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.