eddie mckenzie statistics & modelling science university of strathclyde glasgow scotland

Eddie McKenzieStatistics & Modelling Science

University of Strathclyde

Glasgow

Scotland

Everette S. Gardner JrBauer College of Business

University of Houston

Houston, Texas

USA

Damped Trend Forecasting:

You know it makes sense!

A trend is a trend is a trend, But the question is, will it bend?

Will it alter its course Through some unforeseen force And come to a premature end?

Sir Alec Cairncross, in Economic Forecasting, 1969

11

11

)1()())(1(

)(ˆ

tttt

tttt

ttt

bmmbbmXm

hbmhX

Linear Trend Smoothing (Holt)

)1(ˆ

)(ˆ

1

1

11

ttt

ttt

tttt

ttt

XXe

ebbebmm

hbmhX

Linear Trend Smoothing (Holt)

)1(ˆ

)(ˆ

1

1

11

ttt

ttt

tttt

ttt

XXe

ebbebmm

hbmhX

Past Present Future

Past Present Future

Past Present Future

Past Present Future

Exponential Smoothing

Past Present Future

Exponential Smoothing

Past Present Future

Exponential Smoothing

Damped Trend Forecasting

)1(ˆ

)(ˆ

1

1

11

1

ttt

ttt

tttt

h

k

kttt

XXe

ebbebmm

bmhX

10

Strong Linear Trend in Data usual Linear Trend forecast

Erratic/Weak Linear Trend Trend levels off to constant

No Linear Trend Simple Exponential Smoothing

1

10

0

Demonstrated (1985-89) on a large database of time series that using the method on all non-seasonal series gave more accurate forecasts at longer horizons, but lost little, if any accuracy, even at short ones.

Damping trend may seem – perhaps sensibly conservative – but arbitrary.

However, works extremely well in practice…. …. two academic reviewer comments from large empirical studies…

“… it is difficult to beat the damped trend when a single forecasting method is applied to a collection of time series.” (2001)

Damped Trend can “reasonably claim to be a benchmark forecasting method for all others to beat.” (2008)

Reason for Empirical Success?

Pragmatic View

Projecting a Linear Trend indefinitely into the future is simply far too optimistic (pessimistic) in practice.

Damped Trend is more conservative for longer-term, more reasonable, and so more successful, but ……

…….. leaves unanswered the question:

How can we model what is happening in the observed time series that makes Damped Trend Forecasting a successful approach?

Modelling View:

Amongst models used in forecasting, can we find one which has intuitive appeal

and for which Trend –Damping yields an optimal approach?

SSOE State Space Models

Linear Trend model:

ttt

tttt

tttt

vbbvbmm

vbmX

)1()1(

1

11

11

SSOE State Space Models

Linear Trend model ….

Reduced Form is an ARIMA(0,2,2)

212 )()1( tttt vvvXB

Damped Linear Trend model:

ttt

tttt

tttt

vbbvbmm

vbmX

)1()1(

1

11

11

Reduced Form: ARIMA(1,1,2)

21)()1)(1( tttt vvvXBB

Strong Linear Trend in Data usual Linear Trend forecast

Erratic/Weak Linear Trend Trend levels off to constant

No Linear Trend Simple Exponential Smoothing

1

10

0

Our Approach: use as a measure of the persistence of the linear trend, i.e. how long any particular linear trend persists, before changing slope ……

ttt vbb )1(1

Have RUNS of a specific slope with each run ending as the slope revision equation RESTARTS anew.

tttt vbAb )1(1

where are i.i.d. Binary r.v.s with

)0(1)1( tt APAP

tA

New slope revision equation form

tttt

ttttt

ttttt

vbAbvbAmm

vbAmX

)1()1(

1

11

11

A Random Coefficient

State Space Model

for Linear Trend

21)( )1)(1(

ttttt

tt

vAvAvXBBA

Reduced version is a

Random Coefficient ARIMA(1,1,2)

212 )( tttt vvvX

1 ttt vvX

with probability :

with probability :)1(

Has the same correlation structure as the standard ARIMA(1,1,2)

2211)1)(1( tttt aaaXBB

…and hence same MMSE forecasts

… and so Damped Trend Smoothing offers an optimal approach

Optimal for a wider class of models than originally realized, including ones allowing gradient to change not only smoothly but also suddenly. Argue that this is more likely in practice than smooth change, and so Damped Trend Smoothing should be a first approach. (rather than just a reasonable approximation)

Another – but clearly related – possibility is that the approach can yield forecasts which are optimal for so many different processes that every possibility is covered.

To explore both ideas, used the method on the M3 Competition database of 3003 time series, and noted which implied models were identified.

ParameterParameter ValuesValues Method IdentifiedMethod Identified InitialInitial ValuesValues

LocalLocal GlobalGlobal

Level Trend Damping %-ages %-ages

1 Damped Trend 43.0 27.8

2 1 Linear Trend 10.0 1.8

3 0 SES with Damped Drift 24.8 23.5

4 0 1 SES with Drift 2.4 11.6

5 0 0 SES 0.8 0.6

6 1 0 RW with Damped Drift 7.8 9.6

7 1 0 1 RW with Drift 2.5 8.4

8 1 0 0 RW - Random Walk 0.0 0.0

9 0 0 Modified Expo Trend 8.3 8.7

10 0 0 1 Straight Line 0.1 7.9

11 0 0 0 Simple Average 0.3 0.0

10 10 10 10 10

10 10

10

10

10

10

ParameterParameter ValuesValues Method IdentifiedMethod Identified InitialInitial ValuesValues

LocalLocal GlobalGlobal

Level Trend Damping %-ages %-ages

1 Damped Trend 43.0 27.8

2 1 Linear Trend 10.0 1.8

3 0 SES with Damped Drift 24.8 23.5

4 0 1 SES with Drift 2.4 11.6

5 0 0 SES 0.8 0.6

6 1 0 RW with Damped Drift 7.8 9.6

7 1 0 1 RW with Drift 2.5 8.4

8 1 0 0 RW - Random Walk 0.0 0.0

9 0 0 Modified Expo Trend 8.3 8.7

10 0 0 1 Straight Line 0.1 7.9

11 0 0 0 Simple Average 0.3 0.0

10 10 10 10 10

10 10

10

10

10

10

Series requiring Damping: 84% 70%

ParameterParameter ValuesValues Method IdentifiedMethod Identified InitialInitial ValuesValues

LocalLocal GlobalGlobal

Level Trend Damping %-ages %-ages

1 Damped Trend 43.0 27.8

2 1 Linear Trend 10.0 1.8

3 0 SES with Damped Drift 24.8 23.5

4 0 1 SES with Drift 2.4 11.6

5 0 0 SES 0.8 0.6

6 1 0 RW with Damped Drift 7.8 9.6

7 1 0 1 RW with Drift 2.5 8.4

8 1 0 0 RW - Random Walk 0.0 0.0

9 0 0 Modified Expo Trend 8.3 8.7

10 0 0 1 Straight Line 0.1 7.9

11 0 0 0 Simple Average 0.3 0.0

10 10 10 10 10

10 10

10

10

10

10

Series with some kind of Drift or Smoothed Trend term 98.9% 99.4%

ParameterParameter ValuesValues Method IdentifiedMethod Identified InitialInitial ValuesValues

LocalLocal GlobalGlobal

Level Trend Damping %-ages %-ages

1 Damped Trend 43.0 27.8

2 1 Linear Trend 10.0 1.8

3 0 SES with Damped Drift 24.8 23.5

4 0 1 SES with Drift 2.4 11.6

5 0 0 SES 0.8 0.6

6 1 0 RW with Damped Drift 7.8 9.6

7 1 0 1 RW with Drift 2.5 8.4

8 1 0 0 RW - Random Walk 0.0 0.0

9 0 0 Modified Expo Trend 8.3 8.7

10 0 0 1 Straight Line 0.1 7.9

11 0 0 0 Simple Average 0.3 0.0

10 10 10 10 10

10 10

10

10

10

10

btXbXbX

bXX

t

tt

tttt

2

1

11

2

1. SES with Drift:

2. SES with Damped Drift:

t

ttt

tt

t

ttt

tt

qp

Xbb

XbX

bXX

21

1

11

3. Random Walk with Drift & Damped Drift: 0as 1 & 2 above with

4. Modified Exponential Trend: tt

t bX

btXbXbX

bXX

t

tt

tttt

2

1

11

2

1. SES with Drift:

2. SES with Damped Drift:

t

ttt

tt

t

ttt

tt

qp

Xbb

XbX

bXX

21

1

11

3. Random Walk with Drift & Damped Drift: 0as 1 & 2 above with

Both correspond to random gradient coefficient models in which the drift term or slope satisfies

.. As before, but with no error. Thus, slope is subject to changes of constant values at random times

1 ttt bAb

4. Modified Exponential Trend: tt

t bX

Additive Seasonality (period: n)

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tttt

ttttt

tnttttt

vSSvbAb

vbAmmvSbAmX

)1()1(

1

11

11

tntn vBXBB )()1)(1( 1

with probability :

with probability :)1(

tntn vBXB )()1(

State Space Models:

Non-constant variance models

ttttt

tttt

tttt

vbmbbvbmm

vbmX

))(1())1(1)((

)1)((

111

11

11

ttttttt

ttttt

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vbAmbAb

vbAmm

vbAmX

))(1(

))1(1)((

)1)((

111

11

11

Random Coefficient version:

212 )( ttttX

1 tttX

with probability :

with probability :)1(

tttt vbm )( 11

ttt vm 1

where

where