A Multiplicative Fixed-Effects Model of Consumer Choice
Authors: Papatla Purushottam 
Published: October 2008

The issue of consumer heterogeneity in discrete choice analysis has been attracting much attention recently. Research has suggested that heterogeneity can result in biased parameter estimates which, in turn, can lead to incorrect conclusions. Among the many methods proposed in the literature to handle heterogeneity, fixed effects models seem to be the most attractive from a substantive point of view. However, in order to provide consistent estimates, these models typically require long purchase histories. This difficult constraint has prevented their widespread use.

In this paper, we propose a new model which offers the benefits of fixed effects models without requiring long purchase histories. Our approach differs from the classic formulation in two ways. First, we calibrate the common and fixed-effects sequentially rather than simultaneously. This two-stage estimation permits us to obtain unbiased estimates of the common parameters even when the sample has households with very few observations. Second, we incorporate the fixed effects in a multiplicative rather than in an additive form.

Our assumptions regarding the error term result in a probit model with heteroskedastic and temporally correlated random utilities. We develop a method of moments based estimator to calibrate parameters of models with this type of error structure. This procedure exploits the property that the method of moments yields estimates which are asymptotically as efficient as the maximum likelihood estimates given an appropriate starting point and a matrix of instruments. This is a very general procedure in that its mechanics are not tied to the proposed model. It can be readily applied to other problems where probit models are used to analyze data with serial correlation. This is a methodological contribution of the paper.

The model and estimation procedure are illustrated on the A. C. Nielsen scanner data base for the liquid detergent category. Our results indicate that the proposed model can provide a better fit than random effects and loyalty based models and better predictive performance than random effects models. From a substantive perspective, our analysis provides the following findings.

(a) the distribution of price sensitivities has a low variance—thus, most households have very similar price sensitivity in this category;

(b) the promotion response distribution has a high variance—suggesting that there are some households that are highly responsive to promotions while there are quite a few households that are less influenced by promotions;

(c) preferences and price sensitivity have a negative covariance—which is an indication that a reduction in preference for a brand is associated with an increase in price sensitivity and vice versa;

(d) preferences and promotion response have a positive covariance—thus, as the preference for a brand increases, response to promotions by the brand also increases; and

(e) price sensitivity and promotion response have a negative covariance—implying that, as households become more price sensitive, purchases may be driven more by price than by promotions; in other words, rather than responding to promotions such as displays and features as signals of a reduced price, price sensitive households actively evaluate prices in arriving at a choice.

The proposed approach can be used by theoretical as well as applied researchers of brand choice behavior. Specifically, estimates of fixed-effects provided by the model can be used to cluster households into groups with similar parameters. Profiling the resulting groups in terms of demographics would provide interesting insights—from a theoretical perspective—regarding the household characteristics that are associated with different types of response behavior. It would also help managers to better target their brands.

Using the proposed approach for cross-category analyses of fixed-effects would be an interesting area of future research. For instance, given a sample of households and their purchase histories in multiple categories, the proposed model can be used to explore the relationship between household behavior and category characteristics. In particular, issues such as the following can be investigated by comparing household-level estimates across categories:

(a) are households that are highly price sensitive in one category also price sensitive in other categories?

(b) what household characteristics are associated with high (low) price sensitivity or high (low) promotion response across categories? and

(c) in what types of categories are households likely to exhibit high (low) price sensitivity or high (low) promotion response?