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Article Excerpt
This paper proposes the Additive Risk Model (ARM), first used by Aalen (1980), to explain households' interpurchase times. Unlike the Proportional Hazard Model (PHM), first proposed by Cox (1972), the ARM incorporates the effects of covariates on the individual hazard function in an additive (as opposed to multiplicative) manner. While a large number of previous studies on interpurchase timing have dealt with the question of correctly specifying the parametric distribution for interpurchase times, no study has explicitly investigated the question of correctly specifying the effects of covariates in the model. This study looks at this issue.
We propose an ARM that is suitable for purchase-timing data, and compare its empirical performance to that of the PHM and the Accelerated Failure Time Model (AFTM) using scanner panel data on laundry detergents, paper towels, and toilet tissue. We find that the ARM not only estimates and validates the observed interpurchase times better than existing models, but also recovers a time-varying price elasticity and shows a high degree of robustness in the estimated covariate effects to alternative parametric specifications of the baseline hazard. The estimates of covariate parameters under the PHM, on the other hand, are highly sensitive to alternative parametric specifications of the baseline hazard.
Key words: purchase timing models; additive risk model; proportional hazard model; accelerated failure time model; log-logistic hazard
History: This paper was received March 25, 2002, and was with the authors 1 month for 2 revisions; processed by Pradeep Chintagunta.
1. Introduction
There exists a sizable body of empirical research that deals with the estimation of purchase-timing decisions of households, using statistical models. These studies characterize households' temporal decisions of when to buy a particular product over time. A number of these studies use panel data to estimate the parameters of the purchase-timing model (for a recent study, see Seetharaman and Chintagunta 2003). Early empirical studies, mostly in the sixties and seventies, modeled households' interpurchase times using statistical distributions such as Erlang-2, negative binomial, etc. (see, for example, Chatfield and Goodhardt 1973). Later studies, mostly in the eighties, modeled households' interpurchase times using econometric models--such as the logistic regression model--in order to account for the effects of marketing variables (such as price, advertising, etc.). A statistical model that is well suited to capture both of these features of purchase-timing data, i.e., intrinsic temporal purchase patterns (captured in the early statistical models such as the NBD) and the effects of marketing variables (captured in the later econometric models such as the Logit), is the Proportional Hazard Model, first proposed by Cox (1972) and used in an extensive range of marketing applications over the past decade (see, for example, Jain and Vilcassim 1991, Helsen and Schmittlein 1993, Wedel et al. 1995, Chintagunta 1998).
In the Proportional Hazard Model (PHM henceforth), the construct of interest is a household's instantaneous probability of making a purchase in a product category, conditional on the elapsed time since the household's previous purchase in the product category. This conditional probability, also called the hazard function, is multiplicatively decomposed into two components: One, the baseline hazard captures the household's intrinsic temporal purchase pattern; two, the covariate function captures the influence of marketing variables. In other words, the baseline hazard characterizes the distribution of the household's interpurchase times after controlling for the effects of marketing variables. In the absence of marketing variables, suitable assumptions on the baseline hazard and unobserved heterogeneity across households in the PHM's parameters yield statistical purchase-timing models--such as the Erlang, NBD, CNBD, etc.--that have been extensively used to describe purchase-timing data for over three decades (Chatfield and Goodhardt 1973, Schmittlein and Morrison 1983, Wheat and Morrison 1990). Given its ability to handle time-varying covariates and nest traditional purchase-timing models as special cases, the PHM has become the most popular model to analyze interpurchase times of households.
In the past, researchers working with the PHM have tested a variety of parametric specifications for the baseline hazard. Some commonly used distributions are exponential, Erlang-2, Weibull, log-logistic, Gompertz, etc. Flexible baseline hazard specifications that permit a variety of different shapes for the baseline hazard have also been used. One such flexible specification is the Quadratic Box-Cox specification, used to analyze interpurchase times by Jain and Vilcassim (1991). Another flexible specification is the Expo-Power specification, recently proposed by Saha and Hilton (1997) to study equipment failure times, and used in a purchase timing context by Seetharaman and Chintagunta (2003). The weekly nature of households' shopping trips, coupled with the fact that not all shopping trips involve purchase of the product, has made the discrete-time version of the PHM (also called "Grouped PHM") more suitable for purchase-timing data than the continuous-time version (for an application of the discrete-time PHM, see Helsen and Schmittlein 1993).
Despite extensive work on testing alternative specifications of the baseline hazard, no empirical study on purchase timing has tested alternative specifications of the covariate function. For example, the plausibility of the PHM's assumption that marketing variables influence purchase timing by acting in a multiplicative manner on the baseline hazard has never been empirically tested using data. An alternative specification of covariate effects is the Accelerated Failure Time Model (AFTM henceforth), in which marketing variables are allowed to directly influence the shape parameter of the baseline hazard (Chintagunta 1998). A third possibility could be that marketing variables act on the baseline hazard in an additive manner (instead of a proportional manner as in the PHM, and a shape-shifting manner as in the AFTM). Such an assumption yields a purchase-timing model called the Additive Risk Model (1) (ARM henceforth),1 first proposed by Aalen (1980)...
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