In the previous sections, we learned that regression is a powerful tool for capturing a variety of systematic effects in time series data. In practice, when we need forecasts that are as accurate as possible, regression and other sophisticated time series methods are commonly employed. However, it is often not practical or necessary to pursue detailed modeling for each and every one of the numerous time series encountered in business. One popular alternative approach used by business practitioners is based on the strategy of “smoothing” out the random or irregular variation inherent to all time series. By doing so, we gain a general feel for the longer-term movements in a time series.

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Perhaps the most common method used in practice to smooth out short-term fluctuations is the moving-average model. A moving average can be thought of as a rolling average in that the average of the last several values of the time series is used to forecast the next value.


The moving-average forecast model uses the average of the last values of the time series as the forecast for time period . The equation is


The number of preceding values included in the moving average is called the span of the moving average.


Some care should be taken in choosing the span for a moving-average forecast model. As a general rule, larger spans smooth the time series more than smaller spans by averaging many ups and downs in each calculation. Smaller spans tend to follow the ups and downs of the time series.


Consider again the annual average water levels of Lakes Michigan and Huron studied in Example 13.23 (page 687). Figure 13.45 displays moving-average one-step ahead forecasts based on a span of three years and moving averages based on a span of 15 years.


The 15-year moving averages are much more smoothed out than the three-year moving averages. The 15-year moving averages provide a long-term perspective of the cyclic movements of the lake levels. However, for this series, the 15-year moving average model does not seem to be a good choice for short-term forecasting. Because the 15-year moving averages are “anchored” so many years into the past, this model tends to lag behind when the series shifts in another direction. In contrast, the three-year moving averages are better able to follow the larger ups and downs while smoothing the smaller changes in the time series.


The lake series has 96 observations ending with 2013. Here is the computation for the three-year moving-average forecast of the lake level for 2014:


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Figure 13.45: FIGURE 13.45 Time series plot of annual lake levels (green) with 3-year (red) and 15-year (blue) moving average forecasts superimposed.

When dealing with seasonal data, it is generally recommended that the length of the season be used for the value of . In doing so, the average is based on the full cycle of the seasons, which effectively takes out the seasonality component of the data.


Figure 13.46 displays the quarterly number of U.S. passengers (in thousands) using light rail as a mode of transportation. The series begins with the first quarter of 2009 and ends with the first quarter of 2014.26 We can see a regularity to the series: the first quarter’s ridership tends to be lowest; then there is a progressive rise in ridership going into the second and third quarters, followed by a decline in the fourth quarter. Superimposed on the series are the moving-average forecasts based on a span of . Notice that the seasonal pattern in the time series is not present in the moving averages. The moving averages are a smoothed-out version of the original time series, reflecting only the general trending in the series, which is upward.


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Figure 13.46: FIGURE 13.46 Time series plot of quarterly light rail usage (first quarter 2009 through first quarter 2014) along with moving-average forecasts based on a span of and prediction limits.

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Even though the moving averages help highlight the long-run trend of a time series, the moving-average model is not designed for making forecasts in the presence of trends. The problem is that the moving average is derived from past observations all the while the process is trending away from those observations. So, the moving averages are always lagging behind. Figure 13.46 also shows the moving-average model forecasts and prediction limits projected into the future. Notice that the moving-average model makes no accommodation for the trend in its forecasts.