Chapter 11 Times Series 457
The output shown in Figure 11-16 consists of three columns: the observa-
tion numbers, the recorded mean annual temperatures, and the temperatures
forecasted for each year based on the smoothing model. The values are then
plotted on the chart. It appears that the forecasted values generally underes-
timated the mean annual temperatures in the last decades of the twentieth
century. This may indicate that temperatures are warming faster than
expected. The lower forecasted values might also reflect the effect of the
slight dip in temperature values that occurred during the middle decades of
the century. The standard error of the forecasts, 0.250864, indicates that the
typical forecasting error was about 0.25 degrees Fahrenheit points per year.
The one-parameter exponential smoothing only uses weighted averages
of previous observations to calculate future results. It does not assume a par-
ticular trend for the time series, but it is apparent from the data that the tem-
perature values have been increasing over the time interval being studied.
We can insert a trend assumption into our model by using two-parameter
exponential smoothing.
EXCEL TIPS
Excel’s Analysis ToolPak also includes a command to perform
one-parameter exponential smoothing. To run the command,
select Exponential Smoothing from the list of analysis tools in
the Data Analysis ToolPak.
Two-Parameter Exponential Smoothing
To explore how to add a trend assumption to exponential smoothing let’s
fi rst express one-parameter exponential smoothing in terms of the following
equation for yt, the value of the y variable at time t:
yt 5 b 0 1et
where b 0 is the location parameter that changes slowly over time, and
et is the random error at time t. If b 0 were constant throughout time, you
could estimate its value by taking the average of all the observations. Using
that estimate, you would forecast values that would always be equal to
your estimate of b 0. However, if b 0 varies with time, you weight the more
recent observations more heavily than distant observations in any forecasts
you make. Such a weighting scheme could involve exponential smoothing.
How could such a situation occur in real life? Consider tracking crop yields
over time. The average yield could slowly change over time as equipment
or soil science technology improved. An additional factor in changing the
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