(Fifth in a series)
We pick up on our discussion of exponential smoothing methods, focusing today on double exponential smoothing. Single exponential smoothing, which we discussed in detail last week, is ideal when your time series is free of seasonal or trend components, which create patterns that your smoothing equation would miss due to lags. Single exponential smoothing produces forecasts that exceed actual results when the time series exhibits a decreasing linear trend, and forecasts that trail actual results when the time series exhibits an increasing trend. Double exponential smoothing takes care of this problem.
Two Smoothing Constants, Three Equations
Recall the equation for single exponential smoothing:
Ŷ_{t+1 }= αY_{t} + (1α) Ŷ_{t}_{
}
Where: Ŷ_{t+1} represents the forecast value for period t + 1
Y_{t} is the actual value of the current period, t
Ŷ_{t} is the forecast value for the current period, t
and α is the smoothing constant, or alpha, 0≤ α≤ 1
To account for a trend component in the time series, double exponential smoothing incorporates a second smoothing constant, beta, or β. Now, three equations must be used to create a forecast: one to smooth the time series, one to smooth the trend, and one to combine the two equations to arrive at the forecast:
C_{t} = αY_{t }+ (1α)(C_{t1} + T_{ t1})
T_{t} = β(C_{t }– C_{t1}) + (1 – β)T_{ t1
}
Ŷ_{t+1 }= C_{t }+ T_{t
}
All symbols appearing in the single exponential smoothing equation represent the same in the double exponential smoothing equation, but now β is the trendsmoothing constant (whereas α is the smoothing constant for a stationary – constant – process) also between 0 and 1; C_{t} is the smoothed constant process value for period t; and T_{t} is the smoothed trend value for period t.
As with single exponential smoothing, you must select starting values for C_{t} and T_{t}, as well as values for α and β. Recall that these processes are judgmental, and constants closer to a value of 1.0 are chosen when less smoothing is desired (and more weight placed on recent values) and constants closer to 0.0 when more smoothing is desired (and less weight placed on recent values).
An Example
Let’s assume you’ve got 12 months of sales data, shown in the table below:
Month t

Sales Y_{t}

1

152

2

176

3

160

4

192

5

220

6

272

7

256

8

280

9

300

10

280

11

312

12

328

You want to see if there is any discernable trend, so you plot your sales on the chart below:
The time series exhibits an increasing trend. Hence, you must use double exponential smoothing. You must first select your initial values for C and T. One way to do that is to again assume that the first value is equal to its forecast. Using that as the starting point, you set C_{2} = Y_{1}, or 152. Then you subtract Y_{1} from Y_{2} to get T_{2}: T_{2} = Y_{2} – Y_{1} = 24. Hence, at the end of period 2, your forecast for period 3 is 176 (Ŷ_{3} = 152 + 24).
Now you need to choose α and β. For the purposes of this example, we will choose an α of 0.20 and a β of 0.30. Actual sales in period 3 were 160, and our constantsmoothing equation is:
C_{3} = 0.20(160) + (1 – 0.20)(152 + 24)
= 32 + 0.80(176)
= 32 + 140.8
= 172.8
Next, we compute the trend value with our trendsmoothing equation:
T_{3} = 0.30(172.8 – 152) + (1 – 0.30)(24)
= 0.30(20.8) + 0.70(24)
= 6.24 + 16.8
=23.04
Hence, our forecast for period 4 is:
Ŷ_{4} = 172.8 + 23.04
= 195.84
Then, carrying out your forecasts for the 12month period, you get the following table:


Alpha=

0.2

Beta=

0.3

Month t

Sales Y_{t}

C_{t}

T_{t}

Ŷ_{t}

Absolute Deviation

1

152





2

176

152.00

24.00

152.00


3

160

172.80

23.04

176.00

16.00

4

192

195.07

22.81

195.84

3.84

5

220

218.31

22.94

217.88

2.12

6

272

247.39

24.78

241.24

30.76

7

256

268.94

23.81

272.18

16.18

8

280

290.20

23.05

292.75

12.75

9

300

310.60

22.25

313.25

13.25

10

280

322.28

19.08

332.85

52.85

11

312

335.49

17.32

341.36

29.36

12

328

347.85

15.83

352.81

24.81





MAD=

20.19

Notice a couple of things: the absolute deviation is the absolute value of the difference between Y_{t} (shown in lavender) and Ŷ_{t} (shown in light blue). Note also that beginning with period 3, Ŷ_{3} is really the sum of C and T computed in period 2. That’s because period 3’s constant and trend forecasts were generated at the end of period 2 – and onward until period 12. Mean Absolute Deviation has been computed for you. As with our explanation of single exponential smoothing, you need to experiment with the smoothing constants to find a balance that most accurate forecast at the lowest possible MAD.
Now, we need to forecast for period 13. That’s easy. Add C_{12} and T_{12}:
Ŷ_{13} = 347.85 + 15.83
= 363.68
And, your chart comparing actual vs. forecasted sales is:
As with single exponential smoothing, you see that your forecasted curve is smoother than your actual curve. Notice also how small the gaps are between the actual and forecasted curves. The fit’s not bad.
Exponential Smoothing Recap
Now let’s recap our discussion on exponential smoothing:
 Exponential smoothing methods are recursive, that is, they rely on all observations in the time series. The weight on each observation diminishes exponentially the more distant in the past it is.
 Smoothing constants are used to assign weights – between 0 and 1 – to the most recent observations. The closer the constant is to 0, the more smoothing that occurs and the lighter the weight assigned to the most recent observation; the closer the constant is to 1, the less smoothing that occurs and the heavier the weight assigned to the most recent observation.
 When no discernable trend is exhibited in the data, single exponential smoothing is appropriate; when a trend is present in the time series, double exponential smoothing is necessary.
 Exponential smoothing methods require you to generate starting forecasts for the first period in the time series. Deciding on those initial forecasts, as well as on the values of your smoothing constants – alpha and beta – are arbitrary. You need to base your judgments on your experience in the business, as well as some experimentation.
 Exponential smoothing models do not forecast well when the time series pattern (e.g., level of sales) is suddenly, drastically, and permanently altered by some event or change of course or action. In these instances, a new model will be necessary.
 Exponential smoothing methods are best used for shortterm forecasting.
Next Week’s Forecast Friday Topic: Regression Analysis (Our Series within the Series!)
Next week, we begin a multiweek discussion of regression analysis. We will be setting up the next few weeks with a discussion of the principles of ordinary least squares regression (OLS), and then discussions of its use as a timeseries forecasting approach, and later as a causal/econometric approach. During the course of the next few Forecast Fridays, we will discuss the issues that occur with regression: specification bias, autocorrelation, heteroscedasticity, and multicollinearity, to name a few. There will be some discussions on how to detect – and correct – these violations. Once the regression analysis miniseries is complete, we will be set up to discuss ARMA and ARIMA models, which will be written by guest bloggers who are wellexperienced in those approaches. We know you’ll be very pleased with the weeks ahead!
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