Data = Truth + Error A Paradigm for Any Data

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)

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

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

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)

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

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

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.

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

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 t + (1 - s. t-1)

Example 1 Initialize

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 = t-1 + s.c.(forecast

Example 2 Initialize, no error Recursive Form

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

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 }

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

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

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

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

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, …

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 + (1 -

Example 1 Initialize

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 = + (s.c.)(forecast

Example 2 Initialize

Computing Holt’s Linear Trend Smoothing – an Illustration

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)

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

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)

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 + (1 -

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 = + (s.c.)(forecast

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

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

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.

Example: Additive Seasonality

Example: Multiplicative Seasonality

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

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