. 1. Introduction Asymmetric price effects are an active area of researc in marketing. Beginning wit te work of Helson (1964) wo first motivated te usefulness of including reference prices in sales and coice models, researcers ave found asymmetric effects and te presence of price tresolds tat reflect te presence of discontinuities in consumer beavior. Te literature includes findings of asymmetric effect relative to te reference price (Kaneman and Tversky 1979), te disappearance of tese effects in models tat account for eterogeneity in price sensitivity (Cang et al. 1999, Bell and Lattin 2000), and models in wic price as no effect witin a price interval, i.e., a latitude of price acceptance (Gupta and Cooper 1992, Kalwani and Yim 1992, Kalyanaram and Little 1994, Han et al. 2001). However, models wit tresold effects yielding intervals of price acceptance ave yet to be estimated eterogeneously. Tis paper introduces a eterogeneous coice model wit reference prices, asymmetric responses, and price tresolds. It is caracterized as a piecewise linear form so tat consumers switc teir utility structure accordingly as determined by te relationsip between te sticker sock; i.e., te difference between retail price and reference price, and price tresolds. Heterogeneity is incorporated troug a random-effects specification tat allows determination of te relationsip between te tresold values and consumer caracteristics. MCMC algoritms are provided for estimation. Te model is illustrated using a scanner panel data set of instant-coffee purcases. Our model contributes to te modeling literature on discontinuous likelioods. Price tresolds generate unconventional likeliood functions and discreteness at te tresold values tat make it impossible to estimate wit gradient-based metods suc as maximum likeliood. As discussed by Gilbride and Allenby (2004) for models wit consideration sets, te Bayesian metod of data augmentation is ideally suited for andling discontinuities in te likeliood surface. Here, te tresolds are associated wit alternative likeliood functions tat result in an apportionment of ouseold data to alternative models. Tus, te model contributes to te marketing literature on structural eterogeneity (Yang and Allenby 2000) by allowing for abrupt canges among alternative models witin te unit of analysis. Our empirical application adds to te literature on te presence of reference prices and loss aversion tat initially began wit effects uncovered in scanner data, for example, as indicated by Winer (1986), Mayew and Winer (1992), and Putler (1992), and ten Cang et al. (1999) and Bell and Lattin (2000) sowed tat reference price effects disappear wen eterogeneity is incorporated. Our empirical application sows tat te reference effect and loss aversion return at least for te data used in tis study after price tresolds are taken into a eterogeneous model. In doing so, te 384

2 Terui and Daana: Estimating Heterogeneous Price Tresolds Marketing Science 25(4), pp , 2006 INFORMS 385 degree of loss aversion is attenuated relative to results obtained using te omogeneity model witout price tresolds. Te remainder of te paper is organized as follows. Section 2 presents our stocastic utility function and its consequent coice model te tresold Probit model and we develop a ierarcical Bayes modeling for it. Section 3 describes te application of scanner panel data to our model. Section 4 concludes tis paper. Te appendix explains details of te model estimation including te algoritm for ierarcical Bayes modeling via MCMC. Figure 1 (g) β A β (g) (g) β BL Model for Price Tresolds and Market Responses LPA Gain r 1 r 2 Loss 2. Te Model 2.1. Tresold Probit Model and Hierarcical Bayes Modeling To specify te utility function, we assume tat consumer s utility to brand j at time t of purcase, U jt, reflects a linear function of k kinds of explanatory variables. We also suppose tat consumer as a reference price RP jt for brand j, and two price tresolds r 1 and r 2 r 1 < 0 <r 2. Consequently, we define te tree regimes gain g, price acceptance a, loss l utility function to brand j as u g j u a j U jt = + Xa + Xg jt g + g jt if P jt RP jt r 1 ( g) U jt ( a) U jt jt a + a jt if r 1 <P jt RP jt r 2 + l jt if r 2 <P jt RP jt u l j + Xl jt l ( l) U jt were X i jt is te k-dimensional row vector of explanatory variables allocated to regime i according to te level of sticker sock P jt RP jt P jt : te retail price exposed to consumer ) at te occasion, and k-dimensional column vectors i, i = g a l, represent different market responses around te reference price. Finally, let i jt, i = g a l, respectively represent stocastic error components in te utility for eac regime, and tey are assumed to be independent across regimes. Equation (1) is capable of simultaneously reflecting te presence of reference prices (Helson 1964), asymmetric effects (Kaneman and Tversky 1979), and price tresolds (Serif et al. 1958). Asymmetries are present wen te coefficients in te different regions take on different values. Te price tresolds, r s s = 1 2, identify different regimes associated wit te data. Te model captures a latitude of price acceptance (LPA ereafter) L = r 1 r 2 were prices may not affect coice if price is sufficiently small. (1) Model (i) Model (ii) Model (iii) (l) β A (l) β (l) β BL Notes. Model (i): aggregate (omogeneous) reference price Probit model witout a tresold; Model (ii): two-regime eterogeneous reference price Probit model witout a tresold (Bell and Lattin 2000); Model (iii): treeregime eterogeneous reference price Probit model wit tresolds. Figure 1 displays our utility function. In order for te second regime of te utility function (1) to be caracterized as te LPA, and for r 1 and r 2 to be interpreted as price tresolds literally, i.e., for our proposed model to be recognized as a pricetresold model, we impose te restriction of insensitiveness on te price-response parameter in te LPA regime as a p N0a p 2, wic is an element of a. Under te assumption of consumer omogeneity, Kalyanaram and Little (1994) estimated symmetric LPA tat ad an insignificant price-response estimate over tis range. On te oter and, Han et al. (2001) find a significant response parameter estimate in te LPA. We note tat teir framework differs from ours particularly in te respect tat tey use te same data set trougout regimes. In contrast, our modeling metod allocates data into tree regimes. Following te standard multinomial-brand coice model, we assume tat consumer is observed to make a coice at te period t between m alternatives (c t = j. Tat coice is driven by te relative utility from te last brand y jt = U jt U mt among latent utilities U nt, n = 1 2m. We consider a panel of H ouseolds observed over T periods for eac ouseold. Conditional on te vector of tresold r = r 1 r 2 for consumer, te underlying latent structure at period t wen te latent utility induced by te coice belongs to te regime i, te so-called witin subject model for regime i is expressed in te form of m 1 dimensional multiple regression, y i t = Xi t i + i t i t N0i = 1H t= 1T i i= gal (2)

3 Terui and Daana: Estimating Heterogeneous Price Tresolds 386 Marketing Science 25(4), pp , 2006 INFORMS were y i t is te m 1 dimensional relative-utility vector, X i t is te m 1 k + m 1 explanatory variable matrix measured from te last brand, i is k + m 1 dimensional coefficient vector, i t is te m 1 dimensional stocastic-error vector, and we ave te relation T g + T a + T l = T. Consumers eterogeneity is formulated by way of a ierarcical regression model, as proposed by Rossi et al. (1996). Regarding price tresolds, we employ te model r 1 = Z r r 2 = Z r = 1H (3) were Z r is a vector of d kinds of ouseold specific variables. We assume tat r 1 < 0 <r 2 for identification and s N0s 2 for s = 1 2. We also set a ierarcical structure of between subjects model for regime i for te market-response parameter i = i Z + i iid i N ( 0V i ) = 1H i= gal (4) were Z contains anoter vector of d kinds of ouseold specific variables. In particular, we note tat price response a p in te LPA is assumed a priori to ave zero mean in (4). Following te utility function defined as (1), consumer s probability of coosing brand j is written as Pr { y g j =max( y g ) } 1 yg m 1 >0 if P jt RP jt r 1 Pr { y a j Prc =j = =max( y a ) } 1 ya m 1 >0R (5) if r 1 <P jt RP jt r 2 Pr { y l j =max( y l 1 yl if r 2 <P jt RP jt m 1 ) >0 } were Pry a j = maxya 1 ya m 1>0 R indicates te coice probability under te restriction on price response a p N0a p 2 in te LPA regime Discontinuous Likelioods for Tresolds and Teir Modeling Our model includes a tresold variable of sticker sock P jt RP jt in te model, and it induces discontinuity in te likelioods, as sown in Figure 1. Conditional on te value of r = r 1 r 2, te operation generates te latent utility wit tree different likeliood functions for consumer. Te independence assumption of stocastic errors across regimes generates te likeliood function of i i for consumer : { i 1/2 exp { ( 1 i 2 y t ) Xi t i i=g a l t R i r i 1( y i t )} } Xi t i were eac datum is assigned to one regime R i r, and it olds tat R g r R a r R l r = T. In turn, conditional on i i, we take te above as a function of r to compose te likeliood function of price tresolds. Under te assumption of independent coice beavior across consumers, we ave te conditional likeliood function of r by taking products over respective consumers as Lr I t X t i i { { H i 1/2 exp { ( 1 i 2 y t ) Xi t i =1 i=g a l t R i r i 1( y i t )} }} Xi t i (6) were I t means te personal observed coice data. Coupled wit te prior (3) expressed as ierarcical structure, we can apply te Metropolis-Hasting sampling algoritm for price tresolds to obtain conditional posterior r I t X t i i Z. Prior distributions and MCMC estimation procedures for tese ierarcical Bayes models are described in te appendix. Te proposed model is difficult to estimate wit conventional metods because te likeliood is not differentiable in r. Conventional maximum likeliood estimation collapses and classical asymptotic distribution teory is not operative on tis parameter, and it must terefore rely on an adaptive approac: conditioned on some specific value of r, estimation is conducted troug extensive use of goodness-of-fit criterion, like AIC, in te way of a grid searc to find te estimate tat minimizes te criterion. In te price-tresold literature, Kalyanaram and Little (1994) assume symmetric LPA by setting r 1 = r 2 = r (omogeneous), and tey estimate not te tresold but te lengt of LPA, were several possible lengts are specified a priori, and tey seek te LPA tat as an insignificant estimate of price response inside tis range. Kalwani and Yim (1992) take a grid searc approac. Tis approac does not allow statistical inferences suc as testing ypoteses and confidence intervals. Anoter approac is Bayesian inference. For example, Ferreira (1975), Geweke and Terui (1993), and Cen and Lee (1995) provided Bayesian approaces to deal wit discontinuous likelioods in econometrics and nonlinear timeseries models. Our proposed metod, using more updated tools, directly models price tresolds in coice models in a general way and conducts coerent statistical inference on tem under a small-sample situation. Te advantage of our approac is tat it avoids suc specification searc and can accommodate eterogeneity.

4 Terui and Daana: Estimating Heterogeneous Price Tresolds Marketing Science 25(4), pp , 2006 INFORMS Empirical Application to Scanner Panel Data 3.1. Data, Variables, and Model Specification Data and Variables. Video Researc Inc., Japan, supplied scanner panel data for te instant-coffee category. In all, 2,840 records for 197 panels during were available. We assume tat five national brands existed in te market during te tracking period. We deal wit five primary brands in te market: A, B, C, D, and E. Table 1 provides descriptive information about te data. Brand B as te maximum sare over 48.03%; te minimum sare approximately 5.74% is for brand E. We rescale all prices as yen/100g to equalize quantitative differences for eac package of te five brands. Variables included in te analysis are: Explanatory Variables: X = [Constant Price Display Feature Brand Loyalty], were Price is te log(price); Display and Feature are binary values; and Brand Loyalty is a smooting variable over past purcases proposed by Guadagni and Little (1983) as GL jt = GL jt + 1 I j t 1, were a grid searc (Keane 1997) is applied to fix te smooting parameter as 0.75 based on te criterion of minimum marginal likeliood. Houseold Specific Variables: Z r = [Constant Dprone Pfreq ARP BL], were Dprone is te deal proneness defined as te proportion of purcase (of any five brands) made on promotion (Bucklin and Gupta 1992, Han et al. 2001); Pfreq is te sopping frequency (tree categories; H 3 of Kalyanaram and Little 1994: sopping frequency is a factor to affect te price tresolds); ARP represents te measure of average reference-price level as defined by ARP = m j=1 T t=1 logrp jt/t /m; and BL represents te brand loyalty measure defined as BL = max j T t=1 GL jt/t, bot of wic are used by Kalyanaram and Little (1994) and Han et al. (2001). We furter define Houseold Specific Variables: Z = [Constant Hsize Expend], were Hsize is 1 6 (number of ouseold members) and Expend is nine categories (sopping expenditure/mont) used in Rossi et al. (1996). Table 1 Descriptive Statistics for Data Coice Average Time Time Alternative sare price displayed (%) featured (%) Brand A Brand B Brand C Brand D Brand E Specification ofreference Price. Te literature presents some conceptualizations for te reference price RP jt. Following Briesc et al. (1997), we employ a brand-specific reference price. We also consider four kinds of reference prices of two categories: memorybased and stimulus-based. Tat is, (1) Memory-based as (A) RP jt = P j t 1, te price at its last purcase and (B) RP jt = RP j t P j t 1, te smooted price over previous purcases; (2) Stimulus-based as (C) RP jt = P kt, were k means te brand at te last purcase, and (D) RP jt = P rt, were r indicates te price of a brand cosen randomly at te time of purcase. Model Specification. We consider tree models for comparison wit oter candidates: (i) aggregate (omogeneous) reference-price Probit model witout a tresold (Winer 1986, Mayew and Winer 1992, Putler 1992); (ii) two regimes eterogeneous reference price Probit model witout a tresold (Cang et al. 1999, Bell and Lattin 2000); and our proposed (iii) tree-regime eterogeneous reference-price Probit model wit tresolds. Wit four types of reference prices for eac model above, summing to 12 models are considered for comparison. Table 2 describes logs of marginal likeliood for model comparison. Te model (C)(iii): stimulus-based RP(C) tree regime eterogeneous Probit wit tresolds is most supported by marginal likeliood criterion. Consequently, some evidence supports te ypotesis of price-tresold existence tat was discussed by Kalyanaram and Little (1994) even by using te models incorporating eterogeneous consumers. Te summary statistics of parameter estimates for tat model are provided in Table 3. To save space, estimates of constant terms are not listed ere. Our cosen stimulus-based RP is compatible wit previous studies of Hardie et al. (1993), stating tat it seems likely tat caracteristics of te brand last purcased will be more available in memory in particular wit consumer package goods, and moreover Rajendran and Tellis (1994) discuss te reasons of appropriateness for stimulus-based RP in detail. Table 2 Log of Marginal Likeliood (i) (ii) (iii) Memory-based A B Stimulus-based C D (i) Conventional Probit witout bot eterogeneity and tresolds. (ii) Two-regime Probit wit eterogeneity and witout tresolds. (iii) Tree-regime Probit wit bot eterogeneity and tresolds.

5 Terui and Daana: Estimating Heterogeneous Price Tresolds 388 Marketing Science 25(4), pp , 2006 INFORMS Table 3 Model Specification and Parameter Estimates RP jt = P kt Wit eterogeneity (i) Witout (ii) Witout (iii) Wit eterogeneity tresold tresolds Gain regime Price 4251 [0.556] ( 3.875) ( 4.749) Display [0.184] (2.793) (2.251) Feature [0.353] (7.830) (4.021) Brand Loyalty [0.426] (5.161) (1.342) LPA Price ( 1.711) Display (2.969) Feature (7.361) Brand Loyalty (12.601) Loss regime Price 7021 [1.594] ( 3.034) ( 4.725) Display [0.212] (3.135) (2.972) Feature [0.389] (8.044) (7.366) Brand Loyalty [0.460] (8.805) (12.846) Tresolds ( 0.113, 0.138] LML 36, , , (i) Conventional Probit witout bot eterogeneity and tresolds. (ii) Two-regime Probit wit eterogeneity and witout tresolds. (iii) Tree-regime Probit wit bot eterogeneity and tresolds. Significant at 0.95 HPD region. Te number inside [ ] in te model (i) means standard deviation, and ( ) for (ii) and (iii) means te quasi t-value calculated from te frequency distribution of ouseold s estimates Reference Effects, Loss Aversion, and Price Tresolds Our analysis indicates te presence of asymmetric reference effects, loss aversion, and eterogeneity. Tis finding is different from Cang et al. (1999) and Bell and Lattin (2000) were asymmetric effects were negligible wen eterogeneity and price tresolds were incorporated into te model. Corresponding to eac type of RP, (A) (D), we ave four sets of models: (A)(i), (A)(ii), (A)(iii) troug (D)(i), (D)(ii), (D)(iii). Based on te most supported RP of (C), price-response estimates at te aggregated level sow some loss aversion because, in te case of (C)(iii), we ave te estimate of regime (g) gain wic is smaller tan tat of regime (l) loss as (gain: 3142, loss: 4518). Te same applies to te model (i) (gain: 4251, loss: 7021) and (ii) (gain: 1637, loss: 2622). We furter observe te following: (i) aggregate (omogeneous) Probit witout a tresold sows loss aversion most clearly at te aggregated level; (ii) two-regime eterogeneous Probit witout a tresold is te most vague; and (iii) our proposed tree-regime eterogeneous Probit wit tresolds yields performance between (i) and (ii). We found te presence of asymmetric reference effects and loss aversion after price tresolds were incorporated into te model for tis data set. Tis observation is robust relative to oter types of reference price, except for (A)(ii), (B)(ii), and (B)(iii). Following Bell and Lattin (2000, p. 190, Figure 1), we present intuitive reasoning for our results. First, Bell and Lattin (2000) discuss te relation between models (i) and (ii) relative to loss aversion: model (i) as steeper slopes in te loss regime ( l A > l BL as well as te gain regime ( g A > g BL tan model (ii), owever, te difference ( l A l BL is larger in te loss regime tan ( g A g BL is in te gain regime. Terefore, loss aversion would decrease or disappear if eterogeneity were incorporated into te model. Next, we examine te relation between models (ii) and (iii). Te response functions assumed in eac model are sown in Figure 1; we note tat te response function of model (ii), corresponding to te model of Bell and Lattin (2000), as two regimes witout price tresolds. Two response functions are kinked at te zero of sticker sock. Our model (iii), owever, as tree regimes. We suppose tat te true response function as LPA (insensitive range) limited by te price tresolds, as sown in Figure 1, and tat te data are observed along wit tree lines. If we fit only two response functions, as employed in Bell and Lattin (2000), to data over positive and negative sides of sticker-sock domain separately, te fitted slopes ( g will be less steep tan BL l BL te slopes g l of te tree-regime model in te gain-and-loss regimes. Tat is, g BL < g, l BL < l because te fitted slopes are calculated so as to catc up wit te data inside LPA, to wic a flatter line is applied by definition. For tat reason, a model wit price tresolds is likely to be more responsive tan a model witout a tresold for bot loss and gain regimes. Table 3 presents empirical evidence to support tis conjecture, wic is robust wit respect to te coice of reference price. On te oter and, tat discussion indicates no tendency for loss aversion between models wit and witout tresolds. However, relying on Kaneman and Tversky (1979), it would be likely for g l (we define it as te degree of loss aversion) to be positive in most cases because we are trying to model individual consumers responses. In fact, we used our data to calculate te posterior probability Pr Gain Loss data at te aggregated level for models (i), (ii), and (iii). Figure 2 displays te posterior density of te difference Gain Loss. Te distributions agree wit Prospect teory (Kaneman and Tversky 1979), and te degree of loss aversion is strongest for model (i) in te sense tat te posterior probability of loss aversion Pr Gain Loss > 0 data is largest; it is weakest for model (ii). Model (iii) is located between tem. Te relation between (i) and (ii) is consistent wit te results of Bell and Lattin (2000). Te relation between (ii) and (iii) implies tat te degree of loss

6 Terui and Daana: Estimating Heterogeneous Price Tresolds Marketing Science 25(4), pp , 2006 INFORMS 389 Figure 2 Posterior Distribution for Loss Aversion Figure 3 Heterogeneous Distribution of Price Tresolds Relative frequency (1) (3) β p β p Cumulative relative frequency Model (i) Model (ii) Model (iii) Model (i) Model (ii) Model (iii) Loss aversion (1) (3) β p β p aversion would increase after incorporation of eterogeneous price tresolds into te models. Tese results are robust to te selection of reference price, except for (B). Te relationsip between ouseold-specific variables and teir market-response parameters was significantly estimated. However, tey are not reported ere to save space Price Tresolds and Teir Hierarcical Structure Figure 3 sows te frequency distribution of Bayes estimates ˆr 1 = 1H and ˆr 2 = 1H, were ˆr is te mean of posterior distribution of tresold parameters for ouseold. Bot distributions exibit skewness sowing distinct features of eac oter. Te average distances from zero are different: 0113 for te lower tresold ˆr 1 and for te upper tresold ˆr 2. For tat reason, te symmetric LPA around zero could fail to reflect eterogeneity in te brand coice study. Table 4 sows tat all ierarcical Bayes estimates of regression coefficients 1 and 2 of (3) on ouseold-specific variables are significant in te sense tat te 95% HPD region does not include zero. From tese estimates, we first note tat LPAs relative to ouseolds are not symmetric around te zero levels of teir respective sticker socks. We also 8 Frequency Frequency r Next r 2 Next observe te following: (1) For Pfreq versus LPA, te mean level of LPA increases wit te purcase frequency, wic is consistent wit previous empirical findings; e.g., on te ypotesis H 3 of Kalyanaram and Little (1994): Pfreq is a factor to affect price tresolds. (2) For Dprone versus LPA, te mean level of LPAs decreases accordingly as te increase in deal proneness, wic is compatible wit Han et al. (2001). (3) For ARP versus LPA, te estimates sow tat te ouseold wit ig ARP value as wider LPA, wic is consistent wit te result to te ypotesis H 2 of Kalyanaram and Little (1994): ARP is a factor to affect price tresolds. (4) For BL versus Table 4 Hierarcical Regression Coefficients on Houseold Data (Price Tresolds) (C)(iii) Variable r 1 r 2 Const Pfreq Dprone ARP BL Significant at 0.95 HPD region. Te number inside [ ] means posterior standard deviation.

7 Terui and Daana: Estimating Heterogeneous Price Tresolds 390 Marketing Science 25(4), pp , 2006 INFORMS LPA, we observe tat te ig brand-loyalty ouseold as wider LPA, wic is consistent wit results by ypotesis H 4 of Kalyanaram and Little (1994): BL is a factor to explain price tresolds. 4. Concluding Remarks Tis study introduced a tree-regime piecewise-linear stocastic utility function wit two price tresolds, and proposed a class of brand-coice model tresold Probit model under a framework of a continuous mixture modeling for eterogeneous consumers. Price tresolds generate unconventional discontinuous likeliood functions in te analysis, and tey create difficulties in estimation. However, our metod directly models tresolds for coice models in a general manner, and coerent statistical inference on te tresolds can be done particularly wen te number of samples is scarce. Our empirical application, as far as our data set is concerned, sows tat our proposed model is superior to a conventional linear-utility-based Probit model witout price tresolds as well as aggregate Probit models; te reference effect and loss aversion are observed after price tresolds are incorporated into a eterogeneous price-response model, but tat loss aversion is attenuated ten by an aggregate (omogeneous) model witout price tresolds. Moreover, troug ierarcical modeling, we investigated explanatory factors of eterogeneous price tresolds. Some are consistent wit previous studies tat assumed consumer omogeneity. Individual eterogeneity makes it possible to design a customized strategy, wic as been demanded recently as we can see, for example, in Gal-Or and Gal-Or (2005) for customized advertising. As an application of our model, we explored a possibility of customized pricing strategy based on te knowledge of price tresolds for respective ouseolds. Our limited simulation study sowed tat customized pricing could yield greater profits tan flat pricing, and optimal pricing levels are provided at te lower price tresolds for discounting, and at te upper price tresolds for a price ike. However, tis is not reported ere to save space. Our analysis can be generalized furter in several aspects. For example, eterogeneous modeling could be applied to smooting parameters of te reference price derived from exponentially weigted past prices. Moreover, it can be extended to accommodate te dynamic relation on a utility function, for example, suc as Leicty et al. (2005) recently proposed. Togeter wit a compreensive discussion of customized pricing mentioned above, we leave tese problems for future researc. Acknowledgments Nobuiko Terui acknowledges te financial support from te Japanese Ministry of Education Scientific Researc Grant (C) Te autors appreciate elpful comments from te area editor and tree anonymous reviewers. Appendix. Estimation ofte Model: Markov Cain Monte Carlo Algoritm for Hierarcical Bayes Modeling oftresold Probit Model As for model calibration, extending te framework of Rossi et al. (1996), we employ ierarcical Bayes modeling to implement te tresold Probit model (6). Given te value of r, according to te level of consumer s sticker sock P jt RP jt, we first allocate data X t of te explanatory variable to make X i jti= gal at eac purcase occasion. Te corresponding latent utility vector y i t i= gal is generated based on personal coice data I t (te index of observed coices) using te algoritm of te Bayesian Probit model (Rossi et al. 1996, pp ) applied to eac regime. Ten, except for price tresold r, we can use conditional posterior distributions: y i t I t X t i i r, i yi t X i t i i V i z r, i 1 y i t Xi t i r, i i i V z r, and V i 1 i i z r for i = g a l. Te variance a p 2 for a p N0a p 2 in te LPA was fixed as As for ierarcical modeling for te price tresold, te lower (negative) tresold is modeled as r 1 = z ; 1 N01 2 and stacking over all ouseolds = 1H, leads to matrix notation r 1 = Z ; 1 N H 0 1. We first set prior distributions on as 1 r 1 1 N = 0 1 = 001I d 1 1 Wisart 10V = d + 4 V 10 = 10 I H Ten we ave conditional posterior distributions were 1 r 1 z 1 N Z Z =Z Z Z Z and 1 =Z Z 1 Z r r 1z 1 ( Wisart 10 +HV 10 + r 1t Z t 1 r t Z t 1 ) t Te same formulations apply to upper (positive) price tresold r 2. Next, as for conditional posterior r I t X t i i z, we use te likeliood function Lr I t X t i i of (8) in 2 jointly wit te prior pr NZ defined above, and we employ Metropolis-Hastings sampling wit random walk algoritm as te following steps: Denote r = r; ten (1) i = 0. Set r 0. (2) i>1 z N0RW 2 I H and set r = r i 1 + z. (3) { } pr r i 1 Lr I t X t r= min pr i 1 Lr i 1 I t X t 1

8 Terui and Daana: Estimating Heterogeneous Price Tresolds Marketing Science 25(4), pp , 2006 INFORMS 391 (4) Sample u U 0 1, if u r i 1 r ten r i = r oterwise r i = r i 1 Tus, we ave necessary conditional posterior distributions r z Normal distribution (A-1) 1 r z Inverted Wisart distribution r I t X t i i z Metropolis-Hasting sampling Finally, we denote by f i y i t i i i V i r I t X t z te joint posterior density for te regime i, and under te assumption of uncorrelated errors for latent utility equations of eac regime, overall joint posterior density across regimes can be expressed as (A-2) i=g a l ( i f i y t i i i V i r I t X t z ) In terms of sampling algoritms (A-1) and (A-2) for Markov cain Monte Carlo, we can constitute te posterior distribution of eac regime, respectively, to get overall joint posterior density across regimes. References Bell, D. R., J. M. Lattin Looking for loss aversion in scanner data: Te confounding effect of price response eterogeneity. Marketing Sci Briesc, R. A., L. Krisnamurti, T. Mazumdar, S. P. Raj A comparative analysis of reference price models. J. Consumer Res Bucklin, R. E., S. Gupta Brand coice, purcase incidence, and segmentation: An integrated modeling approac. J. Marketing Res Cang, K., S. Siddart, C. B. Weinberg Te impact of eterogeneity in purcase timing and price responsiveness on estimates of sticker sock effects. Marketing Sci Cen, C. W. S., J. C. Lee Bayesian inference of tresold autoregressive models. J. Time Ser. Anal Ferreira, P. E A Bayesian analyis of a switcing regression models: Known number of regimes. J. Amer. Statist. Assoc Gal-Or, E., M. Gal-Or Customized advertising via a common media distributer. Marketing Sci Geweke, J., N. Terui Bayesian tresold autoregressive models for nonlinear times series. J. Time Ser. Anal Gilbride, T., G. M. Allenby A coice model wit conjunctive, disjunctive, and compensatory screening rules. Marketing Sci Guadagni, P. M., J. D. C. Little A logit model of brand coice calibrated on scanner data. Marketing Sci Gupta, S., L. G. Cooper Discounting of discounts and promotion tresolds. J. Consumer Res Han, S. M., S. Gupta, D. R. Lemann Consumer price sensitivity and price tresolds. J. Retailing Hardie, B. G. S., E. J. Jonson, P. S. Fader Modeling loss aversion and reference dependence effects on brand coice. Marketing Sci Helson, H Adaption-Level Teory. Harper & Row, New York. Kaneman, D., A. Tversky Prospect teory: An analysis of decision under risk. Econometrica Kalwani, M. U., C. K. Yim Consumer price and promotion expectations: An experimental study. J. Marketing Res Kalyanaram, G., J. D. C. Little An empirical analysis of latitude of price acceptance in consumer package goods. J. Consumer Res Keane, M. P Modeling eterogeneity and state dependence in consumer coice beavior. J. Bus. Econom. Statist Leicty, J. C., D. K. H. Fong, W. S. DeSarbo Dynamic models incorporating individual eterogeneity: Utility evolution in conjoint analysis. Marketing Sci Mayew, G. E., R. S. Winer An empirical analysis of internal and external reference price using scanner data. J. Consumer Res Newton, M. A., A. E. Raftery Approximating Bayesian inference wit te weigted likeliood bootstrap. J. Roy. Statist. Soc. B Putler, D Incorporating reference price effects into a teory of consumer coice. Marketing Sci Rajendran, K. N., G. J. Tellis Contexual and temporal components of reference price. J. Marketing Rossi, P. E., R. McCulloc, G. Allenby Te value of purcase istory data in target marketing. Marketing Sci Serif, M., D. Taub, C. I. Hovland Assimilation and contrast effects of ancoring stimuli on judgements. J. Experiment. Psyc Winer, R A reference price model of brand coice for frequently purcased products. J. Consumer Res Yang, S., G. M. Allenby A model for observation, structural, and ouseold eterogeneity in panel data. Marketing Lett