450 Statistical Methods
Forecasting with Exponential Smoothing
Exponential smoothing is often used to forecast the value of the next ob-
servation, given the current and prior values. In this situation, you already
know the value of yn and are trying to forecast the next value yn 11. Call the
forecast Sn. The formula for Sn is similar to the one we derived for the expo-
nentially weighted moving average; it isSn 5 wyn 1 w^112 w^2 yn 211 w^112 w^22 yn 221 c 1 w^112 w^2 n^21 y 11112 w^2 ns 0Sn is more commonly written in an equivalent recursive formula, whereSn 5 wyn 1112 w 2 sn 21so that Sn is equal to the sum of the weighted values of the current observa-
tion and the previous forecast. Therefore, to create the forecasted value, an
initial forecasted value S 0 is required. One option is to let S 0 equal y 1 , the
initial observation. Another choice is to let S 0 equal the average of the fi rst
few values in the series. The examples in this chapter will use the fi rst op-
tion, setting S 0 equal to the fi rst value in the time series.
Once you determine the value of S 0 , you can generate the exponentially
smoothed values as follows:S 15 wy 11112 w^2 S 0S 25 wy 21112 w^2 S 1
(
Sn 5 wyn 1112 w^2 Sn 21and then Sn becomes the value you predict for the next observation in the
time series.Assessing the Accuracy of the Forecast
Once you generate the smoothed values, how do you measure their accuracy
in forecasting values of the time series? One way is to use exponential smooth-
ing to calculate y^t, the predicted value of the time series at time t. Then, for
each value in the time series, compare y^t to the observed value, yt. The mean
square error (MSE), gives the sum of the squared differences between the
forecasted values and the observed values. The formula for the MSE isMSE 5
an
t 51(^1) yt 2 y^t 22
n
By comparing the MSE of one set of smoothed values to another, one can
determine which set does a better job of forecasting the data.