The Econometrics Of The Bass Diffusion Model

We propose a new empirical representation of the Bass diffusion model, in order to estimate the three key parameters, concerning innovation, imitation and maturity. The representation is based on the notion that the observed data may temporarily deviate from the mean path determined by the underlying hazard rate. Additionally, it rests on the idea that uncertainty about the cumulative process should be smaller, the closer it is to the start of the process and to the level of maturity. Taking this into account, we arrive at an extension of the basic representation proposed in Bass (1969), with an additional heteroskedastic error term. The type of heteroskedasticity can be set by the modeler, as long as it obeys certain properties. Next, we discuss the asymptotic theory for this new empirical model, that is, we focus on the properties of the estimators of the various parameters. We show that the parameters, upon standardization by their standard errors, do not have the conventional asymptotic behavior. For practical purposes, it means that the t-statistics do not have an (approximate) t-distribution. Using simulation experiments, we address the issue how these findings carry over to practical situations. In a next set of simulation experiments, we compare the new representation with that of Bass (1969) and Srinivasan and Mason (1986). We document that these last two approaches often seriously overestimate the precision of the parameter estimators. We also shed light on the effects of temporal aggregation and on the effects of a serious and persisent deviation between the actual data and their mean. Finally, we consider the various empirical representations for a monthly series on installed ATMs.