Probabilistic versus random-utility models of state dependence: an empirical comparison

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1 Intern. J. of Research in Marketing 20 (2003) Probabilistic versus random-utility models of state dependence: an empirical comparison P.B. Seetharaman* John M. Olin School of Business, Washington University, St. Louis, MO , USA Received 1 January 2001; received in revised form 1 July 2002; accepted 8 July 2002 Abstract Brand choice models estimated on scanner panel data typically show that a household s brand choice process is characterized by state dependence, i.e. a household s brand choices are serially correlated over time. Two approaches have been employed by marketing researchers to estimate state dependence effects using brand choice data. The first approach is based on probability models such as Markov Chains and Linear Learning Models that directly allow a household s brand choice probabilities to be temporally correlated. The second approach is based on random utility models such as the Multinomial Probit with serially correlated error terms that allow a household s latent utilities for brands to be temporally correlated, and then derive the household s brand choice probabilities as the first-order conditions for the household s utility-maximization problem. The random utility approach has acquired prominence in recent years given the increasing influence of economic models, and hence a utility-based view of consumer decision-making, in marketing. However, the first approach has served a fruitful role for over four decades in terms of accurately tracking and predicting brand choices. In this study, we explicitly compare a probabilistic model versus a random utility model of state dependence both in terms of their ability to explain and predict observed brand choices of households, and in terms of the marketing mix elasticities that they yield. We estimate both models using scanner panel data on households purchases in four different categories of packaged goods. Using either model, we quantify significant state dependence effects along two dimensions. Interestingly, despite the differences in their mathematical foundations, we find both models to be remarkably similar in terms of predicting observed brand choices and in terms of the their recovery of marketing-mix elasticities. D 2003 Elsevier Science B.V. All rights reserved. Keywords: State dependence; Probabilistic models; Random-utility models; Stochastic choice; Brand choice 1. Introduction Brand choice models have a rich history in Marketing. Starting with the early probabilistic brand choice models (Massy, Montgomery, & Morrison, 1970), * Tel.: ; fax: address: seethu@olin.wustl.edu (P.B. Seetharaman). quantifying the dependence of a household s current brand choice on the household s previous brand choices has occupied the interest of empirical marketing researchers for over four decades. Empirical support for such inter-temporal dependencies in brand choices, also known as state dependence effects, has been obtained in various contexts using householdlevel data. State dependence effects have been shown to persist even after accounting for the effects of market /03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi: /s (02)

2 88 P.B. Seetharaman / Intern. J. of Research in Marketing 20 (2003) ing variables and unobserved heterogeneity on brand choices in a flexible manner (Abramson, Andrews, Currim, & Jones, 2000; Ailawadi, Gedenk, & Neslin, 1999; Gupta, Chintagunta, & Wittink, 1997; Seetharaman & Chintagunta, 1998). The estimated state dependence effects have been widely explained using behavioral phenomena such as inertia, variety-seeking, learning, preference renewal at the household-level, reference-dependence effects etc. (Fader & Lattin, 1993; Hardie, Johnson, & Fader, 1993; Seetharaman, Ainslie, & Chintagunta, 1999; Trivedi, Bass, & Rao, 1994). There exists two streams of empirical research on the estimation of state dependence effects in households brand choices. The first stream, also called the stochastic choice approach, employs probabilistic brand choice models to estimate temporal dependencies in a household s brand choices. In these models, the household s probability of buying a brand at time t is assumed to depend directly on either the household s previous brand choice at time t 1 (see Jeuland, 1979), or the household s previous probability of buying the brand at time t 1 (see Kuehn, 1962). The second stream, also called the discrete choice approach, employs random utility models to estimate temporal dependencies in a household s brand choices. In these models, the household s random utility (as opposed to its brand choice probability) for a brand at time t is assumed to depend on either the household s previous brand choice at time t 1 (see Guadagni & Little, 1983), or the household s previous random utility for the brand at time t 1 (see Allenby & Lenk, 1995). In recent years, the discrete choice approach has acquired more prominence in Marketing on account of the increasing recognition given to economic theory while building estimable statistical models. Both streams of research, therefore, have focused on two separate sources of state dependence in a household s brand choices: one, the effect of a lagged brand choice (lagged choice effect); two, the effect of a lagged brand choice probability or a lagged utility (lagged evaluation effect). The difference between these two sources can be explained as follows: Suppose a household s purchase probability or random utility for a brand (say, brand 1) goes up during week t due to the effects of newspaper advertising for the brand. However, suppose the household ends up buying a different brand (say, brand 2) since it was on deal that week. When the same household purchases the next time in the category during week t +1, the effects of the increased purchase probability or random utility for brand 1 may persist even though the brand was not purchased during the previous occasion. This persistence is embodied in the lagged evaluation effect. On the other hand, brand 2 may benefit in week t + 1 from an increased purchase probability on account of the household s previous purchase of brand 2 (in week t). This persistence is embodied in the lagged choice effect. Explicit disentanglement of these two sources of state dependence from each other has been the focus of a recent paper, specifically, Keane (1997). This paper employs a random utility model to separately estimate the two sources of state dependence, i.e. lagged choice effect and lagged evaluation effect (referred to as structural state dependence and habit persistence, respectively, in the labor econometrics literature, see Flaig, Licht, & Steiner, 1993; Heckman, 1981), and shows both effects to be distinct and significant in brand choice data. Estimating the parameters of this random utility model entails numerical simulation since the density function of the error terms, on account of their serial correlation, does not have a closed-form. Given the long history of probabilistic brand choice models in Marketing, it would be of interest to investigate whether estimating the two sources of state dependence is possible within a probabilistic choice framework. Such an investigation would be especially useful if the probabilistic brand choice model involves estimable equations that have an analytical closed-form. If such an easy-to-implement probabilistic brand choice model can be proposed, it would then be of interest to understand whether the results obtained using such a model are consistent with those obtained using the theoretically more appealing random utility brand choice model. These are the primary objectives of this study. We propose a parsimonious, easy-to-estimate probabilistic choice model that explicitly disentangles the lagged choice effect from the lagged evaluation effect. We also propose a random utility model, in the spirit of Keane (1997), which disentangles the two sources of state dependence from each other. We compare the results obtained using

3 P.B. Seetharaman / Intern. J. of Research in Marketing 20 (2003) the two proposed brand choice models in terms of both their ability to fit and predict observed brand choices and their estimated state-dependence and marketing-mix parameters. We find a remarkable degree of consistency between the two models on all fronts. In other words, the computationally easyto-implement, yet theoretically ad-hoc probabilistic brand choice model performs just as well as the economic-theoretic, yet computationally more demanding, random utility brand choice model. These findings generalize across four different categories of packaged goods. This demonstrates the contemporary relevance and versatility of probabilistic brand choice models, pioneered by the likes of Bass, Morrison, etc., as far back as the 1960s and 1970s. 2. Model and estimation We propose two brand choice models in this section: first, a probabilistic brand choice model that explicitly disentangles the two sources of state dependence-lagged choice effect and lagged evaluation effect from each other, while also accounting for the effects of marketing variables and unobserved heterogeneity; second, a random utility brand choice model with identical objectives Probabilistic brand choice model with state dependence Kuehn (1962) proposed a binary 1 brand choice model that allows for inter-temporal dependencies in a household s brand choices. According to this model, a household s probability of purchasing a brand (also called the focal brand) is greater if the same brand had been purchased in the previous purchase occasion than otherwise. Mathematically, if P t and P t 1 refer to the household s choice probabilities for the focal brand during purchase occasions t and t 1, respectively, and Y t 1 is a dummy variable that takes the value of 1 if the focal brand was purchased during 1 It models the household s purchase of a focal brand versus all other brands in the product category (lumped together as the other brand). occasion t 1, and takes the value of 0 otherwise, then the relationship between P t and P t 1 is as shown below. P t ¼ d þ cy t 1 þ kp t 1 ; where d, c and k are parameters such that d; c; kz0; d þ c þ kv1: ð1þ ð2þ This model implies that P t is greater by c if the focal brand were purchased at the previous purchase occasion than otherwise. However, this model is applicable for a two-brand market only, and ignores the effects of marketing variables and unobserved heterogeneity across households. We relax these restrictions next. In the multi-brand context, Kuehn s (1962) model can be modified to have brand subscripts. For example, in a product category with J brands: P j;t ¼ d j þ cy j;t 1 þ kp j;t 1 ; ðja1;...; JÞ; ð3þ where P j,t, P j,t 1 and Y j,t 1 are defined as before, except that they now pertain to brand j (among a set of J brands). Also, the parameter d is now brand-specific (hence the brand subscript j). Since the brand choice probabilities across the J brands must sum to one, this introduces an additional constraint as shown below. d j þ c XJ which reduces to d j þ c þ k ¼ 1: Y j;t 1 þ k XJ P j;t 1 ¼ 1; ð4þ ð5þ In order to satisfy this constraint, we re-parameterize d j as shown below d j ¼ q ea j e a j ; ð6þ

4 90 P.B. Seetharaman / Intern. J. of Research in Marketing 20 (2003) where qa[0, 1] and aar. The parameter a j captures the household s intrinsic preference for brand j (much like the intercept term in random utility models of brand choice). The multi-brand version of Kuehn s (1962) model can be written as follows. P j;t ¼ q ea j e a j þ cy j;t 1 þ kp j;t 1 ; ðja1;...; JÞ; where q, c and k are parameters such that q; c; kz0; q þ c þ k ¼ 1: ð7þ ð8þ This model has the following attractive interpretation as a probability mixture model at the observational level (as in Allenby & Yang, 2000): With probability q the household buys according to an unchanging vector of purchase probabilities, with probability c the household repeat purchases the previously bought brand, and with probability k the household buys according to its vector of purchase probabilities at its previous purchase occasion. If c = k = 0, the model implies zero-order behavior. If k = 0, the model implies first-order (Markov) behavior. If k p 0, the model implies infinite-order behavior (as long as either q or c is non-zero) 2. These three possibilities are mutually exclusive and collectively exhaustive. Therefore, the three corresponding probabilities add up to one. The first two possibilities (i.e. zero-order and first-order behavior) are easy to understand. For example, c = k = 0 reduces the model to the multinomial choice model of Bass (1974), while k =0 reduces the model to the inertia model of Jeuland (1979). To understand the third possibility (i.e. infinite-order behavior), we examine the mathematical implications of the third term (that contains the previous period s purchase probability vector). Clearly, this term makes the model recursive, since 2 We explain this in the next paragraph. P j,t 1 depends on P j,t 2, which in turn depends on P j,t 3 and so on. If one performs this recursive back-substitution, the model reduces to the following equation. P j;t ¼ q ea j e a j ð1 þ k þ k 2 þ...þ k t 1 Þ þ cðy j;t 1 þ ky j;t 2 þ k 2 Y j;t 3 þ... þ k t 1 Y j;0 Þþk t P j;0 : ð9þ From this equation, we see that for sufficiently large k, the order of the brand choice process will be much larger than one, 3 i.e. the current brand choice probability not only depends on the most recent brand choice but also on more distant brand choices in the past. This way, the proposed model parsimoniously allows us to mathematically approximate 4 probabilistic brand choice models of any arbitrary order. We allow the brand choice probabilities to be nonstationary by incorporating the effects of marketing variables as shown below (see the first term on the right-hand side of the equation that has a component identical to the Multinomial Logit likelihood). P j;t ¼ q e a jþx j;t b e a kþx k;t b þ cy j;t 1 þ kp j;t 1 ; ðja1;...; JÞ; ð10þ where X j,t stands for the vector of marketing variables characterizing brand j at time t, and b stands for the corresponding vector of response coefficients. If k = 0, we get the Inertia Model of Seetharaman and Chintagunta (1998) and Jeuland (1979). If k =0, c = 0 and q = 1, we get the Multinomial Logit Model of Gensch and Recker (1979). Ifk =0, c =0, q =1 3 The parameter k looks similar to the discount factor in optimal control problems and to the carryover parameter in Koyck models, both of which also measure the order (or time horizon) of dynamic processes. 4 Technically, for k p 0 it is always an infinite-order effect. However, the recovered estimate of k allows us to understand how many of the infinite past periods actually matter.

5 P.B. Seetharaman / Intern. J. of Research in Marketing 20 (2003) and b = 0, we get the Multinomial Choice Model of Bockenholt (1993). Ifb =0, S = 1 and J = 2, we get the Linear Learning Model of Kuehn (1962). Rewriting the proposed probabilistic choice model in Eq. (10) using recursive back-substitution yields the following equation. P j;t ¼ q e a jþx j;t b e a kþx k;t b þ k 2 þ k t 1 þ k e a jþx j;t 2 b e a kþx k;t 2 b e a jþx j;t b e a kþx k;t b e a jþx j;t 1 b e a kþx k;t 1 b þ ::: þ cðy j;t 1 þ ky j;t 2 þ k 2 Y j;t 3 þ...þ k t 1 Y j;0 Þþk t P j;0 : ð11þ From this equation, we see that both the history of lagged brand choice probabilities and the history of lagged brand choices have identical decaying effects on the current brand choice probabilities. While a preponderant number of previous probabilistic brand choice models have focused on the effects of lagged brand choices, they have typically ignored the effects of lagged brand choice probabilities, i.e. lagged brand evaluations. Our proposed probabilistic brand choice model contributes in this regard Random utility brand choice model with state dependence A random utility brand choice model that incorporates the effects of lagged choices and lagged evaluations on households random utilities for brands is shown below. U jt ¼ a j þ X jt b þ cðy jt 1 þ Y jt 2 v þ Y jt 3 v 2 where U jt stands for the household s random utility for brand j at time t, c and v are parameters used to operationalize the effects of lagged choices, k is an MA(l) parameter capturing serial correlation in the error terms, and e jt is an error term that is distributed iid type-i extreme value. The lagged choice effect in Eq. (12) is given by cðy jt 1 þ Y jt 2 v þ Y jt 3 v 2 þ Y jt 4 v 3 þ...þ ð13þ and the lagged evaluation effect in Eq. (12) is given by u jt ¼ e jt þ e jt 1 k þ e jt 2 k 2 þ e jt 3 k 3 þ... ¼ e jt ð1 þ /L þ /L 2 þ...þ¼e jt ð1 /LÞ 1 ð14þ where L is the lag operator that satisfies the relation Le j,t = e j,t 1. Remembering that the brand loyalty measure of Guadagni and Little (1983) is given by BL jt ¼ cv½ð1 vvþy jt 1 þ vvbl jt 1 Š ¼ cv½ð1 vvþy jt 1 þ vvfcv½ð1 vvþy jt 2 þ vvbl jt 2 ŠgŠ ] ¼ cv½ð1 vvþðy jt 1 þ Y jt 2 vvþ Y jt 3 v 2V þ...þ Y jo v t 1V Þ ð15þ it is clear that our random utility specification of the lagged choice effect is identical to the brand loyalty specification of Guadagni and Little (1983), where v = vv, c = cv(1 vv). Also, our measure of the lagged evaluation effect in Eq. (14) can rewrite this as 5 ð1 /LÞu jt ¼ e jt ; ð16þ which is identical to the first-order autoregressive process, AR(1), used by Allenby and Lenk (1995) and Keane (1997). Therefore, this random utility þ Y jt 4 v 3 þ...þþðe jt þ e jt 1 k þ e jt 2 k 2 þ e jt 3 k 3 þ...þ ð12þ 5 An infinite-order moving average (MA) process can always be written as a finite-order autoregressive (AR) process (Johnston, 1991).

6 92 P.B. Seetharaman / Intern. J. of Research in Marketing 20 (2003) model, while faithful to previously proposed operationalizations of the lagged choice effect and the lagged evaluation effect, explicitly disentangles the two effects from each other. If k = 0, we get the random utility model of Guadagni and Little (1983). If v = 0, we get the habit-persistence model of Heckman (1981). Ifk =0, v = 0, we get the Multinomial Logit Model of Gensch and Recker (1979), etc. This random utility brand choice model is directly comparable to the probabilistic brand choice model in Eq. (10). While c and k capture the lagged choice effect and the lagged evaluation effect respectively in both models, v is an additional source of the lagged choice effect in the random utility model Estimation The likelihood function at the household-level can be written as shown below. ( ) L h ¼ YN h Y J ; ð17þ t¼1 j¼1 P d hjt h;j;t where the subscript h is used to qualify household h, N h stands for the number of purchase occasions corresponding to household h, and d hjt is an indicator variable that takes the value 1 if brand j is purchased by household h at time t, and P hjt stands for the brand choice probability of household h for brand j at time t. In the probabilistic brand choice model, P hjt has an analytical closed form given by Eq. (10). In the random utility brand choice model, P hjt does not have a closed form and has to be evaluated by numerical simulation using the following scheme (Landwehr, Matalas, & Wallis, 1979). e i ¼ lnf lnuðz i Þg; qffiffiffiffiffiffiffiffiffiffiffiffiffi z i ¼ /z i 1 þ 1 / 2 u i ; ð18þ Table 1 Descriptive statistics Brand Price (US$/oz.) Display Feature Share (%) Heinz Control Hunts Heinz DelMonte Heinz Heinz Heinz Number of households = 529. Number of purchases = model. Our estimation results are meant to reveal whether this additional computational effort is warranted, i.e. whether the fit, predictive ability and the parameter estimates of the probabilistic brand choice model are comparable to those of the random utility brand choice model. Finally, we incorporate unobserved heterogeneity by allowing the model parameters to follow a multivariate, discrete distribution across households (Kamakura & Russell, 1989). This yields the following sample likelihood function. " ( L ¼ YH X "! S Y N h Y # )# J p s P h;s;jo ;1 P h;s;j;t ; h¼1 s¼1 t¼2 j¼1 ð19þ where S refers to the number of mass points characterizing the multivariate discrete distribution, p s refers to the densities corresponding to these mass points, P h,s,j,t stands for the choice probability for brand j for household h at time t using the parameter vector corresponding to mass point s, and j o stands for the brand purchased by the household on its first purchase occasion. We address the problem of initial conditions by assuming that the household s first brand choice ( j o ) is characterized by the Multinomial Logit probability 6 (for a comprehensive study of the where U is the standard normal cdf, / is the firstorder autocorrelation parameter, and u i is a standard normal variate. In other words, the random utility brand choice model is computationally more demanding to estimate than the probabilistic brand choice 6 We also re-estimate the proposed model by ignoring the initial conditions, and computing each household s likelihood function starting from its second brand choice only. The results are shown to be insensitive to either choice of specification for the initial conditions.

7 P.B. Seetharaman / Intern. J. of Research in Marketing 20 (2003) initial conditions problem in state dependence models, see Degeratu, 1999). As a benchmark for our two proposed brand choice models of state dependence, we estimate the state dependence model of Roy, Chintagunta, and Haldar (1996), which is based on the following transition probabilities for the Markov chain that characterizes households brand-switching behavior. P jtait 1 ¼ / þð1 /Þ e a jþx jt bþy jt 1 c X K e a kþx kt bþy jt 1 c ; ð20þ where brands i and j are purchased by the household at occasions t 1andt, respectively, /a[0,1] measures the serial correlation in households utility-maximizing brand choices on successive purchase occasions (referred to as habit-persistence by the authors), and c measures the lagged choice effect (referred to as state dependence by the authors). The parameters / and c are in the same spirit as the lagged evaluation and the lagged choice effects in our proposed brand choice models. 3. Empirical results We employ A.C. Nielsen s scanner panel data on household purchases in four different product categories: ketchup, toilet tissue, laundry detergents and yogurt. Since the empirical findings are remarkably consistent across the four categories, for the sake of brevity, we report and discuss the results only for ketchup in this paper. 7 This dataset covers a period of 2 years from January 1985 to January Choosing households that bought only among the top eight brands (that account for 87% of all product sales) of ketchup yields 3032 households. From these households, we use only those that made at least seven purchases over the study period. 8 This yields a sample of 529 households making a total of 5954 purchases in the category. The largest item is Heinz 32 oz., with a conditional market share of 37.8%. It also enjoys the 7 The results for the remaining three product categories are available upon request. 8 This was done to exclude irregular buyers of the product. Table 2 In sample and out-of-sample fits (log-likelihood values on top, BIC values at bottom) Category Proposed Proposed Roy et al. Multinomial brand choice model brand choice model (1996) logit (probabilistic) (random utility) Ketchup Tissue Detergents Yogurt Log-likelihoods based on the Guadagni and Little (1983) model are 6975, 23079, 3127 and maximum display activity in the category. Descriptive statistics pertaining to the data set are provided in Table 1. We estimate both the proposed probabilistic brand choice model (Eq. (10)) and the random utility brand choice model (Eq. (12)) by maximizing the sample likelihood function (Eqs. (17) and (19)). Additionally, for benchmarking purposes, we estimate the state dependence model of Roy et al. (1996). Given in Table 2 are the results of a fit comparison, both in-sample and out-of-sample, 9 between the three models (for all four product categories) as well as a model that ignores state-dependence effects altogether, i.e. Multinomial Logit. Both of the proposed brand choice models of state dependence-probabilistic and random utility-fit significantly better than the Roy et al. (1996) model in all product categories. This emphasizes the importance of modeling both sources of state dependence-lagged choice effect and lagged evaluation effect as laid out in our formulation while estimating brand choice models. The random utility model outperforms the probabilistic model in terms of fit and prediction in 9 For out-of-sample predictive validation, we hold out 20% of each household s purchases in ketchup and toilet tissue, and 20% of all the households in detergents and yogurt, and use the estimates from the included observations to predict the outcomes in the excluded observations.

8 94 P.B. Seetharaman / Intern. J. of Research in Marketing 20 (2003) Table 3 Parameter estimates for ketchup based on a three-support heterogeneity distribution a Parameter Proposed brand choice model (probabilistic) Proposed brand choice model (random utility) Roy et al. (1996) Multinomial logit model a 1 (Heinz 32) 2.81, 2.21, , 2.79, , 3.02, , 3.20, 0.82 a 2 (Control 32) 0.55, 0.82, , 0.10, , 1.82, , 3.31, 2.54 a 3 (Hunts 32) 2.10, 0.64, , 2.02, , 2.55, , 3.19, 0.62 a 4 (Heinz 28) 2.69, 2.06, , 2.89, , 2.43, , 2.22, 1.15 a 5 (D. Mont 28) 1.59, 0.09, , 1.79, , 1.94, , 2.53, 3.05 a 6 (Heinz 14) 1.63, 0.96, , 1.64, , 1.58, , 1.66, 0.94 a 7 (Heinz 44) 0.85, 0.74, , 1.04, , 0.61, , 0.25, 0.94 a 8 (Heinz 64) Price 5.09, 2.66, , 4.22, , 3.26, , 2.43, 0.92 Display 0.52, 0.53, , 0.56, , 0.46, , 0.46, 0.57 Feature 0.53, 0.29, , 0.32, , 0.42, , 0.21, 0.30 c (lag choice) 0, 0.24, , 0.85, , 0.15, 0.54 NA v NA 0.76, 0.35, 0.84 NA NA k (lag eval.) 0.16, 0.63, , 0.22, 0.84 NA NA Support prob. 0.35, 0.54, , 0.31, , 0.48, , 0.34, 0.15 LL a All reported estimates are statistically significant ( p < 0.01). Standard errors have been suppressed for readability. three out of the four product categories. This is not surprising since the random utility model has an additional parameter (v) to capture the lagged choice effect, compared to the probabilistic model. We present the parameter estimates for the two proposed models as well as the benchmark models, i.e. Roy et al. (1996) and the MNL, in Table 3. The estimates are based on a three-support distribution for unobserved heterogeneity (which was found to be optimal from the standpoint of segment interpretability since the four-support solution yielded one support with negligible mass). It is clear from both of the proposed state dependence models that state dependence manifests itself not only in terms of the lagged choice effect (c), but also in terms of the lagged evaluation effect (k). In both models, higher price sensitivity is associated with lower inter-temporal dependence in brand choices, i.e. a support with a higher absolute value of b price is associated with a lower value of c. For example, in the probabilistic brand choice model, support 1 has the highest absolute value of price sensitivity (b price = 5.09) and the lowest value of the lagged choice effect (c = 0). We report some measures of managerial interest in Table 4. Specifically, we report two measures pertaining to the largest brand in each product category (Heinz 32 oz. for ketchup, Wisk for detergents and SF for toilet tissue): the own price-elasticity and the equilibrium market share. We find that price-elasticities are consistently understated, while equilibrium market shares are consistently overstated, in models that either ignore or understate state dependence effects (i.e. the MNL and the Roy et al. (1996) model, respectively). Further, both measures appear Table 4 Own price-elasticities and equilibrium market shares for the largest brands in each product category Parameter Proposed brand choice model (probabilistic) Proposed brand choice model (random utility) Roy et al. (1996) Multinomial logit Ketchup 4.30, , , , 0.41 Detergents 5.30, , , , 0.39 Tissue 4.57, , , , 0.39

9 P.B. Seetharaman / Intern. J. of Research in Marketing 20 (2003) to be quite insensitive to the nature of the mathematical specification probabilistic versus random utility of the lagged choice and lagged evaluation effects. In other words, the two state dependence models yield similar estimates of equilibrium market shares of brands as well as price elasticities despite differences in their mathematical foundations. Next, we conduct a numerical simulation to quantify the long-term effects of a sales promotion, specifically a 20% price cut accompanied by a store display and newspaper feature advertisement on the largest brand (assuming no competitive reactions from other brands). Given in Fig. 1 are the results of such a simulation. Both the probabilistic model and random utility model predict a significantly larger incremental sales effect in the long-term compared to the Roy et al. (1996) model. This indicates that under-specifying the sources of state dependence in brand choices is likely to understate the long-term effectiveness of promotions (as discussed in Blattberg, Briesch, & Fox, 1995). The multinomial logit model, which entirely ignores state dependence effects, severely understates the effectiveness of the sales promotion by predicting a single-period increment only. The remarkable consistency in the predicted long-term impact of promotions between the two proposed models again underscores the ability of the parsimonious probabilistic model to effectively mimic the inferences obtained from the theoretically sounder, yet computationally more demanding, random utility model. 4. Conclusions Fig. 1. Quantifying the long-term effects of a promotion. (A) Ketchup. (B) Detergents. (C) Toilet tissue. In this study, we compare the empirical performance of two state dependence models, one of which is a probabilistic brand choice model (that is theoretically ad-hoc, but computationally easy to estimate) and the other is a random utility brand choice model (that is economic theoretic but more difficult to estimate), in terms of explaining observed brand choice data. Both models operationalize two distinct sources of state dependence the lagged choice effect and the lagged evaluation effect while accommodating non-stationarity due to the effects of time-varying marketing variables as well the effects of unobserved heterogeneity across households. We find that statistical and managerial inferences from the two models are quite similar to each other. This finding is of value to marketing practitioners whose desire to implement quick-and-easy models (such as our probabilistic model) may overwhelm their need for theoretically sounder models (such as our random utility model).

10 96 P.B. Seetharaman / Intern. J. of Research in Marketing 20 (2003) This finding also vindicates the continuing use of stochastic brand choice models in the marketing literature (as in, for example, Seetharaman & Chintagunta, 1998; Trivedi et al., 1994; etc.). It will be of interest to investigate whether the uncovered similarities in the empirical performance of the probabilistic and random utility models of state dependence would generalize to contexts where state dependence is a function of product attributes (Erdem, 1996; McAlister, 1982). It would also be of interest to investigate whether probabilistic and random utility models perform similarly in other domains of consumer choice modeling, such as multi-category purchase incidence behavior of households (as modeled in Chib, Seetharaman, & Strijnev 2002). We leave these as interesting avenues for future research. Acknowledgements We thank Dick Wittink and Chakravarthi Narasimhan for useful comments. 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