Difference Equations in Forecasting Formulas

Abstract

The deviation of actual sales (or other time-dependent statistics) from a model of sales will give rise to forecasting errors that are enhanced, damped out, or shifted in phase, depending on the particular formulas that are used for forecasting. Following R. G. Brown, P. Winters, and Theil, Nerlove, and Wage, we start from a general set of forecasting formulas, but make fewer assumptions than those authors about deviations from the model, and obtain a more extensive collection of results. In particular, we show how to treat errors not serially uncorrelated, and how to investigate forecasting formulas of order higher than second.

Avoiding the bullwhip effect using Damped Trend forecasting and the Order-Up-To replenishment policy

“…Both the stability region and the oscillation region are identical to McClain and Thomas (1973). When 0   and 0   , the SES stability boundary observed   0 2    , is consistent with the result in Brenner et al (1968). It is common practice in exponential smoothing models to restrict the smoothing parameters   ,   to the   0,1 interval (Holt, 2004;Winters, 1960).…”

Complex exponential smoothing

“…If the smoothing parameter $$$ \alpha \in \left(0,1\right) $$$ then the weights decline exponentially with the increase of $$$ j $$$. If it lies in the so called “admissible bounds” (Brenner et al, 1968), that is $$$ \alpha \in \left(0,2\right) $$$, then the weights decline in oscillating manner. Both traditional and admissible bounds have been used efficiently in practice and in academic literature (for application of the latter see for example Gardner & Diaz‐Saiz, 2008; Snyder et al, 2017).…”

“…where ŷt is the calculated value and α is the smoothing parameter which in theory can lie inside the region (0, 2) (Brenner et al, 1968). This exponential smoothing method has an underlying statistical model, ETS(A,N,N) which due to Hyndman et al (2002) has the following statespace form:…”

A Comparison of Standard Forecasting Systems with the Box-Jenkins Approach

“…The one-step ahead forecast is given by (6) This system is commonly used for N = 0, Holt's SEWMA, and N = 1, the adaptation by Holt and Winter to linear trend models. The system has been discussed for general N by Brenner et al (1968) who concern themselves with the problems of choosing the at's for which the difference equations are stable. The transfer function for this system is found to be where W2(Z).…”