time series and arima models
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
APS 425 Winter 2008
me er es na ys s:
ARIMA Models
.585-275-2470
schwert@schwert.simon.rochester.edu
Topics
Typical time series plot
Pattern recognition in auto and partial
autocorrelations
Stationarity & invertibility
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
AR(1): Zt = + 1 Zt-1 + at1 = .9Note: exponential decay
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Lag k
Auto Partial
AR(1): Zt = + 1 Zt-1 + at1 = .5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
AR(1): Zt = + 1 Zt-1 + at1 = -.5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
AR(1): Zt = + 1 Zt-1 + at1 = -.9
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
Note: oscillating autocorrelations
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
4
5
AR(1): Zt = + 1 Zt-1 + at1 = .9
-2
-1
0
1
2
3
-5
-4
-3
0 10 20 30 40 50 60 70 80 90 100
Note: smooth long swings away from mean
4
5
AR(1): Zt = + 1 Zt-1 + at1 = -.9
-1
0
1
2
3
-3
-2
0 10 20 30 40 50 60 70 80 90 100
Note: jagged, frequent swings around mean
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
AR(2): Zt = + 1 Zt-1 + 2 Zt-2 + at1 = 1.4 2 = .45
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
AR(2): Zt = + 1 Zt-1 + 2 Zt-2 + at1 = .4 2 = .45
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
AR(2): Zt = + 1 Zt-1 + 2 Zt-2 + at1 = .7 2 = -.2
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
AR(2): Zt = + 1 Zt-1 + 2 Zt-2 + at1 = -.7 2 = .2
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
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8/8/2019 Time Series and Arima Models
7/20
me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
4
6
AR(2): Zt = + 1 Zt-1 + 2 Zt-2 + at1 = 1.4 2 = .45
-6
-4
-2
0
2
-10
-8
0 10 20 30 40 50 60 70 80 90 100
Note: smooth long swings away from mean
4
5
AR(2): Zt = + 1 Zt-1 + 2 Zt-2 + at1 = -.7 2 = .2Note: jagged, frequent swings around mean
0
1
2
3
-2
-1
0 10 20 30 40 50 60 70 80 90 100
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8/8/2019 Time Series and Arima Models
8/20
me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
Autoregressive Models:Summary
u ocorre a ons ecay or osc a e
2) Partial Autocorrelations cut-off after lag p, forAR(p) model
3) Stationarity is a big issue very slow decay in autocorrelations should you difference?
MA(1): Zt = + at - 1 at-11 = -.9
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
Note: autocorrelations cut off,pacf oscillates
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8/8/2019 Time Series and Arima Models
9/20
me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
MA(1): Zt = + at - 1 at-11 = -.5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
MA(1): Zt = + at - 1 at-11 = .5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
-
8/8/2019 Time Series and Arima Models
10/20
me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
MA(1): Zt = + at - 1 at-11 = .9
-0.4
-0.2
0
0.2
0.4
0.6
0.8
utocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
4
5
MA(1): Zt = + at - 1 at-11 = -.9
-1
0
1
2
3
-3
-2
0 10 20 30 40 50 60 70 80 90 100
Note: smooth long swings away from mean
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8/8/2019 Time Series and Arima Models
11/20
me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
4
5
MA(1): Zt = + at - 1 at-11 = .9
-1
0
1
2
3
-3
-2
0 10 20 30 40 50 60 70 80 90 100
Note: jagged, frequent swings around the mean
Moving Average Models: Summary
1) Autocorrelations cut off after lag q for MA(q)model
2) Partial autocorrelations decay or oscillate
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
Autoregressive Moving Average Models
ARMA(p,q): Zt = + 1 Zt-1 + . . . + p Zt-p +
at 1 at-1 . . . q at-q
Combines both AR & MA characteristics:
e uivalent to infinite order MA rocess
if stationary
equivalent to infinite order AR process
if invertible
1 Note: exponential decay, starting after lag 1
ARMA(1,1): Zt = + 1 Zt-1 + at - 1 at-11 = .9, 1 = .5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Autocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
1
ARMA(1,1): Zt = + 1 Zt-1 + at - 1 at-11 = .5, 1 = .9
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Autocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
1
ARMA(1,1): Zt = + 1 Zt-1 + at - 1 at-11 = -.5, 1 = .5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Autocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
1
ARMA(1,1): Zt = + 1 Zt-1 + at - 1 at-11 = -.9, 1 = .9
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Autocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
4
6
ARMA(1,1): Zt = + 1 Zt-1 + at - 1 at-11 = .9, 1 = .5
-4
-2
0
2
-8
-6
0 10 20 30 40 50 60 70 80 90 100
Note: smooth long swings around the mean
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
2.5
3
ARMA(1,1): Zt = + 1 Zt-1 + at - 1 at-11 = -.9, 1 = .9
0
0.5
1
1.5
2
-1
-0.5
0 10 20 30 40 50 60 70 80 90 100
Note: jagged, frequent swings around the mean
Autoregressive Moving AverageModels: Summary
1) Autocorrelations decay or oscillate
2) Partial Autocorrelations decay or oscillate
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
Autoregressive Integrated MovingAverage ARIMA(p,d,q) Models
1) ARMA model in the dth differences of the data
2) First step is to find the level of differencing
necessary
3) Next steps are to find the appropriate ARMA model
for the differenced data
4) Need to avoid overdifferencing
1
ARIMA(0,1,1):(Zt - Zt-1) = at - .8 at-1 [T=150]
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Autocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
Note: autocorrelations decay very slowly(from moderate level); pacf decays at rate .8
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
1
ARIMA(0,1,1):(Zt - Zt-1) = at - .5 at-1 [T=150]
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Autocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
Note: autocorrelations decay very slowly(from moderate level); pacf decays at rate .5
1
ARIMA(0,1,1):(Zt - Zt-1) = at + .5 at-1 [T=150]
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Autocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
Note: autocorrelations decay very slowly(from moderate level); pacf decays at rate -.5
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8/8/2019 Time Series and Arima Models
18/20
me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
1
ARIMA(0,1,1):(Zt - Zt-1) = at + .8 at-1 [T=150]
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Autocorrelation
-1
-0.8
-0.6
Lag k
Auto Partial
Note: autocorrelations decay very slowly
(from moderate level); pacf decays at rate -.8
1
2
ARIMA(0,1,1)(Zt - Zt-1) = at - .8 at-1
-3
-2
-1
0
-5
-4
0 10 20 30 40 50 60 70 80 90 100
Note: slowly wandering level of series, with lotsof variation around that level
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Prof. G. William Schwert, 2002-2008
20
ARIMA(0,1,1)(Zt - Zt-1) = at + .8 at-1
0
5
10
-10
-5
0 10 20 30 40 50 60 70 80 90 100
Note: slowly wandering level of series, withsmooth variation around that level
Integrated Moving Average Models:Summary
initial level is determined by how close MA parameter
is to one
2) Partial Autocorrelations decay or oscillate determined by MA parameter
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8/8/2019 Time Series and Arima Models
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me Series - ARIMA Models APS 425 - Advanced Managerial
An
Links
Return to APS 425 Home Page:
http://schwert.simon.rochester.edu/A425/A425main.htm
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