Exponential smoothing

This is a widely used forecasting technique in retailing, even though it

has not proven to be especially accurate.

Why is exponential smoothing so popular?

It's easy—the exotic term notwithstanding.

Data storage requirements are minimal (even though this is not the problem it once was due to plunging memory prices).

It is very cost effective when forecasts must be made for a large number of items--hence it has extensive use in retailing.

 

The basic algorithm

1)1( ttt LXL

Where:

•Lt is the forecast for the current period;

•Xt is the most recent observation of the time series variable—such as, for example, sales last month of part #000897

•Lt-1 is the most recent forecast; and

is the smoothing constant, where 0 < < 1

(1)

New Forecast = (New Data) + (1 - )Most Recent Forecast

Equation (1) can be

written as follows:

Exponential smoothing is weighted moving average process

To demonstrate, let

211 )1( ttt LXL

Substitute (2) into (1):

22

1

21

)1()1(

])1()[1(

ttt

tttt

LXX

LXXL

But notice that:

322 )1( ttt LXL

Substitute (4) into (3) to obtain:

33

22

1 )1()1()1( ttttt LLXXL

If we continue to substitute recursively, we get:

33

22

1 )1()1()1( ttttt XXXXL

(4)

Notice that

,)1(,)1(,)1(),1(, 432

are the weights attached to past values of X. Since < 1, the weights attached to earlier or more remote observations of X are diminishing.

You don’t have to go through this recursive process each time you

do a forecast. The process is summarized

in the most recent forecast.

Selecting the smoothing constant ()

•The range of possible values is zero and one.

•If you select a value of close to 1, that means you are attaching a large weight to the most recent observation. This is not indicated if your series is very erratic (swings widely from period to period). For example, suppose you were forecasting the demand for part #56 in month t.

Sale

s of

part

#5

6

Montht-1 tt-2

If you attached too much weight to the observation for t-1, you will have a large forecast error for month t.

We will now forecast

sales of liquor and floor covering using this technique. We have monthly data for each variable

beginning in January 1999 and

running through July of 2007.

4000

4500

5000

5500

6000

6500

7000

1000

2000

3000

4000

5000

99 00 01 02 03 04 05 06 07

Parts, Accessories, Tires Beer, Wine, Liquor

Exponential Smoothing Demonstration

Year/Month

Millions of Dollars

Beer, Wine, Liquor Parts, Accessories, Tires

Mean 2653.35922 Mean 5568.1068Standard Error 49.798155 Standard Error 54.5247066Median 2567 Median 5546Mode 2232 Mode 5613Standard Deviation 505.396076 Standard Deviation 553.365335Sample Variance 255425.193 Sample Variance 306213.194Kurtosis 1.88359717 Kurtosis -0.2735442Skewness 1.19777269 Skewness 0.23599223Range 2770 Range 2411Minimum 1818 Minimum 4503Maximum 4588 Maximum 6914Sum 273296 Sum 573515Count 103 Count 103

The ratio of the standard deviation to the mean gives us a nice measure of the

amplitude or volatility of a series month-to-month (or day-to-day, quarter-to-quarter, as

the case may be).

X

sAmplitude Beer, Wine, Liquor =

0.1904Parts, Tires, etc. = 0.099

•Pricey time series forecasting software, such as EViews, use an algorithm to select the value of the smoothing constant that minimizes mean square error for in-sample forecasts.

•If you lack this software, you can use a trial and error process.

Selecting the smoothing constant

1500

2000

2500

3000

3500

4000

4500

5000

99 00 01 02 03 04 05 06 07

Actual Smoothed

Beer, Wine, and Liquor Sales, Smoothed (Alpha = 0.1280)

Year/Month35.405$MSE

4000

4500

5000

5500

6000

6500

7000

99 00 01 02 03 04 05 06 07

Actual Smoothed

Sales of Auto Parts, Accessories, and Tires, Smoothed (Alpha = 0.69)

Year/Month

millions of dollars

56.347$MSE

Year Month Actual Smoothed2006 7 6493 6642.082006 8 6914 6539.212006 9 6245 6797.822006 10 6419 6416.372006 11 6072 6418.192006 12 5900 6179.322007 1 5628 5986.592007 2 5526 5739.162007 3 6608 5592.082007 4 6144 6293.062007 5 6702 6190.212007 6 6619 6543.342007 7 6538 6595.55

Auto Parts, Accessories, and Tires (Alpha = .69)

Beer, Wine, and Liquor (Alpha = .1280)

Year Month Actual Smoothed2006 7 3322 2994.102006 8 3228 3036.072006 9 3212 3060.642006 10 3120 3080.012006 11 3359 3085.132006 12 4588 3120.192007 1 2710 3308.082007 2 2748 3231.522007 3 3176 3169.632007 4 3037 3170.442007 5 3459 3153.362007 6 3578 3192.482007 7 3547 3241.83

Forecasts for August, 2007

Remember our basic algorithm

1)1( ttt LXL

Hence to parts, accessories, and tire sales (PAT) for August, 2007:

To forecast beer, wine, and liquor sales (BWL):

84.555,6$)]55.595,6)(69.1[()]538,6)(69[(.07 AUGPAT

89.280,3$)]83.241,3)(1280.1[()]547,3)(1280[(.07 AUGBWL