The Econometrics of the Bass Diffusion Model

Abstract

We propose a new empirical representation of the Bass diffusion model, in order to estimate thethree key parameters, concerning innovation, imitation and maturity. The representation isbased on the notion that the observed data may temporarily deviate from the mean pathdetermined by the underlying hazard rate. Additionally, it rests on the idea that uncertaintyabout the cumulative process should be smaller, the closer it is to the start of the process and tothe level of maturity. Taking this into account, we arrive at an extension of the basicrepresentation proposed in Bass (1969), with an additional heteroskedastic error term. The typeof heteroskedasticity can be set by the modeler, as long as it obeys certain properties. Next, wediscuss the asymptotic theory for this new empirical model, that is, we focus on the properties ofthe estimators of the various parameters. We show that the parameters, upon standardizationby their standard errors, do not have the conventional asymptotic behavior. For practicalpurposes, it means that the t-statistics do not have an (approximate) t-distribution. Usingsimulation experiments, we address the issue how these findings carry over to practicalsituations. In a next set of simulation experiments, we compare the new representation withthat of Bass (1969) and Srinivasan and Mason (1986). We document that these last twoapproaches often seriously overestimate the precision of the parameter estimators. We alsoshed light on the effects of temporal aggregation and on the effects of a serious and persisentdeviation between the actual data and their mean. Finally, we consider the various empiricalrepresentations for a monthly series on installed ATMs.

Keywords: Bass diffusion model, representation, estimation

JEL Classification: M, M31, C44, C51