Limitations of moving averages
We used moving averages to smooth out the seasonal variation in a time series, but this method has serious limitations.
Estimating a common seasonal effect
If we assume that the underlying seasonal pattern is the same in each year, this common seasonal pattern can be estimated. Our estimates are based on the residuals from the moving averages that were used to smooth out the seasonal variation. (The residuals are the differences between the actual values and the smoothed values.)
For monthly data, we can estimate a 'January effect' by averaging the difference between actual January values and their smoothed values. For example, if January values were on average 5 higher than their smoothed values, the seasonal effect for January would be 5. Negative values for the seasonal effect correspond to months where the values are, on average, lower than their smoothed values.
International tourists in Hawaii
The time series plot below shows international tourist numbers in Hawaii and a 12-point moving average that smooths out the seasonal variation.
Click on any value on the time series plot to highlight the residuals for that month in each year. The average of these residuals is the seasonal effect for that month.
The plot of the seasonal effect at the bottom of the diagram clearly shows the seasonal variation in tourist numbers.