Technical Note—Nonlinear Least Squares Estimation of New Product Diffusion Models

Schmittlein and Mahajan (Schmittlein, D. C., V. Mahajan. 1982. Maximum likelihood estimation for an innovation diffusion model of new product acceptance. (Winter) 57–78.) made an important improvement in the estimation of the Bass (Bass, F. M. 1969. A new product growth model for consumer durables. (January) 215–227.) diffusion model by appropriately aggregating the continuous time model over the time intervals represented by the data. However, by restricting consideration to only sampling errors and ignoring all other errors (such as the effects of excluded marketing variables), their Maximum Likelihood Estimation (MLE) seriously underestimates the standard errors of the estimated parameters. This note uses an additive error term to model sampling and other errors in the Schmittlein and Mahajan formulation. The proposed Nonlinear Least Squares (NLS) approach produces valid standard error estimates. The fit and the predictive validity are roughly comparable for the two approaches. Although the empirical applications reported in this paper are in the context of the Bass diffusion model, the NLS approach is also applicable to other diffusion models for which cumulative adoption can be expressed as an explicit function of time.