# Exponential smoothing

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```					        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

Lt  Xt  (1   ) Lt         1    (1)

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
Equation (1)
can be
written as
follows:

New Forecast = (New Data) + (1 - )Most Recent Forecast
Exponential smoothing is weighted moving average
process

To demonstrate, let
Lt  1  Xt  1  (1   ) Lt  2

Substitute (2) into (1):

Lt  Xt  (1   )[Xt  1  (1   ) Lt  2] 
Xt   (1   ) Xt  1  (1   ) 2 Lt  2
But notice that:

Lt  2  Xt  2  (1   ) Lt  3      (4)

Substitute (4) into (3) to obtain:

Lt  Xt   (1   ) Xt  1  (1   ) 2 Lt  2  (1   ) 3 Lt  3

If we continue to substitute recursively, we get:

Lt  Xt   (1   ) Xt  1   (1   ) 2 Xt  2   (1   ) 3 Xt  3  
Notice that

 ,  (1   ),  (1   ) 2 ,  (1   ) 3 ,  (1   ) 4 , 

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.
If you attached too
Sales of part #56

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

t-2   t-1   t        Month
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.
Exponential Smoothing Demonstration
Millions of Dollars
5000

4000

7000                                                            3000

6500
2000
6000
5500                                                            1000

5000
4500
4000
99    00     01    02     03   04     05    06    07
Year/Month
Parts, Accessories, Tires      Beer, Wine, Liquor
Beer, Wine,
Liquor                       Parts, Accessories, Tires

Mean              2653.35922   Mean                          5568.1068
Standard Error    49.798155    Standard Error                54.5247066
Median               2567      Median                           5546
Mode                 2232      Mode                             5613
Standard
Deviation         505.396076   Standard Deviation            553.365335
Sample Variance   255425.193   Sample Variance               306213.194
Kurtosis          1.88359717   Kurtosis                      -0.2735442
Skewness          1.19777269   Skewness                      0.23599223
Range                2770      Range                            2411
Minimum              1818      Minimum                          4503
Maximum              4588      Maximum                          6914
Sum                 273296     Sum                             573515
Count                 103      Count                             103
s
Amplitude                Beer, Wine, Liquor = 0.1904
X              Parts, Tires, etc. = 0.099

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).
Selecting the
smoothing constant
•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.
Beer, Wine, and Liquor Sales, Smoothed (Alpha = 0.1280)
5000

4500

4000

3500

3000

2500

2000

1500
99   00     01     02    03    04     05     06    07
Year/Month
MSE  \$405.35              Actual       Smoothed
Sales of Auto Parts, Accessories, and T ires, Smoothed (Alpha = 0.69)
millions of dollars
7000

6500

6000

5500

5000

4500

4000
99   00   01    02    03   04    05    06    07
Year/Month
MSE  \$347.56           Actual     Smoothed
Auto Parts, Accessories, and Tires (Alpha = .69)

Year   Month    Actual   Smoothed
2006     7       6493     6642.08
2006     8       6914     6539.21
2006     9       6245     6797.82
2006    10       6419     6416.37
2006    11       6072     6418.19
2006    12       5900     6179.32
2007     1       5628     5986.59
2007     2       5526     5739.16
2007     3       6608     5592.08
2007     4       6144     6293.06
2007     5       6702     6190.21
2007     6       6619     6543.34
2007     7       6538     6595.55
Beer, Wine, and Liquor (Alpha = .1280)

Year   Month   Actual   Smoothed
2006     7      3322     2994.10
2006     8      3228     3036.07
2006     9      3212     3060.64
2006    10      3120     3080.01
2006    11      3359     3085.13
2006    12      4588     3120.19
2007     1      2710     3308.08
2007     2      2748     3231.52
2007     3      3176     3169.63
2007     4      3037     3170.44
2007     5      3459     3153.36
2007     6      3578     3192.48
2007     7      3547     3241.83
Forecasts for August, 2007
Remember our basic algorithm
Lt  Xt  (1   ) Lt             1

Hence to parts, accessories, and tire sales (PAT) for
August, 2007:
PAT AUG07  [(. 69 )( 6,538 )]  [(1  .69 )( 6,595 .55 )]  \$6,555 .84

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

BWL AUG07  [(. 1280 )(3,547 )]  [(1  .1280 )(3,241 .83)]  \$3,280 .89

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