EC473 Forecasting Seminar

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• EC473 Forecasting Seminar
• Fall 2007
• New Forecasting Technique Simple Exponential
  Smoothing

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Recession Shading RECSHADE.prg
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Moving Averages

• m-period backward moving average
• Uses only current past values of Y.
• Choice of m is arbitrary.
• Lose m observations from the beginning of the
  series.
• EViews command _at_movav(series,periods)

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Centered Moving Average

• The larger the value of m the smoother the
  series.
• m observations are lost at the beginning and at
  the end of the smoothed series.
• Works for non-seasonal data.
• Provides an estimate of the trend component.

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EViews Command To Generate a Centered Moving
Average of Italian Imports

• genr imports_cma
• (importsit(-6) importsit(-5)
• importsit(-4) importsit(-3)
• importsit(-2) importsit(-1) importsit
• importsit(1) importsit(2) importsit(3)
• importsit(4) importsit(5) importsit(6)) / 13

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Moving Averages With Seasonal Data

• Consider a monthly time series, Yt.
• Each set of 12 consecutive observations will
  contain exactly one value from each month.
• Therefore, a 12-period moving average of Y should
  be largely free of seasonality.
• Problem The resulting time series will be
  centered between time period t and t1.

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2 x 12 Moving Average

• Solution Averaging adjacent pairs of smoothed
  values will re-center the series.
• Equivalently, we can construct a 2-point moving
  average of a 12-point moving average.
• The result is an estimate of the trend-cycle
  component.

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Seasonal Adjustment with Moving Averages

• Assume that the monthly time series Xt can be
  decomposed into additive trend-cycle, seasonal,
  and irregular components.
• Step 1. Compute the trend as a 2 x 12 moving
  average of X.
• Step 2. Subtract this from the original
  observations on X leaving a series which contains
  the sum of the seasonal and irregular components.
• Step 3. The seasonal component at time t, St, can
  be estimated by

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Exponential Smoothing

• Smoothing refers to a procedure of taking
  weighted sums of data in order to smooth out
  very short term irregularities in a time series.

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Characteristics

• Simple, naïve forecasting methods that are used
  widely in the areas of sales, inventory and
  production management, and quality control.
• Univariate technique based on the mathematical
  projection of past patterns into the future.

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Characteristics

• Computationally easy, requiring relatively little
  historical data.
• Can be useful for short-term forecasting
  problems.
• Single, double, and triple exponential smoothing
  models can be appropriate for horizontal, linear,
  and quadratic time series, respectively.

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Characteristics

• Judgment on the part of the forecaster plays an
  important role in selecting a smoothing model.
• Check for reasonableness by plotting historical
  and forecasted values on the same graph.

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Potential Problems

• Outliers may cause distortions in the smoothed
  series.
• Cyclical peaks and troughs are rarely followed
  smoothly by exponential smoothing.

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Single Exponential Smoothing

• a is the smoothing constant.
• Can rewrite the basic equation above as
• Interpretation The forecast of Y for the next
  period equals the forecast for the prior period
  adjusted by a fraction (a) of the most recent
  forecast error (Yt-St).

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Can express St as a weighted average with
exponentially decreasing weights

• Substituting (2) (3) into (1) yields

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• By substituting successively for St-2,St-3,St-4,
  
• where S0 is an initial estimate of the smoothed
  value.
• S0 may be obtained from historical data using a
  simple average of recent observations or by
  setting S0 Y0.

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Interpretation

• For the exponential smoothing model, the forecast
  is expressed as a weighted average of actual
  observations that go back to an initial starting
  value.
• The most recent observations have the largest
  contribution to the forecast while the effects of
  distant observations diminish with time.

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Consider a 0.8

• The weights in the smoothing equation are 0.2,
  0.04, 0.008, 0.0016, for Yt, Yt-1, Yt-2, Yt-3,
  , respectively.
• Periods past Yt-3 have almost no effect on the
  forecast of Y.

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Choosing a

• For large values of a, contributions of past
  observations quickly become unimportant. If a1
  St1Yt.
• For small values of a, the contributions of
  past observations to the forecast of Y dampens
  out more slowly.
• Setting a near 0 means you believe there is a
  good deal of information in past values of Y that
  is useful for forecasting future values.

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Series with Trends

• Simple exponential smoothing will consistently
  under forecast time series with upward trends and
  over forecast time series with downward trends,
  no mater how large a value for a you choose.
• Alternative exponential smoothing models are
  available for series with significant trend or
  seasonal components.

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Next Forecast Assignment

• Use decomp.prg to decompose your series into its
  trend-cycle, seasonal, irregular components.
• Use simple exponential smoothing to forecast the
  trend-cycle component.
• Extrapolate the seasonal index over the forecast
  horizon.
• Forecast your series by reassembling the
  components over the forecast horizon and assuming
  zero values for the forecast of the irregular
  component.

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EViews Example

• Execute decomp.prg importsit 7401 0607
• smpl 19741 200606
• show trend
• Point and click on the Proc button and select
  the Exponential Smoothing option.
• Point and click on the Single button.
• Set the Estimation sample 7401 0607
• Point and click on the OK button.

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Result

• Sample 1974M07 2005M12
• Included observations 378
• Method Single Exponential
• Original Series TREND
• Forecast Series TRENDSM
• Parameters Alpha 0.9990
• Sum of Squared Residuals 163175.4
• Root Mean Squared Error 20.77693
• End of Period Levels Mean 2648.629

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Now Put the Pieces Together

• smpl 200607 200812
• genr index index(-12)
• genr importsf trendsmindex
• smpl 197401 200812
• show importsit trendsm importsf

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Loose Ends

• Note that you lose 6 important observations off
  the end of TRENDSM when using this trend
  smoothing procedure.
• Thus, the forecast begins 6 months too early.
• The results can be messy.
• Think about creative solutions.

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Odds Ends

• Experiment with different smoothing constants
  (ALPHA) to help you understand whats going on
  with this forecasting method.
• The simple exponential smoothing method produces
  a flat line forecast. Can you explain why?
• Think carefully about a proper way to produce a
  measure of ex post forecast accuracy.
• Different values for a may produce a more
  accurate forecast even though the historical fit
  is not as good.