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.
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Why is exponential smoothing so popular?

• It's easythe 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.
•  

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The basic algorithm
(1)

• Where
• Lt is the forecast for the current period
• Xt is the most recent observation of the time
  series variablesuch as, for example, sales last
  month of part 000897
• Lt-1 is the most recent forecast and
• ? is the smoothing constant, where 0 lt ? lt 1

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Equation (1) can be written as follows
New Forecast ?(New Data) (1 - ?)Most Recent
Forecast
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Exponential smoothing is weighted moving average
process
To demonstrate, let
(2)
Substitute (2) into (1)
(3)
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But notice that
Substitute (4) into (3) to obtain
If we continue to substitute recursively, we get
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Notice that
are the weights attached to past values of X.
Since ? lt 1, the weights attached to earlier or
remoter observations of X are diminishing.
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You dont have to go through this recursive
process each time you do a forecast. The process
is summarized in the most recent forecast.
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Selecting the smoothing constant (?)
?alpha?

• 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 choppy. For example, suppose you
  were forecasting the demand for part 56 in month
  t.

If you attached too much weight to the
observation for t-1, you will have a large
forecast error for month t.
Sales of part 56
t-1
t-2
t
Month
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Application
We will now forecastsales of liquor and floor
covering using this technique. We have monthly
data for each variable beginning in January 1995
and running through July of 2000.
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Summary statistics for monthly sales of floor
covering and liquor sales, 19951 to 20007 (in
millions of dollars)
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Liquor 0.169Floor covering 0.127
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).
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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.

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• The first set of estimates for monthly floor
  covering and liquor were produced by using the
  algorithm that selects the best performing value
  of the smoothing constant (?) for in-sample
  forecasts.
• The second set of estimates is based on values of
  alpha (?) arbitrarily selected by the instructor.

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Computer algorithm selects alpha to minimize MSE
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Actual and smoothed values of floor covering,
19977 to 20007 (all data in millions of dollars)
Alpha 0.706
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Alpha selected arbitrarily
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Statistics for the floor covering estimates
Data is for 19951 to 20007
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Computer algorithm selects alpha to minimize MSE
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Actual and smoothed values of liquor sales,
19977 to 20007 (all data in millions of dollars)
Alpha 0.122
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Alpha selected arbitrarily
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Statistics for the liquor estimates
Data is for 19951 to 20007
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Forecasts for August, 2000
Remember our basic algorithm
Hence to forecast floor covering sales for
August, 2000
Floor CoveringAUG(0.706)(1420) (1 -
.706)(1375) 1406.77
To forecast liquor sales
LiquorAUG(0.122)(2560) (1 - .122)(2349)
2374.72