## 1. INTRODUCTION

It is important for store managers to understand the influence of the marketing mix on purchase timing. This information can, for example, be used to determine the optimal time between promotional activities and for active stock management. To describe purchase timing several models have been proposed in the literature; see Gupta (1991), Jain and Vilcassim (1991), Helsen and Schmittlein (1993), Seetharaman (2004), and Seetharaman and Chintagunta (2003) for a recent overview. One usually aims at describing the relation between interpurchase times and various explanatory variables. These explanatory variables can be divided into two groups. The first group corresponds to household-specific variables, like household size and family income, but also variables such as the current stock of the product and the time since last purchase within the product category. These variables can be directly linked to the interpurchase times. The second group contains marketing-mix variables, like price and the presence of promotional activities. These variables cannot be directly linked to the interpurchase times, as marketing-mix variables are observed at the brand level and purchase timing is modeled at the category level.

In the ideal case, we would have knowledge of the preferred brand of each household at every moment in time. To explain purchase timing we could then use the marketing mix of the brand that is purchased or would be purchased at any moment in time. In practice this is, of course, not feasible. First of all, data collection would be practically impossible. Secondly, the household may not have a unique preferred brand at every point in time. Therefore the researcher should either somehow summarize the marketing efforts of all brands into a category-level index or construct an integrated model of choice and purchase timing. This final task is exactly the research question we address in this paper. The key question can be summarized as: how to include marketing-mix variables of individual brands in a category-level interpurchase time model?

One may think that the answer to this question is just to use the marketing mix of the purchased brand. The problem is, however, that brand choice is only revealed at purchase occasions and is not available at non-purchase moments. One may opt to use the marketing mix of the previously purchased brand for the non-purchase moments, but this is likely to be suboptimal as households may switch brands; see, for example, Vilcassim and Jain (1991). In fact, a household may change preferences several times between two purchases, especially if the marketing mix changes in this period, for example, due to promotions.

We are, of course, not the first to notice these problems in modeling interpurchase time. In every purchase timing study the researcher will have to decide how to construct category-level marketing-mix variables. An often-used solution is to consider a weighted average of brand-specific marketing-mix variables. The weights are usually household specific and are obtained from choice shares of the particular household; see, for example, Gupta (1988, 1991) and Chib *et al.* (2002). A disadvantage of weighting the marketing mix using choice shares is that household-specific information is required to obtain the weights. This approach is therefore less suitable for out-of-sample forecasting. Furthermore, as choice shares are by definition constant over (periods) of time the model does not take into account that preferences may change. A simple solution is to weigh the marketing mix with brand choice probabilities following from a logit specification, although there is a more elegant solution, which we will propose below.

Another popular approach amounts to using the so-called inclusive value from a brand choice model as a summary statistic for the marketing efforts in a category; see, among others, Bucklin and Gupta (1992), Chintagunta and Prasad (1998), and Bell *et al.* (1999). The inclusive value has the interpretation of the expected maximum utility over all brands in the category. The inclusive value naturally depends on the marketing mix of all brands. A large expected utility is likely to be positively correlated with the probability of a purchase in the category. Although theoretically appealing, this specification is rather restrictive. In the corresponding purchase timing model there is only one parameter that relates all marketing efforts of all brands to purchase timing: that is, the coefficient corresponding to the inclusive value. Moreover, the effects of marketing variables are restricted to be similar on choice as on purchase timing. Another problem may be that the relation between the inclusive value and purchase incidence may only hold within households. Between households there may be substantial differences in inclusive value that are not related to differences in purchase timing. A household with a strong brand preference may have a larger inclusive value than a household with less pronounced preferences. Of course, one cannot conclude from this that the former household will on average have shorter interpurchase times. The between-household differences will be even more pronounced when unobserved heterogeneity in brand preferences is incorporated in the brand choice model.

To meet the limitations of the above-mentioned solutions, we introduce in this paper a new model. The idea behind this model is to use brand choice probabilities as indicators of brand preferences. The brand choice probabilities are the best information we have on the preferences of households. In the model we combine brand-specific hazard functions using choice probabilities. During non-purchase weeks the brand choice is treated as a latent variable. We only observe this variable at the purchase occasion.

This idea not only potentially improves the purchase timing model, but it could also add to the performance of the part of the model related to brand choice. The fact that a household does not make a purchase in a particular week also reveals information about the preferences of this household. Although this information may be very useful, it is usually ignored when modeling brand choice. For example, consider the situation where a household frequently purchases a certain brand that is also frequently promoted. Assume that this household never purchases other brands when they are promoted. If one only considers purchase occasions one may overestimate the effect of promotions on brand choice as the non-purchase promotional activities are completely ignored. The fact that the household does not purchase the other brands even when they are promoted implies that it has a strong base preference for the frequently purchased brand.

Integration of the interpurchase time model and brand choice model could therefore lead to a better performance on explaining brand choice as well as interpurchase time. In the resulting model the brand choices of households are revealed at purchase occasions, while at non-purchase occasions the preferred brand is treated as a latent (unobserved) variable. In this way, we also use information revealed by households at non-purchase occasions to describe brand choices and interpurchase timing. We will call this specification the latent preferences purchase timing model. This integrated model is also useful if one is only interested in the purchase timing. In this case the model provides a coherent framework for including marketing-mix variables in the duration model.

The outline of this paper is as follows. In Section 2 we propose the latent preference purchase timing model. We briefly discuss two standard approaches in the literature as well as a third alternative. In Section 3 we compare our new approach with the alternative solutions using data on purchases in the detergent category. We present a comparison based on in-sample fit and out-of-sample forecasting performance. Furthermore, we discuss differences in estimates of key parameters in the different models. Finally, in Section 4 we conclude.