Eddie McKenzie

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Damped Trend Forecasting You know it makes
sense!
Eddie McKenzie Statistics Modelling
Science University of Strathclyde Glasgow Scotland
Everette S. Gardner Jr Bauer College of
Business University of Houston Houston, Texas USA
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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
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Linear Trend Smoothing (Holt)
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Linear Trend Smoothing (Holt)
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Past
Present
Future
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Past
Present
Future
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Past
Present
Future
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Exponential Smoothing
Past
Present
Future
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Exponential Smoothing
Past
Present
Future
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Exponential Smoothing
Damped Trend Forecasting
Past
Present
Future
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Strong Linear Trend in Data ? ? usual
Linear Trend forecast Erratic/Weak Linear Trend
? ? Trend levels off to constant No
Linear Trend ? ? Simple Exponential
Smoothing
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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)
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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
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.. leaves unanswered the question How can we
model what is happening in the observed time
series that makes Damped Trend Forecasting a
successful approach?
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• Modelling View
• Amongst models used in forecasting, can we find
  one
• which has intuitive appeal
• and
• for which Trend Damping yields an optimal
  approach?

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SSOE State Space Models Linear Trend model
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SSOE State Space Models Linear Trend model .
Reduced Form is an ARIMA(0,2,2)
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Damped Linear Trend model
Reduced Form ARIMA(1,1,2)
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Strong Linear Trend in Data ? ? usual
Linear Trend forecast Erratic/Weak Linear Trend
? ? Trend levels off to constant No
Linear Trend ? ? Simple Exponential
Smoothing
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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
Have RUNS of a specific slope with each run
ending as the slope revision equation RESTARTS
anew.
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New slope revision equation form
where are i.i.d. Binary r.v.s with
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A Random Coefficient State Space Model for
Linear Trend
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Reduced version is a Random Coefficient
ARIMA(1,1,2)
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with probability
with probability
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Has the same correlation structure as the
standard ARIMA(1,1,2)
and hence same MMSE forecasts
and so Damped Trend Smoothing offers an optimal
approach
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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.
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Parameter Values Method Identified Initial Values
Local Global
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
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Parameter Values Method Identified Initial Values
Local Global
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
Series requiring Damping 84
70
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Parameter Values Method Identified Initial Values
Local Global
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
Series with some kind of Drift or Smoothed Trend
term 98.9 99.4
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Parameter Values Method Identified Initial Values
Local Global
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
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1. SES with Drift
2. SES with Damped Drift
3. Random Walk with Drift Damped Drift
as 1 2 above with
4. Modified Exponential Trend
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1. SES with Drift
Both correspond to random gradient coefficient
models in which the drift term or slope satisfies
2. SES with Damped Drift
.. As before, but with no error. Thus, slope is
subject to changes of constant values at random
times
3. Random Walk with Drift Damped Drift
as 1 2 above with
4. Modified Exponential Trend
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Additive Seasonality (period n)
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with probability
with probability
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State Space Models Non-constant variance models
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Random Coefficient version
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with probability
where
with probability
where