Exponential Smoothing

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Exponential Smoothing

• Diebold, Chapter 4, Problem 4

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Exponential Smoothing

• single exponential smoothing uses the parameter
  alpha (a) where alpha is chosen to be between 0
  and 1. The formula for single exponential
  smoothing is
• F(t1) F(t) a X(t) F(t) where 0 lt
  a lt 1.
• Initialize by letting F(1) X(1)
• F(t1) one step ahead forecast at time period
  t
• F(t) forecast for time period t
• X(t) sales for time period t

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Exponential smoothing (Eviews Help)

• a simple method of adaptive forecasting.
• effective way of forecasting when you have only a
  few observations on which to base your forecast.

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Smoothing vs. Regression

• Unlike forecasts from regression models which use
  fixed coefficients, forecasts from exponential
  smoothing methods adjust based upon past forecast
  errors.
• For additional discussion, see Bowerman and
  O'Connell (1979).

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Exponential Smoothing in Eviews

• To obtain forecasts based on exponential
  smoothing methods, choose Proc/Exponential
  Smoothing. The Exponential Smoothing dialog box
  appears

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You need to provide the following information

• Smoothing Method. You have the option to choose
  one of the five methods listed.
• Smoothing Parameters. You can either specify the
  values of the smoothing parameters or let EViews
  estimate them.

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Estimating parameters

• To estimate the parameter, type the letter e (for
  estimate) in the edit field.
• EViews estimates the parameters by minimizing the
  sum of squared errors.
• Don't be surprised if the estimated damping
  parameters are close to one-it is a sign that the
  series is close to a random walk, where the most
  recent value is the best estimate of future
  values.

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Choosing your own parameters

• To specify a number, type the number in the field
  corresponding to the parameter.
• All parameters are constrained to be between 0
  and 1 if you specify a number outside the unit
  interval,
• EViews will estimate the parameter.

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Series Name

• Smoothed Series Name. You should provide a name
  for the smoothed series. By default, EViews will
  generate a name by appending SM to the original
  series name, but you can enter any valid EViews
  name.

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Estimation Sample

• You must specify the sample period upon which to
  base your forecasts (whether or not you choose to
  estimate the parameters). The default is the
  current workfile sample. EViews will calculate
  forecasts starting from the first observation
  after the end of the estimation sample.

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Cycle for Seasonal

• You can change the number of seasons per year
  from the default of 12 for monthly or 4 for
  quarterly series. This option allows you to
  forecast from unusual data such as an undated
  workfile. Enter a number for the cycle in this
  field.

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Single Smoothing (one parameter)

• This single exponential smoothing method is
  appropriate for series that move randomly above
  and below a constant mean with no trend nor
  seasonal patterns. The smoothed series is
  computed recursively, by evaluating

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Single Smoothing (one parameter)

• where alpha is the damping (or smoothing) factor.
  The smaller is the alpha, the smoother is the
  forecasted series. By repeated substitution, we
  can rewrite the recursion as

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Single Smoothing (one parameter)

• This shows why this method is called exponential
  smoothing-the forecast is a weighted average of
  the past values of the series, where the weights
  decline exponentially with time.

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Single Smoothing (one parameter)

• The forecasts from single smoothing are constant
  for all future observations. This constant is
  given by

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Single Smoothing (one parameter)

• To start the recursion, we need an initial value
  for and a value for alpha.
• EViews uses the mean of the initial observations
  of to start the recursion.
• Bowerman and O'Connell (1979) suggest that values
  of around 0.01 to 0.30 work quite well.
• You can also let EViews estimate to minimize the
  sum of squares of one-step forecast errors.

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Double Smoothing (one parameter)

• This method applies the single smoothing method
  twice (using the same parameter) and is
  appropriate for series with a linear trend.
  Double smoothing of a series is defined by the
  recursions

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Double Smoothing (one parameter)

• Where S is the single smoothed series and D is
  the double smoothed series. Note that double
  smoothing is a single parameter smoothing method
  with damping factor alpha between 0 and 1.

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Double Smoothing (one parameter)

• Forecasts from double smoothing are computed as

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Double Smoothing (one parameter)

• The last expression shows that forecasts from
  double smoothing lie on a linear trend with
  intercept
• and slope

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Holt-Winters-Multiplicative (three parameters)

• This method is appropriate for series with a
  linear time trend and multiplicative seasonal
  variation. The smoothed series is given by,

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Holt-Winters-Multiplicative (three parameters)

• Where
• a is permanent component (intercept)
• b is trend
• C is multiplicative seasonal factor

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These three coefficients are defined by the
following recursions
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Forecasts are computed by
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Holt-Winters-Additive (three parameter)

• This method is appropriate for series with a
  linear time trend and additive seasonal
  variation. The smoothed series is given by

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The three coefficients are defined by the
following recursions
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Forecasts are computed by
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Holt-Winters-No Seasonal (two parameters)

• This method is appropriate for series with a
  linear time trend and no seasonal variation.
• This method is similar to the double smoothing
  method in that both generate forecasts with a
  linear trend and no seasonal component.
• The double smoothing method is more parsimonious
  since it uses only one parameter, while this
  method is a two parameter method.

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The smoothed series is given by

• where a and b are the permanent component and
  trend as defined above in Equation (11.40).

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Forecasts are computed by
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Holt-Winters Comparison

• It is worth noting that Holt-Winters-No Seasonal,
  is not the same as additive or multiplicative
  with gamma equal to 0. That condition only
  restricts the seasonal factors from changing over
  time so there are still (fixed) nonzero seasonal
  factors in the forecasts.