(Accepted for publication in Applied Stochastic Models and Data An al ysi s)

Transcription

1 Det er mi ni ng t he Effect s of Obser ved and Unobser ved H et er ogenei t y on Consumer Brand Choice Pet er T. L. Popkowski L eszczyc* Frank M. Bass* * August 1995 r evised Sept ember 1996 (Accepted for publication in Applied Stochastic Models and Data An al ysi s) * Assi st ant Pr ofessor of M ar ket i ng, U ni ver si t y of Al ber t a, Depar t ment of M ar ket i ng, Business Economics and L aw, 4-30F Faculty of Business Building, Edmonton, Alberta, Canada T6G 2R6, Tel: (403) ; ppopkows@gpu.srv.ualberta.ca * * U ni ver si t y of Texas Syst em Eugene M cder mot t Pr ofessor of M anagement, University of Texas at Dallas Ack now ledgment The author s gr at eful l y acknowl edge A.C. N i el sen I nc. for pr ovidi ng t he dat a. We al so thank Dipak Jain, and the participants of the U ni ver si t y of Al ber t a semi nar ser i es for helpful comments and suggestions. Funding for this research has been received from the Central Research Fund at the University of Alberta, Edmonton, and the Social Sciences and Humanities Research Council of Canada.

2 Det er mi ni ng t he Effect s of Obser ved and Unobser ved H et er ogenei t y on Consumer Brand Choice Abst r act Thi s paper i nvest i gat es t he effect s of het er ogenei t y i n consumer choi ce behavior. Omi t t ed consumer het er ogenei t y may l ead t o badl y bi ased r esul t s, and wrong i nfer ences concer ni ng mar ket i ng st r at egi es t o fol l ow. I n t hi s r esear ch we st udy t he ext ent and t he cause of t hi s bi as. We di st i ngui sh bet ween obser ved and unobser ved het er ogenei t y, by par t i al i ng out t he effect s of unmeasured het er ogenei ty and modeling it explicitly. The fol l owi ng quest i ons wi l l be addr essed: What i s unobser ved het er ogenei t y and how much of i t can be expl ai ned? H ow shoul d het er ogenei t y be i ncor por at ed i n consumer choi ce model s? A hazar d model i s used for t he anal ysi s. The hazar d model wi l l yiel d pat t er ns of switching among brands, as well as, the effect of marketing mix variables on brand choi ce and purchase t i mi ng. Di ffer ences bet ween swi t cher s and r epeat purchaser s ar e studied and the extent to which brand choice can be expl ai ned. Our model i s est i mat ed using scanner panel data. We find that it is important to include both observed and unobser ved het er ogenei t y i n or der t o obt ai n a bet t er fit of t he model. Our r esul t s show that it may be sufficient to only include obser ved het er ogenei t y t o obt ai n unbi ased par amet er est i mat es. I ncl udi ng obser ved het er ogenei t y al so r educes t he aggr egat i on or heterogeneity bias in the hazard function. 1

3 1. Intr oduction Consumer het er ogenei t y has become an i mpor t ant t opi c i n mar ket ing, and many r esear cher s have i ncl uded het er ogenei t y i n st udying consumer br and choi ce pr ocesses. Basi cal l y het er ogenei t y i s al l omi t t ed and unmeasurabl e var i abl es i n a consumer 's choi ce pr ocess, such as di ffer ent t ast es or pr efer ences, habi t s, demogr aphi cs, et c. Bemmaor and Schmittlein (1991), define the total variance in purchase behavior as the sum of within household variance and between household variance (and residual variance). The within household variance in purchase behavior (for individual househol ds over t i me) i n mar ket i ng i s nor mal l y r efer r ed t o as non-stationarity and the bet ween househol d var i ance as het er ogenei t y. When pool i ng dat a over consumer s i t i s i mpor t ant t o i ncl ude het er ogenei t y, si nce mar ket i ng mi x var i abl es account for onl y par t of t he expl ai ned var i ance i n br and choi ce pr ocesses. Furt her mor e, i gnor i ng omi t t ed het er ogenei t y may l ead t o badl y bi ased r esul t s and wrong i nfer ences concer ni ng mar ket i ng st r at egi es t o fol l ow (see, for exampl e, Fl i nn and H eckman 1982, H eckman and Singer 1984, Pickles and Davies 1984, Davies and Crouchley 1985, Jain and Vilcassim 1991, and Reader 1993). Omi t t ed het er ogenei t y may al so l ead t o spuri ous st at e dependence. When individuals are aggregated, without adjusting for individual differences, these individual di ffer ences ar e absor bed by t he er r or t er m, causi ng ser i al cor r el at i on i n t he r esi dual s l eadi ng t o spuri ous st at e dependence and l ess effici ent est i mat es. The di st i nct i on bet ween het er ogenei t y and st at e dependence i s i mpor t ant and has l ead t o a debat e i n mar ket i ng concer ni ng fir st and zer o or der behavior (see, for exampl e, Bass et al. 1984). Few paper s i n t he l i t er at ure have i ncor por at ed bot h st at e dependence and het er ogenei t y, some exceptions are Jones and Landwehr (1988), Jain and Vilcassim (1991), Vilcassim 2

4 and Jain (1991), and Gupta et al. (1994). Non-st at i onar i t y i s anot her pot enti al source of spuri ous st at e dependence. The ext ent of model i ng non-st at i onar i t y i n consumer choi ce behavior i n mar ket i ng has been most l y l i mi t ed t o t he i ncl usi on of t i me var ying marketing mix variables. Despi t e r ecent advances l i t t l e i s known about het er ogenei t y. M ost pr evious r esear ch has model ed unmeasured het er ogenei t y as compl et el y unobser vabl e, or has been l i mi t ed t o t he i ncl usi on of br and l oyal t y (see, for exampl e, Guadagni and Little 1983). I n t hi s paper we di st i ngui sh bet ween obser ved and unobser ved het er ogenei t y and i ncl ude bot h i n our model. I n par t i cul ar, we par t i al out t he effect s of het er ogenei t y and expl i ci t l y model i t. We det er mi ne how much of buyer behavior and which buyers, are affect ed by mar ket i ng mi x var i abl es, and t o what ext ent consumer behavior i s uncontr ol l abl e i n t he shor t r un. We di scuss t he consequences of omi t t ed het er ogenei t y, di ffer ent t ypes of het er ogenei t y, and di ffer ent ways t o i ncor por at e and est i mat e het er ogenei t y. Obser ved het er ogenei t y i s i ncor por at ed by i ncl udi ng consumer speci fic var i abl es as i ndependent var i abl es, and t he amount of var i ance i n t he unobser ved het er ogenei t y expl ai ned by t hese var i abl es i s det er mi ned. U nobser ved het er ogenei t y i s model ed usi ng a r andom effect s appr oach. We use a hazar d model whi ch i ncl udes st at e dependence, het er ogenei t y and non-stationarity (by including time-varying independent var i abl es and by al l owi ng for t i me dependence i n t he purchase t i mi ng). The r est of t he paper i s st r uct ured as fol l ows. The next sect i on di scusses t he st at e of t he ar t i n model i ng consumer het er ogenei t y, fol l owed by t he model devel opment i n sect i on 3. The r esul t s of t he st udy ar e pr ovided i n sect i on 4. Fi nal l y, i n sect i on 5, we concl ude our st udy and pr ovide suggest i on for future r esear ch. 3

5 2. L i t er at ur e Revi ew of H et er ogenei t y Whi l e t r adi t i onal appr oaches of i ncor por at i ng het er ogenei t y have been devel oped i n economi cs and ot her fiel ds, most r ecent advances have been i n t he ar ea of mar ket i ng or by marketing researchers (see, for example, Gönül and Srinivasan 1993a, Allenby and Lenk 1994, Jain et al. 1994, and Elrod and Keane 1995). A summar y of a sel ect ed number of st udi es i ncl udi ng het er ogenei t y i s pr ovided i n Tabl e 1. Our focus is on het er ogenei t y i n consumer choi ce model s for frequentl y purchased goods wher e dat a ar e aggr egat ed over i ndi vidual s. For a di scussi on of het er ogenei t y i n mar ket st r uct ure model s see El r od (1991). The emphasi s i n t hese paper s has been on al t er nat i ve ways of including unobser ved het er ogenei t y. H owever, whi l e omi t t ed het er ogenei t y may l ead t o bi ased par amet er est i mat es, t her e i s st i l l contr over sy concer ni ng t he condi t i ons when par amet er s wi l l be bi ased and t he sever i t y of t hi s bi as. For exampl e, sever al r esear cher s have cl ai med t hat i gnor i ng het er ogenei t y i n l ogi t model s l eads t o unbi ased par amet er estimates but to biased choice probabilities (Horowitz 1981, and Jones and Landwehr Comment [PTPL1]: Page: 4 Ther e ar e lar ge number of paper s addr essing het er ogeneit y, a few of which in a stochastic framework are: Bass, Jeuland and Wright (1976); Jeuland, Bass and Wright (1980); Morrison and Schmittlein (1989); Jain, Bass and Chen (1990); Fader and Lattin (1993). I n a hazard model framework there ar e paper s by: H elsen and Schmit t lein (1993); Gönül and Srinivasan (1993a, 1993b); Jain and Vilcassim (1991); Vilcassim and Jain (1991); Popkowski Leszczyc (1992); and for t he logit model: Gaudagni and Little (1983); Jones and Landwehr (1988); Steckel and Vanhonacker (1988); Kamakura and Russel (1989); Allenby (1989); Allenby and Rossi (1991); Chintagunta, Jain and Vilcassim (1991); Popkowski Leszczyc (1992); Rossi and Allenby (1993); Gönül and Srinivasan (1993c); Jain et al. (1993); Elrod and Keane (1993); Chintagunta (1992, 1993, 1994); Allenby and Lenk (1994); and in cr ossect ional and t ime ser ies r egr ession models: Bass and Wittink (1975); Parry and Bass (1990); and Jacobson (1990). 1988). Other researchers (Heckman 1981, Crouchley 1987, and Reader 1993) have shown t hat par amet er est i mat es wi l l be bi ased. A possi bl e r eason for t hese di ffer ent r esul t s i s due t o omi t t ed var i abl e bi as. When omi t t ed var i abl es ar e cor r el at ed (uncor r el at ed) wi t h i ncl uded var i abl es t hen par amet er est i mat es ar e bi ased (unbiased), but utility forecasts will be unbiased (biased). Par t of t hi s contr over sy may be due t o t he di ffer ent ways of i ncor por at i ng and est i mat i ng model s wi t h unobser ved het er ogenei t y. Ther efor e, i t i s i mpor t ant t o fir st discuss t he appr oaches used by t hese r esear cher s. I n Tabl e 1, we di st i ngui sh bet ween t he way het er ogenei t y i s i ncl uded i n a model : (i ) het er ogenei t y can be speci fied i nsi de or outsi de t he l i kel i hood funct i on, (i i ) het er ogenei t y can be model ed usi ng a fixed or a 4

6 r andom effect s speci ficat i on, (i i i ) het er ogenei t y can be i ncl uded as r andom i nter cept s and/or r andom coeffici ents, (i v) het er ogenei t y can be model ed par amet r i cal l y or nonparametrically. **** Insert Table 1 about here **** Comment [PTPL2]: Page: 5 (v) St at ic or dynamic model. (st at ionar y vs non-stationary) When het er ogenei t y i s i ncl uded outsi de t he l i kel i hood funct i on, i t i s added t o t he purchase probabilities, by assuming that these aggregate purchase probabilities differ over househol ds accor di ng t o some di st r i buti on funct i on. Exampl es of t hese ar e t he N BD model (Ehrenber g 1972), t he mul t i nomi al Dirichlet model (Bass, Jeuland and Wright 1976, Jeuland, Bass and Wright 1980). Al t hough, t hese model s pr ovide good descr i pt i ons of mar ket behavior under equi l i br i um condi t i ons, t hey pr ovide no i nfor mat i on about t he underlying causal variables. 1 Ther efor e, most r esear cher s have i ncl uded het er ogenei t y within the likelihood function. This is the approach used in this paper. H et er ogenei t y can be i ncl uded i nsi de t he l i kel i hood funct i on, by al l owi ng for househol d speci fic i nter cept s and/or sl opes (see, col umn 3 i n Tabl e 1). The most common appr oach has been t o pool dat a whi l e i ncl udi ng a separ at e i nter cept for each househol d, Comment [PTPL3]: Page: 5 For example, in t he Dir ichlet model individual purchase probabilities are modeled according to a multinomial distribution, obt aining a aggr egat e/mean populat ion vect or of constant purchase probabilities which is assumed t o be dist r ibuted over t he population according to a dirichlet dist r ibution (t he mixing or het er ogeneit y distribution). Comment [PTPL4]: Page: 5 *** Since we ar e inter est ed in par amet er estimates,... in this approach the utility is adjust ed for het er ogeneit y not t he parameters? *** adjust i ng for t he househol d speci fic t i me-i nvar i ant effect s (such as pr efer ences). M or e r ecentl y r esear cher s have al so al l owed for het er ogenei t y i n sl opes, wher e par amet er est i mat es var y over t he popul at i on, adjust i ng for var i at i ons i n, for exampl e, pr i ce or promotional sensitivity among individuals (a random coefficient model). It is well known t hat pool i ng consumer s wi t h het er ogeneous par amet er s may l ead t o bi ased r esul t s (Bass and Wit t i nk 1975). Resul t s of t hese st udi es r epor t ed si gni ficant i mpr ovements i n fit, as 1 Sever al author s have ext ended t hese models t o include independent var iables, Jones and Zufryden 1980, Dunn and Wrigley 1985, and Wagner and Taudes

7 wel l as di ffer ences i n par amet er est i mat es, for model s i ncl udi ng bot h het er ogenei t y i n t he i nter cept s and sl opes. Ther e ar e t wo ways t o model consumer speci fic het er ogenei t y, l eadi ng ei t her t o a fi xed effect s or a r andom effect s model (see col umn 4 i n Tabl e 1). I n t he fixed effect s model an i nter cept or dummy var i abl e i s i ncl uded for each househol d, t o est i mat e t he househol d speci fic effect s. I n t he r andom effect s model t hese househol d speci fic components ar e assumed t o be di st r i buted over t he popul at i on accor di ng t o some distribution function. The household specific effect is a random sampling from a distribution function and instead of estimating an intercept for each household only the mean and t he var i ance of t he assumed di st r i buti on ar e est i mat ed. Thi s appr oach has been ext ended i n t he l ogi t model t o a mul t i var i at e het er ogenei t y distribution, or so called "pr efer ence" het er ogenei t y, whi ch i ncl udes brand speci fic i nter cept s for each househol d (see, col umn 4 i n Tabl e 1). It is i mpor t ant t o di st i ngui sh bet ween t he t wo model s as t hey may gi ve vast l y di ffer ent r esul t s (see, for exampl e, H si ao 1989). I f t he mai n focus i s on t he di ffer ences bet ween i ndi vidual s a fixed effect s model i s most appr opr i at e. On t he ot her hand i f i nter est l ays i n t he var i at i on over househol ds, as i s t he case wi t h scanner panel st udi es, t hen t he r andom effect s model i s mor e sui t abl e. (For a mor e det ai l ed di scussi on on t he di ffer ences see, for exampl e, Popkowski L eszczyc 1992). Ther efor e, we appl y, l i ke most pr evious r esear cher s, a r andom effect s model. When using a random effects model, an assumption concerning the shape of the Comment [PTPL5]: Page: 6 Bot h met hods have limit at ions. The r andom effect s model assumes t hat t he het er ogeneit y is independent fr om t he covar iat es and an assumption concerning the distribution of the het er ogeneit y is needed (bot h const r aints have been relaxed by Kamukura and Russell?? Jain et al. 1993). *** This needs to be expanded/explained * * * A non-parametric est imat ion pr ocedure can be used t o over come t he lat t er pr oblem. This will be discussed in t he next sect ion. The fixed effect s model cannot est imat e complet ely loyal households nor t ime const ant var iables, such as demographics, which are an important par t of obser ved het er ogeneit y. Gönül and Srinivasan (1993b), relaxed the latter constraint. het er ogenei t y or mi xing di st r i buti on i s needed (see col umn 4 i n Tabl e 1). Par amet r i c mixing distributions like the Beta and Gamma distributions have been frequently used in marketing. A drawback of the assumed parametric form is that parameter estimates 6

8 can be bi ased due t o mi sspeci ficat i on of t he mi xing di st r i buti on (H eckman and Si nger 1984, and Tuma and Hannan 1984). An alternative approach is to use a nonparametric, or distribution free, distribution function. The non-parametric method is an empirical estimation technique which does not make any assumptions concerning the shape of the mixing distribution, avoiding possible miss-speci ficat i on. A non-par amet r i c est i mat i on t echni que, t he mass poi nt appr oach has been used most frequentl y. Fi nal l y we l ook at t he empi r i cal r esul t s of t he st udi es i n Tabl e 1 t o det er mi ne t he ext ent of het er ogenei t y bias. Sever al st udi es compar ed model s wi t h and wi t hout het er ogenei t y usi ng si mul at ed dat a. H or owi t z (1981) found t hat par amet er est i mat es for t he homogeneous l ogi t model s wer e not sever el y bi ased, but choi ce pr obabi l i t i es wer e ser i ousl y bi ased. Col ombo and L andwehr (1990), r epor t ed t hat t he het er ogeneous l ogi t model 's (usi ng a non-parametric het er ogenei t y di st r i buti on) par amet er est i mat es wer e cl ose t o t he t r ue val ues, whi l e t he homogenous model 's par amet er s wer e bi ased. Reader (1993), compar ed a model wi t hout and wi t h bot h par amet r i c het er ogenei t y usi ng a Di r i chl et -l ogi st i c model and nonpar amet r i c het er ogenei t y usi ng a mass poi nt appr oach. The model wi t hout het er ogenei t y and t he model wi t h par amet r i c het er ogenei t y pr ovided poor par amet er est i mat es whi l e t he nonpar amet r i c appr oach's est i mat es wer e cl ose t o t he t r ue par amet er s. The model wi t hout het er ogenei t y pr ovided t he wor st fit and even a par amet er wi t h t he wrong si gn. H eckman (1981), compar ed r esul t s of t he r andom effect s and t he fixed effect s model, and a st at i c and a dynami c l ogi t model. H e concl uded t hat t he r andom effect s est i mat es ar e mor e effici ent. H owever, when t he var i ance of t he i ndi vidual speci fic component i s l ar ge, t he r andom effect s est i mat or has a l ar ger bi as t han t he fixed effect s model. Whi l e 7

9 het er ogenei t y bi as was not t hat sever e i n t he static case, het er ogenei t y bi as was consi der abl e for t he r andom and especi al l y for t he fixed effect s dynamic model. I n concl usi on, we have t r i ed t o pr ovide an over view of di ffer ent ways het er ogenei t y has been i ncl uded i nto model s of consumer choi ce behavior. The r esul t s from empi r i cal analysis and simulation studies show that including unobserved heterogeneity leads to a significant improvement of model fit. All studies with an asterisk in column 4 of Table 1, Comment [PTPL6]: Page: 7 Dunn, Reader and Wrigley (1987), compared a Dirichlet-L ogist ic model wit h a nonpar amet r ic logit model. The non-parametric est imat ion t echnique pr ovided super ior r esult s based on likelihood value t est s. H owever, t he pr edict ed pr obabilit ies of bot h models wer e similar. have compar ed model s wi t h and wi t hout het er ogenei t y. For t he st at i c l ogi t model, t he par amet er est i mat es have t he same si gn and si gni ficance l evel, but par amet er est i mat es di ffer i n magni t ude. Di ffer ences i n par amet er est i mat es wer e not ver y l ar ge, however, i n most i nst ances di ffer ences wer e gr eat er t han 3 st andar d deviat i ons. Ther e was no syst emat i c di r ect i on i n t he bi as (for exampl e, we obser ved bi ases i n bot h di r ect i ons for pr i ce par amet er s). I n dynami c model s par amet er s may be significantly biased due to omi t t ed het er ogenei t y. One st udi es usi ng a dynami c model (Jai n and Vi l cassi m 1991) r epor t ed bi ased r esul t s due t o mi sspeci ficat i on of t he het er ogenei t y di st r i buti on. Three st udi es usi ng a l ogi t model di d not r epor t such bi as. M ost r ecent paper s have i ncl uded het er ogenei t y i n t he sl opes and i n t he i nter cept s, l eadi ng t o si gni ficant i mpr ovements i n fit and di ffer ences i n magni t ude of t he par amet er est i mat es. These findi ngs suggest s t hat bot h het er ogenei t y i n i nter cept s and sl opes shoul d be i ncl uded t o over come het er ogenei t y bi as. Par amet er est i mat es ar e mor e sensi t i ve t o t he speci ficat i on of het er ogenei t y i n dynami c choi ce model s. Ther efor e, non-par amet r i c est i mat i on t echni ques shoul d especi al l y be used i n dynami c model s. The l i t er at ure r eview above has focused on i ncl udi ng unobser ved het er ogenei t y t o obt ai n unbi ased par amet er s. L i t t l e at t enti on has been pai d t o t he pot enti al sources of "obser ved" het er ogenei t y. Obser ved het er ogenei t y, such as, consumer demogr aphi cs and 8

10 pr efer ences ar e i ndi vidual speci fic var i abl es whi ch can be i ncl uded i n t he model. M ost r esear cher s i gnor e or i ncl ude a l i mi t ed number of consumer speci fic var i abl es i nto t hei r model s (see col umn 5 i n Tabl e 1). I n t hi s paper we decompose obser ved het er ogenei t y based on concept s of consumer behavior for frequentl y purchased, l ow i nvol vement pr oduct s. I n gener al, consumer s spend l i t t l e t i me i n maki ng a purchase deci si on, l i t t l e i nfor mat i on i s pr ocessed, and consumer s may depend on wel l -for med scr i pt s l eadi ng t o habit forming and routinized behavior (Ortmeyer, Lattin and Montgomery 1991). Consumer purchase behavior i s assumed t o be pr edomi nantl y a zer o-or der pr ocess. These ar e mat ure pr oduct s wi t h st at i onar y mar ket shar es. Ther efor e, i t i s r easonabl e t o assume t hat t he l ar gest par t of t he popul at i on has const ant pr efer ences (see, Bass et al. 1984). Tabl e 2 pr ovides a l i st wi t h components causi ng het er ogenei t y. The fir st ei ght ar e expl i ci t l y i ncor por at ed i nto t he model whi l e t he l ast five and ot her omi t t ed var i abl es ar e capt ured by t he unobser ved het er ogenei t y. ********* Include Table 2 about here ********* I n summar y, we i ncl ude bot h obser ved and unobser ved het er ogenei t y i n our model. U nobser ved het er ogenei t y i s i ncl uded as a househol d speci fic i nter cept, as a r andom effect s speci ficat i on est i mat ed non-par amet r i cal l y. H et er ogenei t y i n par amet er s i s i ncor por at ed by est i mat i ng a compet i ng r i sk model wher e t r ansi t i on or swi t ch speci fic par amet er s ar e est i mat ed. Dat a ar e segmented based on t he pat ters of switching and r epeat purchasi ng, and separ at e par amet er s ar e est i mat ed. 9

11 3. M odel Speci fi cat i on Because our focus i s on st udying unobser ved het er ogenei t y and t he effect s of mar ket i ng mi x var i abl es on consumer choi ce behavior, a st ochast i c model which allows one t o i ncor por at e bot h mar ket i ng mi x var i abl es and unobser ved het er ogenei t y i n a var i et y of ways i s speci fied. The model appl i ed i s a pr opor t i onal hazar d model. The hazar d model has been wi del y used i n bi omedi cal and soci al sci ence r esear ch, for example, Cox (1972), Kalbfleish and Prentice (1980), Tuma and Hannan (1984), Heckman and Singer (1984, 1985), and Kiefer (1988), and recently in Marketing, for example, Jain and Vilcassim (1991), Vilcassim and Jain (1991), Helsen and Schmittlein (1993), and Gönül and Srinivasan (1993b). Within a relatively simple framework it is possible, using the proposed model, to study brand switching and purchase timing, while i ncor por at i ng mar ket i ng mi x var i abl es, st at e dependence, het er ogenei t y, and nonst ationarity, making this model more flexible than traditionally used choice models such as the logit model or the multinomial dirichlet model. I n t he hazar d model dat a ar e uti l i zed on t he sequence, t i mi ng, and number of purchases for an individual; the probability of purchasing a specific brand is a function of t i me. Thi s i s r efer r ed t o as t he hazar d r at e or funct i on. The hazar d r at e i s t he dependent var i abl e, and t he r el at i onshi p bet ween t hi s r at e and a var i et y of obser ved and unobser ved var i abl es i s model ed. A compet i ng-risk specification (or multiple spell multiple state) of the proportional hazar d model i s used i n our anal ysi s. The model i s based on t he wor k by Fl i nn and Heckman (1982), Heckman and Singer (1984), and Vilcassim and Jain (1991). In the compet i ng r i sk model hazar d r at es ar e est i mat ed for swi t ches bet ween al l di ffer ent brands. For example, in a market with four brands, 16 hazard functions are estimated Comment [PTPL7]: Page: 10 Advances in Econometrics (1982), Basmann & Rhodes jr. Comment [PTPL8]: Page: 10 Econometrica 10

12 (t r ansi t i ons t o and from each br and), obt ai ni ng t wel ve cr oss el ast i ci t i es. H ence t he model can account for asymmetric switching and cross elasticities. This model is an individual level, continuous-t i me Semi -M ar kov model. The conti nuous t i me appr oach has t he advantage of avoi di ng t he t i me i nter val bi as as i t frees t he model of t he t i mi ng of obser vat i ons. A Semi -M ar kov pr ocess i s di ffer ent from t he M ar kov model as t he t i me bet ween t r ansi t i ons i s an ar bi t r ar y r andom var i abl e. H ence t he model al so al l ows for het er ogenei t y and non-stationarity in the interpurchase time distributions. H et er ogenei t y i s i ncor por at ed i n t he hazar d model i n t he fol l owi ng ways: (i ) obser ved het er ogenei t y, i ndi vidual char act er i st i cs l i ke; househol d si ze, i ncome, and brand loyalty and frequency of usage, (ii) unobser ved or omi t t ed het er ogenei t y a r andom effect s speci ficat i on for t he er r or t er m and ot her omi t t ed var i abl es, (i i i ) by di st i ngui shi ng bet ween r epeat purchaser s and swi t cher s and measuri ng t r ansi t i on pr obabi l i t i es between all brands allowing for asymmetric crosselasticities. (iv) allowing for different purchase timing distributions for different brands. (i), (ii) and (iii) include heterogeneity in the brand choice process and (iv) i ncor por at es het er ogenei t y i n t he purchase t i mi ng pr ocess. The fir st 2 ar e consumer speci fic het er ogenei t y components. Al l but t he unobser ved het er ogenei t y (i i ) can be easily included in the model (by explicitly including the components) The H azar d Model The continuous hazard function gives the instantaneous rate of buying a brand at t i me per i od t, gi ven t hat no purchase was made up t o t i me t ; i t i s t he unobser ved r at e at which events occur. Popkowski Leszczyc (1992), Vilcassim and Jain (1991), and Gönül and Srinivasan (1993b), have applied the competing risk model to study brand switching 11

13 behavior. 2 The hazar d r at e for t hi s model i s t he unobservable, continuously distributed, time-depen d en t pu r ch a se pr open si t y for a consumer, for a par t i cul ar brand. The hazard rate for a transition from brand j to brand k is specified as follows: (1) h jk (t) = f jk (t)/s j (t) = f jk (t)/(1-fj(t)), wher e, f(t ), t he pr obabi l i t y densi t y funct i on of t he durat i on, i s t he l i kel i hood of a purchase at time t. F(t), the cumulative distribution function, is the probability that a consumer di d not buy a br and befor e t (t he end of t he st udy per i od). S(t ) i s cal l ed t he survival funct i on. A pr opor t i onal hazar d model i s used t o model t he t r ansi t i ons bet ween br ands. The pr opor t i onal hazar d model speci ficat i on (Cox 1972) conject ures t hat t he l og of t he condi t i onal hazar d i s l i near i n funct i ons of t he basel i ne hazar d (t ), and a vect or of covar i at es (X), and het er ogenei t y (Θ). (2) h jk (t X,Θ) = h ojk (t) g jk (X) z jk (Θ), wher e h jk (t X,Θ) = t r ansi t i on r at es from st at e j t o k condi t i onal on X and Θ. h ojk (t) = baseline hazard function, modeling the interpurchase time distribution. g jk (X) = a funct i on of a vect or of covar i at es. z jk (Θ) = a funct i on for t he unobser ved het er ogenei t y. The covar i at es, ar e i ncl uded as exp(α jk + X k β jk ), wher e, α jk and β jk, ar e t he i nter cept and coeffici ents for t he t r ansi t i on from br and j t o br and k, X k i s a vect or of covar i at es of br and k for t he t r ansi t i on from br and j t o br and k, t hese can be t i me- 2 One limitation of the compet ing r isk model is t hat it only consider s exit s t o differ ent st at es, only swit ches ar e consider ed. We follow t he appr oach t ak en by Vilcassim and Jain (1991), who add an ext r a dummy st at e for each br and, t o t ak e into account r epeat purchases. 12

14 varying or t i me i nvar i ant. Exponenti at i on pr ohi bi t s t he pr edi ct i on of negat i ve hazar d r at es, and does not put r est r i ct i ons on t he par amet er s. Pr evious r esear ch has i ndi cat ed sensi t i vit y of par amet er est i mat es due t o t he speci ficat i on of t he basel i ne hazar d function (Trussel and Richards 1985, and Heckman and Walker 1987). We use an approach where the baseline hazard is nested within the Box-Cox t r ansfor mat i on. For furt her det ai l s see, Fl i nn and H eckman (1982), H eckman and Singer (1985), Jain and Vilcassim (1991), Helsen and Schmittlein (1993). The fol l owi ng basel i ne hazar d model, consi st i ng of an i nter cept and t hree durat i on t er ms, i s used for est i mat i on purposes: (3) h ojk (t) = exp( λ ojk + λ 1jk ln t + λ 2jk t + λ 3jk t 2 ). Dependi ng on t he val ues of t he λ, di ffer ent di st r i buti on funct i ons ar e obt ai ned. For example, an exponential hazard function is obtained when λ 1 = 0, λ 2 =0 and λ 3 =0, and the Weibull is obtained when λ 1 =0, λ 3 =0. For different distributions, like the Ehrlang distribution see, Tuma and Hannan (1984), and Jain and Vilcassim (1991). Finally, we allow data to be right censored. This is the probability that the (N+1) th purchase occasi on has not occurr ed by t he end of t he sampl e pat h, speci fied by the survivor function. Assuming censoring is independent, we obtain the following l i kel i hood of a purchase, for househol d i (L i ): wher e Ri dr 1 (4) Li f jk t X i d = (, Θ ) S j ( t X, Θ ) l= 1 ri 13

15 d r i 1, if the R = 0, if the R th th spell of spell of the i the i th th household is not censored household is censored and l = 1,... Ri, t he N umber of spel l s for t he i th househol d. (5) f jk = K j= 1 j k h jk (t jk t jk )exp h 0 jk (u)du K t j (6) S j (t) = exp - h j (u) du, j=1 0 j = 1, 2,..K, the number of brands j k (no sel f t r ansi t i ons) Fi r st t he unobser ved het er ogenei t y (Θ) needs t o be i ntegr at ed out of t he l i kel i hood function. As mentioned pr eviousl y, we appl y a non-par amet r i c est i mat i on t echni que; t he mass poi nt appr oach. A st andar d i t er at i ve maximum l i kel i hood pr ocedure i s used, addi ng suppor t poi nts unti l t he val ue of t he l i kel i hood funct i on does not i ncr ease (see, for exampl e, H eckman and Singer 1984). Then summing the likelihood function over N househol ds, and adjust i ng for het er ogenei t y we obt ai n: N M (7) L( ξ X) = Li ( ξ X, Θ ) Pm i= 1 m= 1 wher e M = t he number of mass poi nts P m = t he wei ght of t he m th mass point H ence, Pm is t he pr opor t i on of t he popul at i on wi t h het er ogenei t y Θm. This could be t hought of as di vidi ng t he popul at i on i nto m segments based on het er ogenei t y. The Conti nuous Ti me M odel (CTM) pr ogr am devel oped by Geor ges Yat es of t he nat i onal opi ni on center i s used t o est i mat e t he par amet er s (see, Yi, H onor e and Wal ker 1987). 14

16 4. The Empirical Analysis The dat a anal yzed ar e scanner panel dat a for househol d purchases of K et chup from t wo t est mar ket s pr ovided by A.C. N i el sen I nc. Scanner panel data is obtained from a panel of consumer s who make t hei r gr ocer y purchases usi ng a speci al cr edi t car d, whi ch i s r ecor ded by scanner s. Dat a ar e avai l abl e for al l ket chup purchases for a per i od of t hree year s. I t i ncl udes i nfor mat i on about t he br and and si ze sel ect ed, pr i ce pai d, exist ence of pr i ce speci al s, coupon usage, and consumer demogr aphi cs. For our estimation sample we use data which covers a period of two years, 1987 and 1988 (1986 was used to calibrate the brand loyalty variable), for 1474 consumers with purchase occasi ons (compl et ed spel l s) and 1452 censor ed spel l s. 3 Typically studies using scanner panel s ar e based upon smal l sampl es of a few hundr ed househol ds. Tabl e 3, pr ovides t he aggr egat e br and swi t chi ng mat r i x for t he ket chup mar ket, and the average interpurchase times. The large numbers on the diagonal indicate a high l evel of r epeat purchasi ng. 77 per cent (5774 purchases) of H ei nz's sal es came from r epeat purchases ver sus 48 per cent for Gener i cs, 38 per cent for H unts and 30 per cent for Del monte. Based on t he absol ute number s, swi t chi ng bet ween br ands i s vir t ual l y symmet r i c consumer s swi t ched from H ei nz (395 t o Del monte, 899 t o H unts and 425 t o Gener i cs), and 1719 swi t ched t o H ei nz. H ei nz, Del monte and Gener i cs al l l ost some cust omer s t o H unts. H owever, r el at i vel y t he swi t chi ng mat r i x r eveal s t he asymmet r y i n t he swi t chi ng bet ween br ands. For 13 per cent of H ei nz's purchases consumers switched to Hunts, while 39.5 percent of purchases of Hunts resulted in a 3 Consumer s mak ing less t han 5 purchases during t he per iod of t he st udy wer e delet ed. 15

17 swi t ch t o H ei nz. Ther e i s l i t t l e di ffer ence i n t he mean purchase i nter val bet ween br ands, and bet ween swi t cher s and r epeat purchaser s. The var i abl es used i n our model ar e descr i bed i n Tabl e 4. The onl y ot her i nfor mat i on needed i s t he br and chosen and t he time per i od bet ween t wo successi ve purchase occasi ons of a househol d. ***** Insert Tables 3 and 4 about here ***** 4.1. I ncludi ng obser ved and unobser ved het er ogenei t y i n H azar d M odels To st udy t he det er mi nants and ext ent of het er ogenei t y ei ght di ffer ent model s ar e estimated, including the explanatory variables listed in Table 4. Starting with a simple model wi t hout expl anat or y var i abl es and het er ogenei t y, and addi ng bot h obser ved and unobser ved het er ogenei t y (see Tabl e 5). These r esul t s ar e for t r ansi t i ons from t he br ands. L i kel i hood r at i os ar e est i mat ed and t he i mpr ovement i n fit due t o added var i abl es i s det er mi ned. Tabl e 5, pr ovides t he negat i ve l ogl i kel i hood val ue, and a measure of "fit " (R) for t he di ffer ent model s. The measure of fit i s defined as fol l ows: L n Pr esent model - L n Base model (8) R = L n L east r est r i ct ed model - L n Base model wher e Ln = t he l og l i kel i hood val ue Base model = a model wi t h durat i on t er ms onl y Pr esent model = t he model t o be t est ed L east r est r i ct ed model = is the model including all variables Thi s measure of fit can be used t o l ook at t he i mpr ovement i n t he l i kel i hood val ue over t he base model, wi t hout covar i at es and het er ogenei t y, compar ed t o t he l east r est r i ct ed model wi t h all covar i at es and unobser ved het er ogenei t y. I n t hi s way i t i s 16

18 possi bl e t o det er mi ne t he i mpr ovement i n fit of model s and compar e nonnest ed model s, which are nested within t he l east r est r i ct ed model. 4 Bentler and Bonett (1980) discuss di ffer ent measures of goodness of fit for covar i ance st r uct ure anal ysi s, and pr ovide sever al i ncr emental fit i ndexes si mi l ar t o equat i on 8. 5 The fact or st r uct ure al l ows for t he est i mat i on of a ful l y sat urat ed model, agai nst whi ch mor e r est r i ct ed model s can be compar ed. H owever, i t i s not our object i ve t o compar e t he di ffer ent model agai nst t he ful l y sat urat ed model (whi ch i s obviousl y al so an i nter est i ng compar i son), we ar e i nter est ed i n measuri ng t he i ncr emental fit r el at i ve t o model 8 (t he l east r est r i ct ed model we est i mat e wi t h bot h obser ved and unobser ved het er ogenei t y). Table 5, shows that marketing mix variables (model 2) only explain a small amount of t he i mpr ovement i n t he t r ansi t i ons from br ands, compar ed wi t h t he base model (model 1). M odel 3, consi st s of covar i at es whi ch r emai n const ant over t i me; i ncome, fami l y si ze and mar ket, (t hese wi l l be r efer r ed t o as demogr aphi c var i abl es). The model contai ni ng t he demogr aphi c var i abl es pr ovides a sl i ghtl y bet t er fit t han model 2 wi t h t he (t i me var ying) mar ket i ng mi x var i abl es. M odel 4 i ncl udes onl y unobser ved het er ogenei t y. Thi s model has a bet t er fit t han ei t her model s 2 or 3, and a si mi l ar fit as model 5 wi t h bot h mar ket i ng mi x and demographic variables. Including loyalty (model 6) l eads t o a si gni ficant i ncr ease i n fit, and so does t he i ncl usi on of a dummy var i abl e for 4 This measure of fit cannot be inter pr et ed in t he same way as t he R 2 measure, as it does not est imat e t he amount of var iance explained by t he model. 5 We thank t he r eviewer for pointing t his out t o us. 17

19 t he heavy user s (model 7). Fi nal l y t he l east r est r i ct ed model (model 8) adds unobser ved het er ogenei t y t o model 7, which leads to a significant improvement in fit in all instances. ***** Insert Table 5 about here ***** L i kel i hood r at i o t est s ar e per for med for al l nest ed model s and i n al l i nst ances t he si mpl er model s ar e r eject ed i n favor of t he mor e compl et e model s (see Tabl e 5). A nonnest ed t est i s needed t o compar e model s wi t h and wi t hout unobser ved het er ogenei t y. Schwar t s cr i t er i a (SC) was used t o t est model 1 ver sus model 4, and model 7 ver sus model 8. 6 I n al l i nst ances t he model i ncl udi ng het er ogenei t y pr ovides a better fit. To what ext ent do t he covar i at es adjust for het er ogenei t y, and how much het er ogenei t y r emai ns after i ncl udi ng covar i at es? To det er mi ne t he expl anat or y power of t he covar i at es anot her measure of fit, pr oposed by L ancast er (1979) i s uti l i zed. Thi s measure det er mi nes t he amount of var i ance i n t he hazar d funct i on expl ai ned by t he i ncl uded covar i at es. (9) R 2 = Var ln(x k β jk ) / {Var ln(x k β jk ) + Var(zjkΘ)} Table 6, shows the amount of explained variance in the hazard function due to all covar i at es, for t he di ffer ent t r ansi t i ons. Ther e appear s t o be no syst emat i c di ffer ences bet ween swi t cher s and r epeat purchaser s. M ost of t he t r ansi t i ons ar e asymmet r i c, for exampl e, t he t r ansi t i on from Del monte t o Gener i cs (D -> G) has an R 2 of.17, while G -> D has an R 2 of.61. Onl y a smal l per centage of t he var i ance i s expl ai ned by t he covar i at es for r epeat purchases of H ei nz, whi l e a l ar ge ext ent of var i ance i s expl ai ned by 6 SC = -lnl -[(lnn x m)/2]; wher e, N = t he sample size, m = # of par amet er s. The model wit h t he highest SC value pr ovides t he best fit (see, for example, H eck man and Walk er 1987). 18

20 covar i at es for r epeat purchases of Del monte. We al so l ooked at t he per centage of expl ai ned var i ance of t he di ffer ent covar i at es (r esul t s ar e not pr ovided her e). These ar e t he per centages of t he t r ansi t i on speci fic var i ance expl ai ned by each covar i at e. The l ar gest per centage i s expl ai ned by l oyal t y, fol l owed by usage, t he t est mar ket and pr i ce. Family size and income only explain a small percentage. In most instances only a small per centage of t he t ot al var i ance i n purchase behavior i s expl ai ned by t he mar ket i ng mi x variables. ***** Include Table 6 about here ***** Vilcassim and Jain (1991) found that more variance in switching was explained by t he mar ket i ng mi x var i abl es whi l e r epeat purchaser s, who ar e mor e br and l oyal, ar e l ess sensitive to marketing mix variables. A difference with our analysis is that we include br and l oyal t y i n our model. Ther efor e, i t i s not expect ed t hat al l swi t ches wi l l have a higher fit than the repeat purchases, as t he r esul t s i n Tabl e 6 i ndi cat e. H owever, we find t hat r epeat purchases of H ei nz have a ver y l ow R 2. Hence even after including brand l oyal t y (and ot her obser ved het er ogenei t y) a l ar ge ext ent of unobser ved het er ogenei t y remains. This is an important findi ng si nce most r esear cher s pr esumed t hat t he mai n source of het er ogenei t y i s di ffer ences i n consumer pr efer ences. The r emai ni ng het er ogenei t y may be due t o non-st at i onar i t y i n t he consumer pr efer ences or het er ogenei t y i n t he purchase t i mi ng pr ocess. Or due t o a combi nat i on or dependence bet ween t he purchase t i mi ng and br and choi ce (for exampl e, dependence wi l l exist when a particular brand uses a promotional strategy which attracts more frequent buyers). Thi s i s an ar ea of future r esear ch. 19

21 4.3. Effect s of het er ogenei t y on par amet er est i mat es Fol l owi ng we wi l l di scuss t he r esul t s of t he par amet er est i mat es of t he compet i ng r i sk model. A syst em of si xt een t r ansi t i ons r at es i s est i mat ed bet ween t he 4 br ands (4 for r epeat purchases and 12 for swi t chi ng bet ween br ands). To det er mi ne t he effect of het er ogenei t y on par amet er est i mat es, t he par amet er s for t he ei ght model s from Tabl e 5 ar e est i mat ed. To pr eser ve space onl y r esul t s for t he t r ansi t i ons from H ei nz t o ot her brands are provided in Tables 7a-d. Par amet er est i mat es di ffer consi der abl y dependi ng on t he var i abl es i ncl uded. We compar e al l r esul t s t o t he l east r est r i ct ed model wi t h unobser ved het er ogenei t y (model 8). I t has been wel l est abl i shed i n t he l i t er at ure t hat a model which includes unobser ved het er ogenei t y pr ovides bet t er par amet er est i mat es (see, for example, Heckman and Singer 1984, and Heckman and Walker 1987). I n addi t i on, model 8 has t he best fit for t he dat a as shown i n t he r esul t s of Tabl e 5, and al l coeffici ents have t he cor r ect si gn. M odel s 4 and 8 ar e t he onl y model s t o i ncl ude unobser ved het er ogenei t y. I n al l i nst ances t wo mass or suppor t poi nts wer e needed t o appr oximat e t he empi r i cal unobser ved het er ogenei t y di st r i buti on (see equat i on 7). Befor e compar i ng t he par amet er est i mat es of t he di ffer ent model s, we br i efly summar i ze t he r esul t s for al l si xt een t r ansi t i ons for model 8. Al l t he het er ogenei t y coeffici ents ar e si gni ficant at t he.05 l evel of confidence. M ost br and l oyal t y par amet er s ar e si gni ficant, posi t i ve for t he r epeat purchaser s and negat i ve for t he swi t cher s. Fami l y size is significant for all but two transitions. Income is negatively related with brand choi ce, except for t he t r ansi t i ons from Gener i cs. The dummy var i abl e for t he t est mar ket (indicating market 1 or 2), is also significant for most transitions. For the marketing mix variables price is significant in several cases, while advertising, price promotions and coupons were insignificant for most transitions. Price is particularly important for 20

22 swi t ches t o Del monte and Gener i cs (t he l ower pr i ced br ands), and for r epeat purchases of Generics. The advertising variable is significant for switching from Hunts to Heinz, and for a switch from Heinz to Hunts. This is local advertising which is mostly gathered t owar ds pr i ce pr omot i ons. Pr i ce speci al s ar e onl y si gni ficant for swi t chi ng from H unts t o H ei nz and from Del monte t o H ei nz, whi l e coupons ar e onl y si gni ficant for r epeat purchases of H ei nz. These r esul t s show asymmet r y i n t he effect s of mar ket i ng mi x var i abl es. I n par t i cul ar for pr i ce wher e al l swi t ches from H ei nz and H unts t o Del monte and Gener i cs wer e si gni ficant, whi l e al l swi t ches from Gener i cs and Del monte t o H unts and H ei nz wer e i nsi gni ficant. Furt her mor e, some di ffer ences exist i n t he val ues of t he par amet er s. The par amet er s for t he househol d speci fic var i abl es ar e far mor e st abl e for different transitions. An exception is income which is insignificant for all switching from Gener i cs, whi l e si gni ficant for most ot her t r ansi t i ons. M odel 1, i s a model without explanatory variables, including only duration terms. M ost of t he durat i on t er ms (see equat i on 3) ar e di ffer ent for each t r ansi t i on and ar e si gni ficant, l eadi ng t o t he r eject i on of common di st r i buti ons l i ke t he exponenti al, t he ehrlang, and t he Wei bul l, i ndi cat i ng t he exist ence of durat i on dependence (t he pr obabi l i t y of a purchase does not r emai n const ant over t i me). The durat i on t er ms r emai n fai r l y const ant i n al l 4 t abl es except when unobser ved het er ogenei t y i s added (models 4 and 8). H et er ogenei t y i n purchase t i mi ng may be caused by a var i et y of var i abl es such as: di ffer ences i n usage r at es (l i ght ver sus heavy user s), consumer demogr aphi cs (fami l y si ze, i ncome, et c.), consumer pr efer ences (l oyal s ver sus swi t cher s), consumer shopping behavior (see, for exampl e, Popkowski L eszczyc and Ti mmer mans 1996), or di ffer ent pr omot i onal st r at egi es. We di d not obser ve si gni ficant di ffer ences i n purchase t i mi ng bet ween r epeat purchaser s and swi t cher s. Some purchase accel er at i on 21

23 was obser ved due t o pr i ce pr omot i ons. Whi l e we i ncor por at e non-stationarity and het er ogenei t y i n t he i nter purchase t i mi ng di st r i buti on het er ogenei t y i n t he basel i ne hazar d funct i ons r emai ns. We al so pl ot t ed t he basel i ne hazar d funct i ons separ at el y for heavy and light user s and compar ed t hese wi t h t he aggr egat e di st r i buti ons. Si gni ficant di ffer ences bet ween t he basel i ne hazar d funct i ons for t he l i ght and heavy user s exist. Ther efor e, we i ncl uded a dummy var i abl e for heavy user s i n our model. Thi s r educes t he aggregation or het er ogenei t y bi as i n t he hazar d funct i on. Compar i ng model s t wo and ei ght we obser ve ser i ous par amet er bi as for t he mar ket i ng mi x var i abl es when het er ogenei t y i s excl uded. For exampl e, for a r epeat purchase of H ei nz (Tabl e 7a) pr i ce and pr i ce speci al are significant for model 2, but insignificant for model 8. For model 2 advertising is significant and has the wrong sign for a swi t ch from H ei nz t o Gener i cs (Tabl e 7d). Once het er ogenei t y i s added adver t i si ng becomes i nsi gni ficant. Besi des t he ei ght model s we al so est i mat ed a model wi t h onl y t he mar ket i ng mi x var i abl es and unobser ved het er ogenei t y, r esul t s wer e si mi l ar t o model 8's results. Model 3 includes consumer demographics, which remain fairly constant when unobser ved het er ogenei t y i s added. M odel 5 includes marketing mix variables and consumer demogr aphi cs. When compar ed t o model 8, t he r esul t s i mpr ove si gni ficantl y over model 2's r esul t s. H owever, for a r epeat purchase of H ei nz (Tabl e 7a) coupon i s i nsi gni ficant for model 5 but becomes si gni ficant once unobser ved het er ogenei t y i s added. Model 6 adds brand loyalty and model 7 adds usage rate (indicating heavy or light usage). Bot h model s pr ovide r esul t s si mi l ar t o model 8's r esul t s. H ence, our r esul t s indicate that it is sufficient to include obser ved het er ogenei t y t o over come bi as i n t he par amet er s of t he mar ket i ng mi x var i abl es. Addi ng unobser ved het er ogenei t y does l ead t o an i mpr ovement i n fit of t he model. 22

24 **** Insert Tables 7a-d about her e * * * * 5. Conclusion and Future Research The object i ve of t hi s r esear ch has been t o find out mor e about het er ogenei t y, what i t i s and how t o model i t. We di st i ngui shed bet ween obser ved and unobser ved het er ogenei t y, and par t i al ed out t he effect s of het er ogenei t y. We t r i ed t o det er mi ne how much of buyer behavior and which buyers, are affected by marketing mix variables. We find t hat par amet er s of a model i ncl udi ng onl y mar ket i ng mi x var i abl es ar e ser i ousl y bi ased. Our r esul t s show t hat i ncl udi ng bot h obser ved and unobser ved het er ogenei t y l eads t o a si gni ficant i mpr ovement i n t he l i kel i hood val ue of t he model. H owever, i ncl udi ng unobser ved het er ogenei t y does not have a l ar ge i mpact on t he par amet er est i mat es. Addi ng consumer demogr aphi cs, l oyal t y and usage, adjust s for t hi s bi as. These consumer speci fic t i me per si st ent var i abl es when omi t t ed l ead t o cor r el at i ons over t i me wi t h t he er r or t er m, and cause par amet er bi as. The r emai ni ng unobser ved het er ogenei t y (t he l ast 5 var i abl es i n Tabl e 2) i s mor e r andom and does not ser i ousl y bi as parameter est i mat es. Ther efor e, i f i nter est l ays i n est i mat i ng t he mar ket i ng mi x var i abl es i t may be suffici ent t o i ncl ude onl y obser ved het er ogenei t y. Thi s consi der abl y si mpl i fies t he model and est i mat i on pr ocedure. Brand loyalty and usage rate and regional di ffer ences wer e found t o be t he most i mpor t ant contr i butor s t o obser ved het er ogenei t y i n our st udy. H ousehol d demogr aphi cs and marketing mix variables have a significantly smaller impact (the amount purchased was not significant). Non-stationarity in purchase timing was found to be significant due t o t he r eject i on of t he exponenti al basel i ne hazar d funct i on. We found t hat after 23