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![Page 1: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/1.jpg)
Eddie McKenzie
Statistics & Modelling Science
University of Strathclyde
Glasgow
Scotland
Everette S. Gardner Jr
Bauer College of Business
University of Houston
Houston, Texas
USA
Damped Trend Forecasting:
You know it makes sense!
![Page 2: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/2.jpg)
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
![Page 3: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/3.jpg)
11
11
)1()(
))(1(
)(ˆ
tttt
tttt
ttt
bmmb
bmXm
hbmhX
Linear Trend Smoothing (Holt)
![Page 4: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/4.jpg)
)1(ˆ
)(ˆ
1
1
11
ttt
ttt
tttt
ttt
XXe
ebb
ebmm
hbmhX
Linear Trend Smoothing (Holt)
![Page 5: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/5.jpg)
)1(ˆ
)(ˆ
1
1
11
ttt
ttt
tttt
ttt
XXe
ebb
ebmm
hbmhX
![Page 6: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/6.jpg)
![Page 7: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/7.jpg)
Past Present Future
![Page 8: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/8.jpg)
Past Present Future
![Page 9: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/9.jpg)
Past Present Future
![Page 10: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/10.jpg)
Past Present Future
Exponential Smoothing
![Page 11: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/11.jpg)
Past Present Future
Exponential Smoothing
![Page 12: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/12.jpg)
Past Present Future
Exponential Smoothing
Damped Trend Forecasting
![Page 13: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/13.jpg)
)1(ˆ
)(ˆ
1
1
11
1
ttt
ttt
tttt
h
k
kttt
XXe
ebb
ebmm
bmhX
10
![Page 14: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/14.jpg)
![Page 15: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/15.jpg)
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
![Page 16: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/16.jpg)
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)
![Page 17: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/17.jpg)
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 ……
![Page 18: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/18.jpg)
…….. leaves unanswered the question:
How can we model what is happening in the observed time series that makes Damped Trend Forecasting a successful approach?
![Page 19: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/19.jpg)
Modelling View:
Amongst models used in forecasting, can we find one
which has intuitive appeal
and
for which Trend –Damping yields an optimal approach?
![Page 20: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/20.jpg)
SSOE State Space Models
Linear Trend model:
ttt
tttt
tttt
vbb
vbmm
vbmX
)1(
)1(
1
11
11
![Page 21: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/21.jpg)
SSOE State Space Models
Linear Trend model ….
Reduced Form is an ARIMA(0,2,2)
212 )()1( tttt vvvXB
![Page 22: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/22.jpg)
Damped Linear Trend model:
ttt
tttt
tttt
vbb
vbmm
vbmX
)1(
)1(
1
11
11
Reduced Form: ARIMA(1,1,2)
21)()1)(1( tttt vvvXBB
![Page 23: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/23.jpg)
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
![Page 24: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/24.jpg)
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.
![Page 25: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/25.jpg)
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
![Page 26: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/26.jpg)
tttt
ttttt
ttttt
vbAb
vbAmm
vbAmX
)1(
)1(
1
11
11
A Random Coefficient
State Space Model
for Linear Trend
![Page 27: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/27.jpg)
21)(
)1)(1(
ttttt
tt
vAvAv
XBBA
Reduced version is a
Random Coefficient ARIMA(1,1,2)
![Page 28: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/28.jpg)
212 )( tttt vvvX
1 ttt vvX
with probability :
with probability :)1(
![Page 29: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/29.jpg)
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
![Page 30: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/30.jpg)
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.
![Page 31: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/31.jpg)
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
![Page 32: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/32.jpg)
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%
![Page 33: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/33.jpg)
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%
![Page 34: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/34.jpg)
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
![Page 35: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/35.jpg)
bt
Xb
XbX
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
![Page 36: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/36.jpg)
bt
Xb
XbX
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
![Page 37: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/37.jpg)
Additive Seasonality (period: n)
tntt
tttt
ttttt
tnttttt
vSS
vbAb
vbAmm
vSbAmX
)1(
)1(
1
11
11
![Page 38: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/38.jpg)
tntn vBXBB )()1)(1( 1
with probability :
with probability :)1(
tntn vBXB )()1(
![Page 39: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/39.jpg)
State Space Models:
Non-constant variance models
ttttt
tttt
tttt
vbmbb
vbmm
vbmX
))(1(
))1(1)((
)1)((
111
11
11
![Page 40: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/40.jpg)
ttttttt
ttttt
ttttt
vbAmbAb
vbAmm
vbAmX
))(1(
))1(1)((
)1)((
111
11
11
Random Coefficient version:
![Page 41: Eddie McKenzie Statistics & Modelling Science University of Strathclyde Glasgow Scotland Everette S. Gardner Jr Bauer College of Business University of.](https://reader036.fdocuments.in/reader036/viewer/2022062716/56649dd35503460f94aca98e/html5/thumbnails/41.jpg)
212 )( ttttX
1 tttX
with probability :
with probability :)1(
tttt vbm )( 11
ttt vm 1
where
where