1. Data = Truth + Error A Paradigm for Any Data

2. Finding Truth in Forecasting 1.Smoothing: Truth can be “approximated” by averaging out data. 2.Standard Models: Truth can be “approximated” by a standard forecasting model (DGP)

3. FM 1: Smoothing How to average out data? How to forecast? Problems ? When most applicable?

4. Notations (NB) 1.Level, L t 2.Trend, T t 3.Season, F t 4.Irregular t ( Equal variability) Not constant

5. When Most Applicable Many items to forecast –E.g. demand for standard items Automatic procedure is needed Excel works well for implementation –(if Eviews is not available)

6. Model for Y t : Y t = L t + irregular t No trend, no seasonality Forecasting of Y (T+h) Pred_Y (T+h|T) = Y T (h) in NB = L T A. Simple Exponential Smoothing

7. Estimation of L T Information set at T Average only the most recent m observations

8. weighted average of all observations: L T = w T Y T + w (T-1) Y (T-1) + … 0 < w t < 1 for all t greater weights for recent data points. Estimation of L T – cont.

9. Weighting Scheme Choose 0 <  < 1 w T =  w (T-1) =  (1-  ) w (T-2) =  (1-  ) 2 and so on. Note:

10. Recursive Form Algorithm L T =  Y T +  Y (T-1) +  2 Y (T-2) +... =  Y T +  L (T-1) L (T-1) =  Y (T-1) +  L (T-2) and so on. Est. for t = (smooth. const.) x Data @ t + (1 - s. c.)(Est. @ t-1)

11. Example 1 Initialize

12. Error Correction Form One Step Ahead Forecast Error –e t = Y t - L (t-1) Error Correction Form –L T =  Y T + (1 -  ) L (T-1) =  (Y T - L (T-1) ) + L (T-1) = L (T-1) +  e T Est. for t = Est.@ t-1 + s.c.(forecast error@t)

13. Example 2 Initialize, no error Recursive Form

14. Selecting  Extreme Values  = 1 L T = Y T  = 0L T = L 1 (initial value) Guide Lines Large  for less volatile series Small  for more volatile series

15. SSE and RMSE SSE = Sum of Squared Residuals –For Exponential Smoothing, SSE = Sum of Squared One Step Ahead Forecasting Errors. RMSE = Root Mean Squared Error –Square Root of { SSE / # of Errors in SSE }

16. Practicality 1. Only information needed to forecast Y (T+1) is { Y T and L (T-1) } Forecast of Y (T+1|T) = L T =  Y T + (1 -  ) L (T-1) 2. Robustness Ref. NB 6.10

17. Two Problems How to determine the initial value? –Use the first observation –Take the average of the first half observations How to determine the best smoothing constant,  ? –Use RMSE as a guide –Do not minimize RMSE

18. Extensions of Simple Exponential Smoothing Data = Trend + Seasonality + Cycle + Irregularity How to Incorporate Trend and Seasonality for Forecast? –B: Holt’s Linear Trend for Trend without Seasonality –C: Holt-Winters for Trend and Seasonality Problems –(1) Initial estimates –(2) smoothing constants – one for each component

19. B. Holt’s Linear Trend Exponential Smoothing     Holt Simple YtYt t T

20. Include Trend Component for Forecast Model for Data: Y t = L t + irregular t L t = L (t-1) +T (t-1) Forecast: Pred_Y(T+1 | T) = L T + T T Pred_Y(T+h | T) = L T +hT T h=1, 2, …

21. Recursive Formula for L t and T t For Level: L t =  Y t + (1 -  )(L (t-1) + T (t-1) ) For Trend: T t =  (L t - L (t-1) ) + (1 -  ) T (t-1) Est. for t = (smooth. const.) x Data@t + (1 - s.c.)(Est.@t-1)

22. Example 1 Initialize

23. Error Correction Form “One Step Ahead” Forecast Error for Y t : e t = Y t - {L (t-1) + T (t-1) } ECF (see page 198 of NB): L t = {L (t-1) + T (t-1) } +  e t T t = T (t-1) +  e t Est. for t = Est.@t-1 + (s.c.)(forecast error@t)

24. Example 2 Initialize

25. Computing Holt’s Linear Trend Smoothing – an Illustration

26. Comparison With Fixed Trend Fixed Trend: Y( T+1| T) =  +  T+1) = L T +  Holt’s Model: Y( T+1| T) = L T +  T (slope variable)

27. Let: s : # of “seasons” in a year Model for Y t = L t +F t + irregular t - additive seasonality Y t = L t F t (irregular t ) - multiplicative seasonality L t = L t-1 + T t-1 C. Holt-Winters Seasonal Exponential Smoothing

28. Additive Seasonality Pred_Y T+1 |T = L T +T T +F (T+1-s) Pred_Y T+h |T = L T +hT T +F (T+h-s) Multiplicative Seasonality Pred_ Y T+1 |T = (L T +T T ) F (T+1-s) Pred_ Y T+h |T = (L T +h T T ) F (T+h-s) Forecasting for Holt – Winters Methods Need to Estimate F t by F (t-s)

29. Recursive Formula - additive seasonality Level: L t =  (Y t - F (t-s) ) + (1 -  ) {L (t-1) + T (t-1) } Trend:T t =  (L t - L (t-1) ) + (1 -  ) T (t-1) Season:F t =  (Y t - L t ) + (1 -  ) F (t-s) Est. for t = (smooth. const.) x Data@t + (1 - s.c.)(Est.@t-1)

30. Error:e t = Y t - (L t-1 + T t-1 + F (t - s) ) ECF: L t = (L (t-1) +  T (t-1) )  e t T t = T (t-1) +  e t F t = F (t-s) +  e t Error Correction Form - additive seasonality Est. for t = Est.@t-1/s + (s.c.)(forecast error@t)

31. Recursive Formula - multiplicative seasonality Level: Trend: Season: T t =  (L t - L (t-1) ) + (1 -  ) T (t-1)

32. Error: e t = Y t - (L (t-1) + T (t-1) ) F (t-s) ECM: L t = L (t-1) +  T (t-1)  e t / F (t-s) T t = T (t-1) +  e t / F (t-s) F t = F (t-s) +  e t / L t Error Correction Form - multiplicative seasonality

33. Determining Initial Values Use the average of the first s observations of data for L 1..L s. Compute the F 1 through F s, using (Y 1, L 1 ) …(Y s, L s ). Set T 1 …T s = 0 Note: This is just one method.

34. Example: Additive Seasonality

35. Example: Multiplicative Seasonality

36. Choosing Smoothing Constants Forecast = f(Data, s.c, initial values) Big Question: must evolve from using the system Recommendation: use small values, say 0.2 to 0.5, to begin with

37. Using Eviews Simple smooth(s,  ) ser_name smooth_name Holt smooth(n,  ) Holt-Winters smooth(a,  additive smooth(m,  multiplicative