Forecasting

Simple Exponential Smoothing

An intuitive approach to the prediction of a series without trend is to use the historic average as prediction for all future periods. That is,

Equal weighting of the observations, irrespective of their age, requires stability of the series over time. [Page 398]Because most data series lack this stability, it is better to either average observations over a short, recent time window, or consider a weighted average that discounts observations according to their age. Simple exponential smoothing uses an exponential (geometric) weighted average:

Ln is the estimate of the level at time n. It becomes the forecast of all future observations. The smoothing constant α(0 ≤α≤ 1) gives more weight to recent observations and less weight to observations in the past. When α = 1, the forecast ^yn(r) = Ln = yn makes use of the last observation only. It is called the “naïve” or “random walk” forecast. When α approaches 0, the resulting forecast ^yn(r) = Ln = (yn + yn−1 puts equal weight on all observations.

The level is updated according to Ln = αyn + (1 − α)Ln−1; only the last observation and the previous level estimate need to be stored. This is important if many series must be predicted. It is common to start the recursion at the beginning of the series, with a certain starting value for the level at Time 1. Typically, one uses the first observation (L1 = y1), or an average of the first few observations. From this starting value, one calculates L2 = αy2 + (1 − α)L1, L3 = αy3 + (1 − α)L2, …, until one reaches Ln = αyn + (1 − α)Ln−1. The last level estimate becomes the forecast of all future observations: ^yn(r) = Ln.

The calculations require a value for the smoothing constant. The smoothing constant is estimated by minimizing the sum of the squared one-step-ahead forecast errors,

We illustrate simple exponential smoothing with the quarterly Iowa nonfarm growth rates in Figure 1. The data (n = 127; from 1948/2 through 1979/4) were first analyzed in Abraham and Ledolter (1983). The time series graph shows no obvious trends, but indicates that the level of the series shifts with time. Table 1 illustrates the calculations with smoothing constant α = 0.1. The iterations start with L1 = y1 =0.50. Repeated use of the updating equation gives L2 = (0.1)(2.65) + (0.9)(0.5) = 0.715, L3 = (0.1)(0.97) + (0.9)(0.715) = 0.741, until L127 = (0.1)(2.35) + (0.9)(2.681) = 2.648. Forecasts of all future observations are given by ^y127(r) = 2.648. The one-step-ahead forecast errors are given in the fifth column: y2 − ^y1(1) = 2.65 − 0.50 = 2.15, y3 − ^y2(1) = 0.97 − 0.715 = 0.255, …, y127 − ^y126 (1) = 2.35 − 2.681 =−0.331; the sum of their squares is SSE = 122.61. The sum of squares depends on the smoothing constant, because a different α implies different forecasts. In this example, any smoothing constant different from 0.1 increases the sum of the squared one-step-ahead forecast errors.

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