FORECASTING:AN EXPONENTIAL SMOOTHING METHOD FOR A LINEAR TREND MODEL

AN EXPONENTIAL SMOOTHING METHOD FOR A LINEAR TREND MODEL

Recall that the constant-level model introduced in Sec. 27.3 assumes that the sequence of random variables {X1, X2, . . . , Xt} generating the time series has a constant expected value denoted by A, where the goal of the forecast Ft+1 is to estimate A as closely as possible. However, as was illustrated in Fig. 27.2b, some time series violate this assumption by having a continuing trend where the expected values of successive random variables keep changing in the same direction. Therefore, a forecasting method based on the constant- level model (perhaps after adjusting for seasonal effects) would do a poor job of forecasting for such a time series because it would be continually lagging behind the trend. We now turn to another model that is designed for this kind of time series.

Suppose that the generating process of the observed time series can be represented by a linear trend superimposed with random fluctuations, as illustrated in Fig. 27.2b. De- note the slope of the linear trend by B, where the slope is called the trend factor. The model is represented by

Xi = A + Bi + ei, for i = 1, 2, . . . ,

where Xi is the random variable that is observed at time i, A is a constant, B is the trend factor, and ei is the random error occurring at time i (assumed to have expected value equal to zero and constant variance).

For a real time series represented by this model, the assumptions may not be completely satisfied. It is common to have at least small shifts in the values of A and B occasionally. It is important to detect these shifts relatively quickly and reflect them in the forecasts. Therefore, practitioners generally prefer a forecasting method that places sub- stantial weight on recent observations and little if any weight on old observations. The exponential smoothing method presented next is designed to provide this kind of approach.

Adapting Exponential Smoothing to This Model

The exponential smoothing method introduced in Sec. 27.4 can be adapted to include the trend factor incorporated into this model. This is done by also using exponential smoothing to estimate this trend factor.

Let

Tt+1 = exponential smoothing estimate of the trend factor B at time t + 1, given the observed values, X1 = x1, X2 = x2, . . . , Xt = xt.

Given Tt+1, the forecast of the value of the time series at time t + 1 (Ft+1) is obtained simply by adding Tt+1 to the formula for Ft+1 given in Sec. 27.4, so

Ft+1 = axt + (1 - a)Ft + Tt+1.

To motivate the procedure for obtaining Tt+1, note that the model assumes that

B = E(Xi+1) - E(Xi), for i = 1, 2, . . . .

Thus, the standard statistical estimator of B would be the average of the observed differ- ences, x2 - x1, x3 - x2, . . . , xt - xt-1. However, the exponential smoothing approach recognizes that the parameters of the stochastic process generating the time series (in- cluding A and B) may actually be gradually shifting over time so that the most recent ob- servations are the most reliable ones for estimating the current parameters. Let

Lt+1 = latest trend at time t + 1 based on the last two values (xt and xt-1) and the last two forecasts (Ft and Ft-1).

27.6 AN EXPONENTIAL SMOOTHING METHOD 27-13

The exponential smoothing formula used for Lt+1 is

Lt+1 = a(xt - xt-1) + (1 - a)(Ft - Ft-1).

Then Tt+1 is calculated as

Tt+1 = {3Lt+1 + (1 - {3)Tt,

where {3 is the trend smoothing constant which, like a, must be between 0 and 1. Cal- culating Lt+1 and Tt+1 in order then permits calculating Ft+1 with the formula given in the preceding paragraph.

Getting started with this forecasting method requires making two initial estimates about the status of the time series just prior to beginning forecasting. These initial estimates are

x0 = initial estimate of the expected value of the time series (A) if the conditions just prior to beginning forecasting were to remain unchanged without any trend;

T1 = initial estimate of the trend of the time series (B) just prior to beginning forecasting.

The resulting forecasts for the first two periods are

F1 = x0 + T1,

L2 = a(x1 - x0) + (1 - a)(F1 - x0),

T2 = {3L2 + (1 - {3)T1,

F2 = ax1 + (1 - a)F1 + T2.

The above formulas for Lt+1, Tt+1, and Ft+1 then are used directly to obtain subsequent forecasts.

Since the calculations involved with this method are relatively involved, a computer commonly is used to implement the method. The Excel files for this chapter in your OR Courseware include two Excel templates (one without seasonal adjustments and one with) for this method. In addition, the forecasting area in your IOR Tutorial includes a proce- dure of this method that also enables you to investigate graphically the effect of making changes in the data.

Application of the Method to the CCW Example

Reconsider the example involving the Computer Club Warehouse (CCW) that was intro- duced in the preceding section. Figure 27.3 shows the time series for this example (rep- resenting the average daily call volume quarterly for 3 years) and then Fig. 27.4 gives the seasonally adjusted time series based on the seasonal factors calculated in Table 27.1. We now will assume that these seasonal factors were determined prior to these three years of data and that the company then was using exponential smoothing with trend to forecast the average daily call volume quarter by quarter over the 3 years based on these data. CCW management has chosen the following initial estimates and smoothing constants:

The Excel template in Fig. 27.5 shows the results from these calculations for all 12 quar- ters over the 3 years, as well as for the upcoming quarter. The middle of the figure shows the plots of all the seasonally adjusted call volumes and seasonally adjusted forecasts. Note how each trend up or down in the call volumes causes the forecasts to gradually trend in the same direction, but then the trend in the forecasts takes a couple of quarters to turn around when the trend in call volumes suddenly reverses direction. Each number in column I is calculated by multiplying the seasonally adjusted forecast in column H by the corresponding seasonal factor in column M to obtain the forecast of the actual value (not seasonally adjusted) for the average daily call volume. Column J then shows the re- sulting forecasting errors (the absolute value of the difference between columns D and I).

Forecasting More Than One Time Period Ahead

We have focused thus far on forecasting what will happen in the next time period (the next quarter in the case of CCW). However, decision makers sometimes need to forecast further into the future. How can the various forecasting methods be adapted to do this?

In the case of the methods for a constant-level model presented in Sec. 27.4, the fore- cast for the next period Ft 1 also is the best available forecast for subsequent periods as well. However, when there is a trend in the data, as we are assuming in this section, it is important to take this trend into account for long-range forecasts. Exponential smoothing with trend provides a straightforward way of doing this. In particular, after determining the estimated trend Tt 1, this method’s forecast for n time periods into the future is