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ISEN 315 Spring 2011 Dr. Gary Gaukler

ISEN 315 Spring 2011 Dr. Gary Gaukler. Forecasting for Stationary Series. A stationary time series has the form: D t = m + e t where m is a constant and e t is a random variable with mean 0 and var s 2 .

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ISEN 315 Spring 2011 Dr. Gary Gaukler

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  1. ISEN 315Spring 2011Dr. Gary Gaukler

  2. Forecasting for Stationary Series A stationary time series has the form: Dt = m + e t where m is a constant and e t is a random variable with mean 0 and var s2 . Two common methods for forecasting stationary series are moving averages and exponential smoothing.

  3. Moving Averages In words: the arithmetic average of the n most recent observations. For a one-step-ahead forecast: Ft = (1/n) (Dt - 1 + Dt - 2 + . . . + Dt - n ) (Go to Example.)

  4. Exponential Smoothing Method A type of weighted moving average that applies declining weights to past data. 1. New Forecast = a (most recent observation) + (1 - a) (last forecast) or 2. New Forecast = last forecast - a (last forecast error) where 0 < a < 1 and generally is small for stability of forecasts ( around .1 to .2)

  5. Comparison of ES and MA • Similarities • Both methods are appropriate for stationary series • Both methods depend on a single parameter • Both methods lag behind a trend • Differences

  6. Two-equation Smoothing Model Add linear trend: • Assume Dt = m + t G + et St = aDt + (1-a ) [St-1 + 1 Gt-1], where Gt -1 = 1-period trend estimate

  7. Two-equation Smoothing Model: Update G by exponential smoothing: Gt = b (St - St-1) +(1 - b)Gt-1 Then forecast is: Ft,t+t = St + tGt

  8. Example Demand: 200, 250, 175 Estimates: S0=200, G0=10 Parameters: a= b=0.1 Estimate demand in weeks 4 - 6

  9. Using Regression for Forecasting • (Linear) regression methods can be used when trend is present • Model: Dt = a + bt, or y = a + bx • How do we find the a and b?

  10. Deriving the Regression Parameters

  11. Deriving the Regression Parameters

  12. Deriving the Regression Parameters

  13. Deriving the Regression Parameters

  14. Deriving the Regression Parameters

  15. Using Regression for Forecasting • Least squares estimates for a and b are computed as follows: • Set Sxx= n2 (n+1)(2n+1)/6 - [n(n+1)/2]2 • Set Sxy = n Σ (i Di)- [n(n + 1)/2] Σ Di • Let b = Sxy / Sxxand a = Davg - b (n+1)/2

  16. Example Assume demand for periods 1 through 5 is as follows: 200, 250, 175, 186, 235 What is the regression forecast for period 7?

  17. The Difficulty with Long-Term Forecasts

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