Data = Truth Error

1
Data Truth Error

• A Paradigm for Any Data

2
Finding Truth in Forecasting

• Smoothing
• Truth can be approximated by averaging out
  data.
• 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)

• Level, Lt
• Trend, Tt
• Season, Ft
• Irregulart
• (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
A. Simple Exponential Smoothing

• Model for Yt
• Yt Lt irregulart
• No trend, no seasonality
• Forecasting of Y(Th)
• Pred_Y(ThT) YT(h) in NB LT

7
Estimation of LT

• Information set at T
• Average only the most recent m observations

8
Estimation of LT cont.

• weighted average of all observations
• LT wT YT w(T-1) Y(T-1) 0 lt wt
  lt 1 for all t
• greater weights for recent data points.

9
Weighting Scheme

• Choose 0 lt a lt 1
• wT a
• w(T-1) a (1-a)
• w(T-2) a (1-a)2 and so on.
• Note

10
Recursive Form Algorithm

• LT a YT a (1-a) Y(T-1) a (1-a)2 Y(T-2)
  ...
• a YT (1-a) L(T-1)
• L(T-1) a Y(T-1) (1-a) L(T-2) and so on.

Est. for t (smooth. const.) x Data _at_ t (1 -
s. c.)(Est. _at_ t-1)
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Example 1
Initialize
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Error Correction Form
Est. for t Est._at_ t-1 s.c.(forecast error_at_t)

• One Step Ahead Forecast Error
• et Yt - L(t-1)
• Error Correction Form
• LT a YT (1 - a) L(T-1) a (YT - L(T-1))
  L(T-1)
• L(T-1) a eT

13
Example 2
Recursive Form
Initialize, no error
14
Selecting a

• Extreme Values
• a 1 LT YT
• a 0 LT L1 (initial value)
• Guide Lines
• Large a for less volatile series
• Small a 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(T1) is YT and L(T-1)
• Forecast of Y(T1T) LT a YT (1 - a) 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, a?
• 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 Holts Linear Trend for Trend without
  Seasonality
• C Holt-Winters for Trend and Seasonality
• Problems
• (1) Initial estimates
• (2) smoothing constants one for each component

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B. Holts Linear Trend Exponential
Smoothing
T
20
Include Trend Component for Forecast

• Model for Data Yt Lt irregulart
• Lt L(t-1) T(t-1)
• Forecast Pred_Y(T1 T) LT TT
• Pred_Y(Th T) LT hTT
• h1, 2,

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Recursive Formula for Lt and Tt
Est. for t (smooth. const.) x Data_at_t (1 -
s.c.)(Est._at_t-1)

• For Level Lt aYt (1 - a)(L(t-1) T(t-1))
• For Trend Tt b(Lt - L(t-1)) (1 - b)
  T(t-1)

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Example 1
Initialize
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Error Correction Form
Est. for t Est._at_t-1 (s.c.)(forecast error_at_t)

• One Step Ahead Forecast Error for Yt
• et Yt - L(t-1) T(t-1)
• ECF (see page 198 of NB)
• Lt L(t-1) T(t-1) a e t
• Tt T(t-1) abe t

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Example 2
Initialize
25
Computing Holts Linear Trend Smoothing an
Illustration
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Comparison With Fixed Trend

• Fixed Trend
• Y( T1 T) a b(T1) LT b
• Holts Model
• Y( T1 T) LT b T (slope variable)

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C. Holt-Winters Seasonal Exponential
Smoothing

• Let s of seasons in a year
• Model for Yt Lt Ft irregulart
• - additive seasonality
• Yt Lt Ft (irregulart)
• - multiplicative seasonality
• Lt Lt-1 Tt-1

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Forecasting for Holt Winters MethodsNeed to
Estimate Ft by F(t-s)

• Additive Seasonality
• Pred_YT1T LTTTF(T1-s)
• Pred_YThT LThTTF(Th-s)
• Multiplicative Seasonality
• Pred_ YT1T (LTTT) F(T1-s)
• Pred_ YThT (LTh TT) F(Th-s)

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Recursive Formula- additive seasonality
Est. for t (smooth. const.) x Data_at_t (1 -
s.c.)(Est._at_t-1)

• Level Lt a (Yt - F(t-s) ) (1 - a) L(t-1)
  T(t-1)
• Trend Tt b (Lt - L(t-1)) (1 - b) T(t-1)
• Season Ft g (Yt - Lt) (1 - g) F(t-s)

30
Error Correction Form- additive seasonality
Est. for t Est._at_t-1/s (s.c.)(forecast
error_at_t)

• Error et Yt - (Lt-1 Tt-1 F(t - s))
• ECF
• Lt (L(t-1) T(t-1)) a e t
• Tt T(t-1) a b et
• Ft F(t-s) g (1-a) e t

31
Recursive Formula- multiplicative seasonality

• Level
• Trend
• Season

Tt b (Lt - L(t-1)) (1 - b) T(t-1)
32
Error Correction Form- multiplicative seasonality

• Error et Yt - (L(t-1) T(t-1) ) F(t-s)
• ECM
• Lt L(t-1) T(t-1) a e t / F(t-s)
• Tt T(t-1) a b e t / F(t-s)
• Ft F(t-s) g (1-a) e t / Lt

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Determining Initial Values

• Use the average of the first s observations of
  data for L1 ..Ls.
• Compute the F1 through Fs, using (Y1, L1) (Ys,
  Ls).
• Set T1Ts 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, a) ser_name smooth_name
• Holt smooth(n, a, b)
• Holt-Winters smooth(a, a, b, g) additive
• smooth(m, a, b, g)
  multiplicative