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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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

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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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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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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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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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me Series - ARIMA Models APS 425 - Advanced Managerial

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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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Links

Return to APS 425 Home Page:

http://schwert.simon.rochester.edu/A425/A425main.htm