Methodology of the Monthly Index of Services Annex B: The Holt-Winters Forecasting Method

This is the simplest form of exponential smoothing and can be used only for data without any systematic trend or seasonal components. Given such a time series, a sensible approach is to take a weighted average of past values. So for a series, y 1 , y 2 ,…, y n , the estimate of the value of y n+1 , given the information available up to time n, is: y w y w y w y or i n i i n n y w y − ∞ = + ∑ = 0 1 ˆ where w i are the weights given to the past values of the series and sum to one. Since the most recent observations of the series are also the most relevant, it is logical that they should be given more weight at the expense of observations further in the past. This is achieved by assigning geometrically declining weights to the series. These decrease by a constant ratio and are of the form: i i w) 1 (α α − = where i=0, 1, 2,… and α is the smoothing constant in the range 0 < α < 1. For example, if α is set to 0.5, the weights will be: w 0 = 0.5 w 1 = 0.25 w 2 = 0.125 and so on. It can be seen that weights taking this form will sum to ~1 for all values of α in the above range. The equation for the estimate of y n+1 then becomes: y y y y α α α α α Since: ...)) 1 ()(1 (ˆ 2 1 1 + − + − + = − − + n n n n n y y y y α α α α α it can be seen that: 1 1 ˆ) 1 (ˆ − + − + = n n n n n y y y α α where 1 ˆ − n n y is simply the previous estimate.