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.

Moving-average models

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.

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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.

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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.

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.

Moving average and seasonal ratios

Figure 13.46 illustrated that we should not rely on moving averages for predicting future observations of a trending series. However, we can use the moving averages to isolate the general trend movement. By comparing the seasonal observations relative to the general trend, we have a means of estimating the seasonal component. Recall that with Examples 13.16 and 13.18, we used seasonal indicator variables in a regression model to estimate the seasonal effect. In practice, there is another approach for estimating the seasonal effect without the use of regression.

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The approach utilizes the moving averages to provide a baseline for the general level of the series, not to provide forecasts. Instead of projecting the moving averages into the future, we use them as a summary of the past. Consider, for example, the first computed moving average on quarterly data:

If we were to use this average as a forecast, then it would forecast period 5. Instead, we look at the average as representing the past level of the time series. A minor difficulty can arise when using the moving average to estimate the past level of the time series. Because the moving average was based on periods 1, 2, 3, and 4, it is technically centered on period . This is problematic because we want to compare the observations that fall on the whole number time periods with the level of the series to estimate the seasonality component. The solution to this difficulty can be recognized by considering the next moving average in the quarter series:

The preceding moving average represents the level of the series at . We now have one moving average representing and another one representing . By taking the average of these two moving averages, we now have a new average that will be centered on . This average is referred to as centered moving average (CMA).

The second step of averaging the averages is only necessary when the number of seasons is even, as with semiannual, quarterly, or monthly data. However, if the number of seasons is odd, then the initial moving average is the centered moving average. As an example, if we had data for the seven days of the week, then we would take a moving average of span 7. The average of the first seven periods will center on . Let’s now illustrate the computation of centered moving averages with an example.

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Now that we have an appropriate baseline estimate for the level of the process historically, we can compute the seasonal component. Probably the most common approach is to compute a seasonal ratio. The ratio approach works on the basis of a multiplicative seasonal model studied earlier, namely,

For the observed data , the basic idea behind seasonal ratios can be seen by rearranging the trend-times-seasonal model:

This ratio says that we can isolate the seasonal component by dividing the data by the level of series as estimated by the trend component. Let’s see how this works by continuing our study of the light rail passenger series.

The seasonality ratios are a snapshot of the typical ups and downs over the course of a year. If you think of the ratios in terms of percents, then the ratios show how each quarter compares to the average level for all four quarters of a given year. For example, the first quarter’s ratio is 0.97, or 97%, indicating that the number of passengers for the first quarter sales are typically 3% below the average for all four quarters. The third quarter’s ratio is 1.022, or 102.2%, indicating that ridership in the third quarter is typically 2.2% above the annual average. Figure 13.49 plots the seasonality factors by quarter. The reference line marked at 1.0 (100%) is a visual aid for interpreting the factors compared to the overall average quarterly ridership. Notice how the seasonality ratios mimic the pattern that is repeated every four quarters in Figure 13.46 (page 693).

The centered moving averages provide us with a historical baseline of the trend level. They, however, are not intended for forecasting the series into the future. With seasonal ratios in hand, the next step is to seasonally adjust the series so that we can estimate the overall trend for forecasting purposes.

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Many economic time series are routinely adjusted by season to make the overall trend, if it exists, in the numbers more apparent. Government agencies often release both versions of a time series, so be careful to notice whether you are analyzing seasonally adjusted data when using government sources.

For the series of Example 13.30, the regression output for the trend fit would show that the trend coefficient is significant with a -value of less than 0.0005. In some cases, we may find that the seasonally adjusted series exhibits no significant trend. In such cases, we shouldn’t impose the use of the trend model unnecessarily. For example, if we find the seasonally adjusted series to exhibit “flat” random behavior, then we would simply project the average of the series into the future and make adjustments upon the average value.

Exponential smoothing models

Moving-average forecast models appeal to our intuition. Using the average of several of the most recent data values to forecast the next value of the time series is easy to understand conceptually. However, two criticisms can be made against moving-average models. First, our forecast for the next time period ignores all but the last observations in our data set. If you have 100 observations and use a span of , your forecast will not use 95% of your data! Second, the data values used in our forecast are all weighted equally. In many settings, the current value of a time series depends more on the most recent value and less on past values. We may improve our forecasts if we give the most recent values greater “weight” in our forecast calculation. Exponential smoothing models address both of these criticisms.

Several variations exist on the basic exponential smoothing model. We look at the details of the simple exponential smoothing model, which we refer to as, simply, the exponential smoothing model. More complex variations exist to handle time series with specific features, but the details of these models are beyond the scope of this chapter. We only mention the scenarios for which these more complex models are appropriate.

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It should be noted that in the time series literature, there is no consistency on the assumption about the range of in terms of inclusion or exclusion of the values of 0 and 1 as choices for . Even with software there are differences. Excel allows to be 1 but not 0. JMP and Minitab allow to be 0 or 1. At the end of this section, we note that statistical software can even allow for a wider range for .

Choosing the smoothing constant in the exponential smoothing model is similar to choosing the span in the moving-average model—both relate directly to the smoothness of the model. Smaller values of correspond to greater smoothing of movements in the time series. Larger values of put most of the weight on the most recent observed value, so the forecasts tend to be close to the most recent movement in the series. Some series are better suited for larger , while others are better suited for smaller . Let’s explore a couple of examples to gain insights.

When successive observations do not show a persistence to be close to each other, a larger value for is no longer a preferable choice, as we can see with the next example.

A little algebra is needed to see how the exponential smoothing model utilizes the data differently than the moving-average model. We start with the forecasting equation for the exponential smoothing model and imagine forecasting the value of the time series for the time period , where is the number of observed values in the time series:

This alternative version of the forecast equation shows exactly how our forecast depends on the values of the time series. Notice that the calculation of can be tracked all the way back to the initial forecast . It is common practice to initiate the exponential smoothing model by using the actual value of the first time period as the forecast for the first time period . Excel and JMP indeed initiate the exponential smoothing model in this manner. However, Minitab’s default is to use the average of the first six observations as the initial forecast. This default can be changed so that the initial forecast is the first observation. We also learn from the backtracking that the forecast for uses all available values of the time series , not just the most recent -values like a moving-average model would. The weights on the observations decrease exponentially in value by a factor of as you read the equation from left to right down to the . This factor of is known as the damping factor.

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While the second version of our forecasting equation reveals some important properties of the exponential smoothing model, it is easier to use the first version of the equation for calculating forecasts.

With the forecasted value for August 2014 from Example 13.33, the forecast for September requires only one calculation:

Once we observe the actual PMI for August 2014 (), we can enter that value into the preceding forecast equation. Updating forecasts from exponential smoothing models only requires that we keep track of last period’s forecast and last period’s observed value. In contrast, moving-average models require that we keep track of the last observed values of the time series.

In Examples 13.31 and 13.32, we illustrated situations in which either larger or smaller smoothing constants are preferable. However, our discussions only revolved around values of 0.1 and 0.7 for . These choices are quite arbitrary and were only used for illustrative purposes. In practice, we want to fine-tune the smoothing constant with the hope to obtain tighter forecasts.

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One approach is to try different values of along with a measure of forecast accuracy and find the value of that does best. For example, we can search for the value of that minimizes mean squared error (MSE); MSE is a measure of forecasting accuracy that was introduced in Exercise 13.48 (page 694). Spreadsheets can be utilized to do the computations on different possible values of to help us hone in on a reasonable choice of . As an alternative, statistical software packages (such as JMP and Minitab) provide an option to estimate the “optimal” smoothing constant. Earlier we mentioned that there is a more general class of time series known as ARIMA models (page 682). It turns out that the exponential smoothing forecast equation is the optimal equation for a very special ARIMA model.²⁹ This special ARIMA model has a certain parameter that is directly related to the smoothing constant . So, by estimating the parameter, we have an estimate for .

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When we defined the exponential smoothing model (page 700), we stated that the smoothing constant is traditionally assumed to fall in the range of . A smoothing constant value in this range has intuitive appeal as being a weighted average of the observation and forecast in which the weights are positive percentages adding up to 100%. For example, a of 0.40 implies 40% weight on the observation value and 60% weight on the forecast value.

However, it turns out that when the exponential smoothing model is viewed from the perspective of being connected to an ARIMA model, then the smoothing constant can possibly take on a value greater than 1. When finding an optimal value for w, Minitab allows to fall in the range of . JMP provides the user with a number of options in its estimation of the smoothing constant: (1) constrained to the range of ; (2) constrained to the range of ; (3) allowing to be any possible value with no constraints. Without going into the technical reasons, it can be shown that as long as is in the range of , then the forecasting model is considered stable. If software estimates to be greater than 1, then there are a couple of options. The first option is to go ahead and use it in the exponential smoothing model. There is nothing technically wrong with doing so. However, there are practitioners who are uncomfortable with a smoothing constant greater than 1 and prefer to have the traditional constraint. In such a case, the next best option in terms of minimizing MSE is to choose a value of 1 for .

Similar to the moving-average model, the simple exponential smoothing model is best suited for forecasting time series with no strong trend. It is also not designed to track seasonal movements. Variations on the simple exponential smoothing model have been developed to handle time series with a trend (double exponential smoothing and Holt’s exponential smoothing), with seasonality (seasonal exponential smoothing), and with both trend and seasonality (Winters’ exponential smoothing). Your software may offer one or more of these smoothing models.