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Why the damped trend works

Everette S. Gardner, Jr. Eddie McKenzie

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Empirical performance of the damped trend

“The damped trend can reasonably claim to be a benchmark forecasting method for all others to beat.” (Fildes et al., JORS, 2008)

“The damped trend is a well established forecasting method that should improve accuracy in practical applications.” (Armstrong, IJF, 2006)

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Why the damped trend works Practice

Optimal parameters are often found at the boundaries of the [0, 1] interval. Thus fitting the damped trend is a means of automatic method selection from numerous special cases.

Theory The damped trend and each special case has an

underlying random coefficient state space (RCSS) model that adapts to changes in trend.

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The damped trend method

Recurrence form )()1( 11 tttt by

11 )1(( ) tttt bb

th by ttht )...(ˆ 2

Error-correction form

tttt eb 11

ttt ebb 1

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Special case when ø = 1

1. Holt method )()1( 11 tttt by

11 )1(( ) tttt bb

tbhy ttht ˆ

1 1

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Special cases when β = 02. SES 3. SES with drift (Hyndman and Billah, IJF, 2003)

10 , 1 : by ttt 1)1(

bhy ttht ˆ

4. SES with damped drift

10 , 10 : by ttt 1)1(

by httht )...(ˆ 2

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Fit periods for M3 Annual series # YB067

0

1,000

2,000

3,000

4,000

5,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Year

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Special cases when β = 0, continued5. Random walk 6. Random walk with drift

1 , 0 : bytt

bhy ttht ˆ

7. Random walk with damped drift

1 , 10 : bytt

by httht )...(ˆ 2

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Special cases when α = β = 0

8. 1 Linear trend

9. 10 Modified exponential trend

10. 0 Simple average

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Fitting the damped trend to the M3 series Multiplicative seasonal adjustment Initial values for level and trend

Local: Regression on first 5 observations Global: Regression on all fit data

Optimization (Minimum SSE) Parameters only Parameters and initial values (no significant difference

from parameters only)

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M3 mean symmetric APE (Horizons 1-18)Makridakis & Hibon (2000) with backcasted initial values 13.6%

Gardner & McKenzie (2010) with local initial values 13.5 with global initial values 13.8

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Method Local Global

Damped trend 43.0% 27.8%Holt 10.0

1.8

SES w/ damped drift 24.8 23.5 SES w/ drift 2.4 11.6SES 0.8 0.6RW w/ damped drift 7.8 9.6RW w/ drift 2.5 8.4RW 0.0 0.0Modified exp. trend 8.3 8.7Linear trend 0.1 7.9Simple average 0.3 0.0

Methods identified in the M3 time seriesInitial values

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Initial values

Components identified in the M3 time series

Component Local Global

Damped trend 51.3% 36.5%

Damped drift 32.6 33.6

Trend 10.1 9.7

Drift 4.9 20.0

Constant level 1.2 0.6

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Method Ann. Qtr. Mon.Damped trend 25.9% 47.1% 47.5%Holt 17.4

14.2 3.6

SES w/ damped drift 17.7 16.7 33.6 SES w/ drift 3.6 3.7 1.1SES 0.2 0.4 1.5RW w/ damped drift 18.3 9.0 1.9RW w/ drift 7.8 2.1 0.4RW 0.0 0.0 0.0Modified exp. trend 9.1 6.0 10.1Linear trend 0.2 0.1 0.1Simple average 0.0 0.8 0.2

Methods identified by type of data(Local initial values)

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Rationale for the damped trend Brown’s (1963) original thinking:

Parameters are constant only within local segments of the time series

Parameters often change from one segment to the next Change may be sudden or smooth

Such behavior can be captured by a random coefficient state space (RCSS) model

There is an underlying RCSS model for the damped trend and each of its special cases

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SSOE state space models for the damped trend

Constant coefficient Random coefficient tttt by 11 ttttt vbAy 11

tttt hb 111 ttttt vhbA *111

ttt hbb 21 tttt vhbAb *21

{At} are i.i.d. binary random variates White noise innovation processes ε and are different Parameters h and h* are related but usually different

ttttt vbAy 11

v

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Runs of linear trends in the RCSS model

With a strong linear trend, {At } will consist of long runs of 1s with occasional 0s.

With a weak linear trend, {At } will consist of long runs of 0s with occasional 1s.

In between, we get a mixture of models on shorter time scales, i.e. damping.

tttt vhbAb *21

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Advantages of the RCSS model Allows both smooth and sudden changes

in trend.

is a measure of the persistence of the linear trend. The mean run length is thus

and

RCSS prediction intervals are much wider than those of constant coefficient models.

)1/( )1( tAP

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Conclusions Fitting the damped trend is actually a means of

automatic method selection.

There is an underlying RCSS model for the damped trend and each of its special cases.

SES with damped drift was frequently identified in the M3 series and should receive some consideration in empirical research.

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ReferencesPaper and presentation available at:www.bauer.uh.edu/gardner