Forecasting using belief functions: An application to marketing econometrics

• A method is proposed to quantify forecast uncertainty using belief functions.
• The method is simple and can be applied to any parametric statistical model.
• The approach does not require to specify a prior distribution on the parameter.
• The Bayesian approach is recovered as a special case when a prior is provided.
• The method is applied to innovation diffusion forecasting using the Bass model.

A method is proposed to quantify uncertainty on statistical forecasts using the formalism of belief functions. The approach is based on two steps. In the estimation step, a belief function on the parameter space is constructed from the normalized likelihood given the observed data. In the prediction step, the variable Y to be forecasted is written as a function of the parameter θ and an auxiliary random variable Z with known distribution not depending on the parameter, a model initially proposed by Dempster for statistical inference. Propagating beliefs about θ and Z through this model yields a predictive belief function on Y. The method is demonstrated on the problem of forecasting innovation diffusion using the Bass model, yielding a belief function on the number of adopters of an innovation in some future time period, based on past adoption data.