A Guadagni–Little Likelihood Can Have Multiple Maxima

Despite many advances in marketing models, the Guadagni–Little (1983) model is still in widespread use by both practitioners and academics. For many new marketing models, the Guadagni–Little model serves as a benchmark. The key variable that allows the Guadagni–Little model to accurately fit data is the loyalty variable, which is an exponential smoothing of past purchases. In this paper, I show that inclusion of this variable in the logit model may result in a likelihood function that can have multiple maxima. I am able to demonstrate this using simulated data and actual household scanner panel data. In addition, I document a systematic relationship between the loyalty coefficient and the loyalty smoothing parameter. Insight for this systematic relationship and the multiple maxima is obtained by recognizing a trade-off between capturing household heterogeneity and state dependence. Finally, in the Guadagni–Little model extreme parameter values capture two different idealized forms of consumer behavior. However, reported studies rarely find these extreme parameter values. I show that procedures commonly used to initialize loyalty biases against these extreme parameter values. This bias offers some explanation for the observed empirical regularity in Guadagni–Little parameter estimates and suggests that researchers should be cautious concluding these parameters capture regularity in consumer behavior.