1. ISEN 315 Spring 2011 Dr. Gary Gaukler

2. Forecasting for Stationary Series A stationary time series has the form: D t =  +  t where  is a constant and  t is a random variable with mean 0 and var    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: F t = (1/n) (D t - 1 + D t - 2 +... + D t - n ) (Go to Example.)

4. Exponential Smoothing Method A type of weighted moving average that applies declining weights to past data. 1. New Forecast =  (most recent observation) + (1 -  (last forecast) or 2. New Forecast = last forecast -  last forecast error) where 0 <  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 –

7. Two-equation Smoothing Model Add linear trend: Assume D t =  + t G +  t S t =  D t + (1-  ) [S t-1 + 1 G t-1 ], whereG t -1 = 1-period trend estimate

8. Two-equation Smoothing Model: Update G by exponential smoothing: G t =  ( S t - S t-1 ) + (1 -  ) G t-1 Then forecast is: F t, t+  = S t +  G t

10. Example Demand: 200, 250, 175 Estimates: S 0 =200, G 0 =10 Parameters:  Estimate demand in weeks 4 - 6

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

15. Deriving the Regression Parameters

20. Using Regression for Forecasting Least squares estimates for a and b are computed as follows: 1) Set S xx = n 2 (n+1)(2n+1)/6 - [n(n+1)/2] 2 2) Set S xy = n Σ (i D i )- [n(n + 1)/2] Σ D i 3) Let b = S xy / S xx and a = D avg - b (n+1)/2

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

22. The Difficulty with Long-Term Forecasts