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Winterís Method





Often, we must predict future values of a time series such as monthly costs or monthly prod- uct revenues. This is usually difficult because the characteristics of any time series are con- stantly changing. Smoothing or adaptive methods are usually best suited for forecasting future values of a time series. In this section, we describe the most powerful smoothing method: Winterís method. To help you understand how Winterís method works, we will use it to fore- cast monthly housing starts in the United States (U.S.). Housing starts are simply the number

of new homes whose construction begins during a month. We begin by describing the three key characteristics of a time series.


Time Series Characteristics


The behavior of most time series can be explained by understanding the following three characteristics: base, trend, and seasonality.


■     The base of a series describes the seriesí current level in the absence of any seasonality. For example, suppose the base level for U.S. housing starts is 160,000. In this case, we believe that if the current month were an ďaverageĒ month relative to other months of the year, then 160,000 housing starts would occur.

■     The trend of a time series is the percentage increase per period in the base. Thus a trend

of 1.02 means that we estimate that housing starts are increasing by 2 percent each month.

■     The seasonality (seasonal index) for a period tells us how far above or below a typical month we can expect housing starts to be. For example, if the December seasonal index

is .8, then December housing starts are 20 percent below a typical month. If the June sea- sonal index is 1.3, then June housing starts are 30 percent higher than a typical month.









Parameter Definitions


After observing month t, we will have used all data observed through the end of month t to estimate the following quantities of interest:


■     Lt=Level of series

■     Tt=Trend of series

■     St=Seasonal index for current month

The key to Winterís method is the following three equations, which are used to update Lt, Tt, and St. In the following formulas, alp, bet, and gam are called smoothing parameters. The values

of these parameters will be chosen to optimize our forecasts. In the following formulas, c

equals the number of periods in a seasonal cycle (c=12 months for example) and xt equals the observed value of the time series at time t.


■     Formula 1: Lt=alp(xt/stĖc)+(1Ėalp)(L1*T1)

■     Formula 2: Tt=bet(Lt/L1)+(1Ėbet)TtĖ1

■     Formula 3: St=gam(xt/Lt)+(1Ėgam)stĖ-c

Formula 1 indicates that our new base estimate is a weighted average of the current observa- tion (deseasonalized) and last periodís base updated by our last trend estimate. Formula 2 indicates that our new trend estimate is a weighted average of the ratio of our current base to last periodís base (this is a current estimate of trend) and last periodís trend. Formula 3 indi- cates that we update our seasonal index estimate as a weighted average of the estimate of the seasonal index based on the current period and the previous estimate. Note that larger values

of the smoothing parameters correspond to putting more weight on the current observation.


We define Ft,k as our forecast (F) after period t for the period t+k. This results in the formula


Text Box: t)


Ft,k=Lt*(T  ks






This formula first uses the current trend estimate to update the base k periods forward. Then the resulting base estimate for period t+k is adjusted by the appropriate seasonal index.


Initializing Winterís Method


To start Winterís method, we must have initial estimates for the series base, trend, and seasonal indexes. We will use monthly housing starts for the years 1986 through 1987 to initialize Winterís method. Then we will choose our smoothing parameters to optimize

our one-month-ahead forecasts for the years 1988 through 1996. See Figure 53-1 and the file House2.xlsx. Weíll use the following process.


Step 1: We will estimate, for example, the January seasonal index as the average of January housing starts for 1986 through 1987 divided by the average monthly starts for 1986



through 1987. Therefore copying from G14 to G15:G25 the formula =AVERAGE(B2,B14)/ AVERAGE($B$2:$B$25) will generate our estimates of seasonal indexes. For example, the January estimate is 0.75 and the June estimate is 1.17.


Step 2: To estimate the average monthly trend, we take the twelfth root of (1987 mean starts divided by the 1986 mean starts). We compute this in cell J3 (and copy it to cell D25) with the formula =(J1/J2)^(1/12)


Figure 53-1    Initialization of Winterís method


Step 3: Going into January 1987, we estimate the base of the series as the deseasonalized

December 1987 value. This is computed in C25 with the formula =(B25/G25).


Estimating the Smoothing Constants


We are now ready to estimate our smoothing constants. In column C, we will update the

series base; in column D, the series trend; and in column G, our seasonal indexes. In column

E, we compute our forecast for next month, and in column F, we compute our absolute per- centage error for each month. Finally, we will use solver to choose smoothing constant values that minimize the sum of our absolute percentage errors. Weíll use the following process.


Step 1: In G11:I11, we enter trial values (between 0 and 1) for our smoothing constants.


Step 2: In C26:C119, we compute the updated series level with (1) by copying from C26 to

C27:C119 the formula =alp*(B26/G14)+(1Ėalp)*(C25*D25).




Step 3: In D26:D119, we use (2) to update the series trend. Copy from D26 to D27:D119 the formula =bet*(C26/C25)+(1Ėbet)*D25.


Step 4: In G26:G119, we use (3) to update the seasonal indexes. Copy from G26 to G27:G119

the formula =gam*(B26/C26)+(1Ėgam)*G14.


Step 5: In E26:E119, we use (4) to compute the forecast for the current month by copying from E26 to E27:E119 the formula =(C25*D25)*G14.


Step 6: In F26:F119, we compute the absolute percentage error for each month by copying from F26 to F27:F119 the formula =ABS(B26-E26)/B26.


Step 7: We compute the average absolute percentage error for the years 1988 through 1996 in

F21 with the formula =AVERAGE(F26:F119).


Step 8: We can now use the Microsoft Office Excel 2007 Solver feature to determine smooth- ing parameter values that minimize our average absolute percentage error. The Solver Parameters dialog box is shown in Figure 53-2.


Figure 53-2    Solver Parameters dialog box for Winterís model


We choose our smoothing parameters (G11:I11) to minimize the average absolute percentage error (cell F21). The Excel Solver ensures we will find the best combination of smoothing con- stants. Smoothing constants must be between 0 and 1. We find that alp=.54, bet=.02, and gam=.29 minimizes our average absolute percentage error. You might find slightly different values of the smoothing constants, but you should obtain a MAPE close to 7.3 percent. In this example, there are many combinations of the smoothing constants that give forecasts having approximately the same MAPE. Our one-month-ahead forecasts are off by an average of 7.3 percent.







■     Instead of choosing our smoothing parameters to optimize one-period forecast errors, we could, for example, have chosen to optimize the average absolute percentage error incurred in forecasting total housing starts for the next six months.


■     Suppose our time series is sales of a software product and we have conducted a major promotion during June 2000. Assume predicted sales for June 2000 were 20,000 units, but we sold 35,000 units. Then a good guess is that the promotion caused 15,000 extra sales during June. When updating the base, trend, and seasonal indexes, however, we should not put in June 2000 sales of 35,000. We should put in June 2000 sales of our forecast (20,000); otherwise, we will incorrectly bump up our forecasts of future sales. When making a forecast for a future month in which there is a promotion similar to the June promotion, we would just bump up the Winterís method forecast by using the for- mula 35,000/20,000=75%!


■     If at the end of month t we wanted to forecast sales for the next four quarters, we would simply add ft,1+ft,2+ft,3+ft,4. If desired, we could choose our smoothing parameters to minimize the absolute percentage error incurred in estimating sales for the next year.




All the data for the following problems is in the file Quarterly.xlsx.


1.    Use Winterís method to forecast one-quarter-ahead revenues for Apple.


2.    Use Winterís method to forecast one-quarter-ahead revenues for Amazon.com.


3.    Use Winterís method to forecast one-quarter-ahead revenues for Home Depot.


4.    Use Winterís method to forecast total revenues for the next two quarters for Home




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