An Estimation Framework for the Influential-Imitator Diffusion

This paper presents an inferential approach for the influential-imitator dynamics, a broadly accepted extension of the traditional Bass model, for cases in which the population has two segments: influentials (who influence each other) and imitators (whose choices are affected by the ones of influentials). Focusing on the parameter’s estimation, the core difficulty of this continuous time diffusion lays on the fact that the solution of the underlying differential equation is not an explicit function of its unknown parameters. We have four main results to address the inferential problem. We develop a truncated series expansion, providing an explicit solution of the differential equation. This results in an asymptotically correct approximation, exhibiting an increasing accuracy when the spontaneous innovation/adoption parameter decreases. We show that a block-decomposition of the underlying parameter matrix leads to a two-truncation of the series that allows explicitly expressing the dependency of the likelihood function with respect to the unknown parameters. We propose a block coordinate descent method for the numerical maximization of the approximated likelihood function. Empirical and computational tests are provided using the Michell and West dataset about the cannabis consumption of a cohort of students over their second, third and fourth year at a secondary school in Glasgow. The estimated imitation pattern confirms well-known hypothesis on peer influences, where the choices of popular children represent the leading effects to determine the habits of others