The evolution of family level sales forecasts into product level forecasts: Modeling and estimation

Abstract

It is frequently the case that sales forecasts are available at the detailed product level for only a relatively short time horizon. For the rest of the forecast horizon, only aggregate sales forecasts at the product family level are available. The problem addressed in this paper is how to fit a forecast simulation model to a history of these aggregate and disaggregate forecasts. Our approach to develop such a model is to combine a forecast update model with a forecast disaggregation model. The forecast update model is called the Martingale model of forecast evolution. The parameters of the two models must be estimated from historical forecast data. It is this statistical parameter estimation problem that occupies the major part of our investigation. We recommend an estimation technique based on the method of moments.

Keywords:

• Forecasting
• disaggregation
• forecast evolution
• Martingale

Notes

¹The upper bound comes from the algebraic manipulation of the second derivative of the objective function. The objective function (11) is convex in r if and only if γ _n ≤ ā _n + _nn /2 −ā ² _n /2 + 0.5 [(t ^1/3−u ^1/3)⁻²−1] holds for all n, where t = 1/2+ √31/6√3, u = −1/2+ √31/6√3. Since 0.5 [(t ^1/3−u ^1/3)⁻²− 1] ≈ 0.5739, we simply replace this expression with 0.574 in the stated bound.

²∑ _{k = 1} ^K ‖r− _k ‖²/K = ∑ _{k = 1} ^K ‖r−+− _k ‖²/K = ∑ _{k = 1} ^K [‖r−‖² + ‖− _k ‖² + 2 (r−) · (− _k )]/K = ‖r− ‖² + ∑ _{k = 1} ^K ‖− _k ‖²/K +2 (r− ) · ∑ _{k = 1} ^K (− _k )/K = ‖r− ‖²+ ∑ _{k = 1} ^K ‖− _k ‖²/K where “·” represents the inner (dot) product.

³The arithmetic ratios are calculated by: r = (N/∑ _{n = 1} ^N n, (N−1)/∑ _{n = 1} ^N n, …, 2/∑ _{n = 1} ^N n, 1/∑ _{n = 1} ^N n)′. The pareto ratios are generated by a geometric distribution r(n) = q(1−q) ⁿ⁻¹, for n = 1, …, N. For N = 10, q would be set to 1−√0.2 to make the distribution follow the 80-20 law.