444 Statistical Methods
temperature). The length of the cycle in this example is 12, indicated in the
ACF by the large positive autocorrelation for lag 12. Because the data in the
time series follow a cycle of length 12, you would expect that values 12 units
apart would be highly correlated with each other. Seasonal time series mod-
els are covered more thoroughly later in this chapter.
The third example shows an oscillating time series. In this case, a large
value is followed by a low value and then by another large value. An ex-
ample of this might be winter and summer sales over the years for a large
retail toy company. Winter sales might always be above average because of
the holiday season, whereas summer sales might always be below average.
This pattern of oscillating sales might continue and could follow the pattern
shown in Figure 11-8. The ACF for this time series has an alternating pat-
tern of positive and negative autocorrelations.
Finally, if the observations in the time series are independent or nearly inde-
pendent, there is no discernible pattern in the ACF and all the auto correlations
should be small, as shown in the fourth example. This is characteristic of a
random walk model in which current values are independent of previous
values and thus you cannot use current values to predict future ones.
There are many other possible patterns of behavior for time series data
besides the four examples shown here.Applying the ACF to the Change in Average
Global Temperature
Having looked at the autocorrelation function for the mean annual tempera-
ture, let’s look at the ACF for the change in the average global temperature.
Does an increase in temperature in one year imply that the next year will
also show an increase? Or is the opposite more likely, where years that show
a large increase in temperature are followed by years in which the tempera-
ture increase is smaller or is even a decrease? Let’s fi nd out.To calculate the autocorrelation for the change in annual
temperature:1 Click the Temperature sheet tab.
2 Click Time Series from the StatPlus menu and then click ACF Plot.
3 Click the Data Values button, click the Use Range References option
button, and select the range F3:F129.
4 Deselect the Range includes a row of column labels checkbox and
click OK.
You want to deselect this checkbox because this selection does not
include a header row.