Bias and systematic change in the parameter estimates of macro-level diffusion models

Studies estimating the Bass model and other macro-level diffusion models with an unknown ceiling feature three curious empirical regularities: (i) the estimated ceiling is often close to the cumulative number of adopters in the last observation period, (ii) the estimated coefficient of social contagion or imitation tends to decrease as one adds later observations to the data set, and (iii) the estimated coefficient of social contagion or imitation tends to decrease systematically as the estimated ceiling increases. We analyze these patterns in detail, focusing on the Bass model and the nonlinear least squares (NLS) estimation method. Using both empirical and simulated diffusion data, we show that NLS estimates of the Bass model coefficients are biased and that they change systematically as one extends the number of observations used in the estimation. We also identify the lack of richness in the data compared to the complexity of the model (known as ill-conditioning) as the cause of these estimation problems. In an empirical analysis of twelve innovations, we assess how the model parameter estimates change as one adds later observations to the data set. Our analysis shows that, on average, a 10% increase in the observed cumulative market penetration is associated with, roughly, a 5% increase in estimated market size m, a 10% decrease in the estimated coefficient of imitation q, and a 15% increase the estimated coefficient of innovation p. A simulation study shows that the NLS parameter estimates of the Bass model change systematically as one adds later observations to the data set, even in the absence of model misspecification. We find about the same effect sizes as in the empirical analysis. The simulation also shows that the estimates are biased and that the amount of bias is a function of (i) the amount of noise in the data, (ii) the number of data points, and (iii) the difference between the cumulative penetration in the last observation period and the true penetration ceiling (i.e., the extent of right censoring). All three conditions affect the level of ill-conditioning in the estimation, which, in turn, affects bias in NLS regression. In situations consistent with marketing applications, m can be underestimated by 20%, p underestimated by the same amount, and q overestimated by 30%. The existence of a downward bias in the estimate of m and an upward bias in the estimate of q, and the fact that these biases become smaller as the number of data points increases and the censoring decreases, can explain why systematic changes in the parameter estimates are observed in many applications. A reduced bias, though, is not the only possible explanation for the systematic change in parameter estimates observed in empirical studies. Not accounting for the growth in the population, for the effect of economic and marketing variables, or for population heterogeneity is likely to result in increasing m̂ and decreasing q̂ as well. In an analysis of six innovations, however, we find that attempts to address possible model misspecification problems by making the model more flexible and adding free parameters result in larger rather than smaller systematic changes in the estimates. The bias and systematic change problems we identify are sufficiently large to make long-term predictive, prescriptive and descriptive applications of Bass-type models problematic. Hence, our results should be of interest to diffusion researchers as well as to users of diffusion models, including market forecasters and strategic market planners.