First try: AR(1) Use MINITAB’s ARIMA-procedure.

First try: AR(1)

ttt ayy 1

Use MINITAB’s ARIMA-procedure

ARIMA Model

ARIMA model for CPIChnge

Final Estimates of Parameters

Type Coef StDev T P

AR 1 0,8247 0,1048 7,87 0,000

Constant 0,7634 0,3347 2,28 0,030

Mean 4,354 1,909

Number of observations: 33

Residuals: SS = 111,236 (backforecasts excluded)

MS = 3,588 DF = 31

Modified Box-Pierce (Ljung-Box) Chi-Square statistic

Lag 12 24 36 48

Chi-Square 25,9 32,2 * *

DF 10 22 * *

P-Value 0,004 0,075 * *

Ljung-Box statistic:

where

n is the sample size

d is the degree of nonseasonal differencing used to tranform original series to be stationary. Nonseasonal means taking

differences at lags nearby, which can be written (1–B)d

rl2(â) is the sample autocorrelation at lag l for the residuals of the estimated model.

K is a number of lags covering multiples of seasonal cycles, e.g. 12, 24, 36,… for monthly data

K

ll arldndndnKQ

1

2* )ˆ(2

Under the assumption of no correlation left in the residuals the Ljung-Box statistic is chi-square distributed with K – nC degrees of freedom, where nC is the number of estimated parameters in model except for the constant

A low P-value for any K should be taken as evidence for correlated residuals, and thus the estimated model must be revised.

ARIMA Model



Type Coef StDev T P

AR 1 0,8247 0,1048 7,87 0,000

Constant 0,7634 0,3347 2,28 0,030

Mean 4,354 1,909



MS = 3,588 DF = 31


Lag 12 24 36 48

Chi-Square 25,9 32,2 * *

DF 10 22 * *

P-Value 0,004 0,075 * *

Low P-value for K=12.

Problems with residuals at nonseasonal level

Study SAC and SPAC for the original series:

PACF look not fully consistent with AR(1)

More than one significant spike (2 it seems)

If an AR(p)-model is correct, the ACF should decrease exponentially (montone or oscillating)

and PACF should have exactly p significant spikes

Try an AR(2)

ARIMA Model



Type Coef StDev T P

AR 1 1,1872 0,1625 7,31 0,000

AR 2 -0,4657 0,1624 -2,87 0,007

Constant 1,3270 0,2996 4,43 0,000

Mean 4,765 1,076



MS = 2,9540 DF = 30


Lag 12 24 36 48

Chi-Square 19,8 25,4 * *

DF 9 21 * *

P-Value 0,019 0,231 * *

OK!

Still not OK

Might still be problematic.

Could it be the case of an Moving Average (MA) model?

MA(1):

1 ttt aay

ta are still assumed to be uncorrelated and identically distributed with mean zero and constant variance

MA(q):

qtqttt aaay 11

• always stationary

• mean=

• is in effect a moving average with weights

q ,,,1 ,21

for the (unobserved) values qttt aaa ,,, 1

Index

AR(1

)_0.2

200180160140120100806040201

5

4

3

2

1

0

Time Series Plot of AR(1)_0.2

Index

AR(1

)_0.8

200180160140120100806040201

14

13

12

11

10

9

8

7

6

5

Time Series Plot of AR(1)_0.8

Index

MA

(1)_

0.2

3002702402101801501209060301

3

2

1

0

-1

-2

-3

Time Series Plot of MA(1)_0.2

Index

MA

(1)_

0.8

3002702402101801501209060301

4

3

2

1

0

-1

-2

-3

-4

Time Series Plot of MA(1)_0.8

Index

MA

(1)_

(-0.5

)

3002702402101801501209060301

4

3

2

1

0

-1

-2

-3

Time Series Plot of MA(1)_ (-0.5)

Index

AR(1

)_(-

0.5

)

200180160140120100806040201

5

4

3

2

1

0

-1

-2

-3

Time Series Plot of AR(1)_ (-0.5)

Try an MA(1):

ARIMA Model



Type Coef StDev T P

MA 1 -0,9649 0,1044 -9,24 0,000

Constant 4,8018 0,5940 8,08 0,000

Mean 4,8018 0,5940



MS = 3,361 DF = 31


Lag 12 24 36 48

Chi-Square 33,8 67,6 * *

DF 10 22 * *

P-Value 0,000 0,000 * *

Still seems to be problems with residuals

Look again at ACF and PACF of original series:

The pattern corresponds neither with AR(p), nor with MA(q)

Could it be a combination of these two?

Auto Regressive Moving Average (ARMA) model

ARMA(p,q):

qtqttptptt aaayyy 1111

• stationarity conditions harder to define

• mean value calculations more difficult

• identification patterns exist, but might be complex: exponentially decreasing patterns or sinusoidal decreasing patterns in both ACF and PACF (no cutting of at a certain lag)

Index

ARM

A(1

,1)_

(0.2

)(0.2

)

3002702402101801501209060301

3

2

1

0

-1

-2

-3

Time Series Plot of ARMA(1,1)_ (0.2)(0.2)

Index

ARM

A(1

,1)_

(-0.2

)(-0

.2)

3002702402101801501209060301

3

2

1

0

-1

-2

-3

Time Series Plot of ARMA(1,1)_ (-0.2)(-0.2)

Index

ARM

A(2

,1)_

(0.1

)(0.1

)_(-

0.1

)

3002702402101801501209060301

3

2

1

0

-1

-2

-3

-4

Time Series Plot of ARMA(2,1)_ (0.1)(0.1)_ (-0.1)

Always try to keep p and q small.

Try an ARMA(1,1):

ARIMA Model


Unable to reduce sum of squares any further


Type Coef StDev T P

AR 1 0,6513 0,1434 4,54 0,000

MA 1 -0,9857 0,0516 -19,11 0,000

Constant 1,5385 0,4894 3,14 0,004

Mean 4,412 1,403



MS = 2,0613 DF = 30


Lag 12 24 36 48

Chi-Square 9,6 17,0 * *

DF 9 21 * *

P-Value 0,386 0,713 * *

Calculating forecasts

For AR(p) models quite simple:

1)1(211

)2(2)1(1

)2(2112

)1(1211

ˆˆˆˆˆˆˆˆ

ˆˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆ

tpptptpt

tpptptpt

ptpttt

ptpttt

yyyy

yyyy

yyyy

yyyy

kta is set to 0 for all values of k

For MA(q) ??

MA(1):

1ˆˆˆ ttt aay

If we e.g. would set and equal to 0

the forecast would constantly be .

which is not desirable.

ta 1ta

Note that

ˆ)ˆ1(ˆˆ

)1(

0

1

1

2

1

211

ttt

ttt

t

ttt

ttt

yya

yya

a

aay

aay

Similar investigations for ARMA-models.

First try: AR(1) Use MINITAB’s ARIMA-procedure.

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Transcript of First try: AR(1) Use MINITAB’s ARIMA-procedure.