The General ARIMAp,d,qP,D,QS models

1
The General ARIMA(p,d,q)(P,D,Q)S models
? The Model is
? where
is an appropriate pre-differencing transformation
Do not need pre-differencing transformation
Yt Original Data
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Example ARIMA(0,1,1)(0,0,1)12 model
Yt Original Data
? The Model is
Zt Yt Yt-1
? where
and
Therefore,
Mean (Zt)
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General Theoretical ACF and PACF of ARIMA
Multiplicative Seasonal Models with L (length of
seasonal period).
Seasonal
Non-seasonal
multiplicative
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Example Theoretical ACF and PACF of ARIMA
Multiplicative Seasonal Models with 12 (length of
seasonal period).
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Theoretically of ACF and PACF of The
Multiplicative Moving Average Model or
ARIMA(0,0,1)(0,0,1)12 or MA(1)(1)12
The model
Zt ? at
?1 at-1 ? ?1 at-12 ?1.?1 at-13 , where ?
? ? Stationarity condition ?1 lt 1 and ?1 lt
1
Theoretically of ACF
Theoretically of PACF
Dies Down at the nonseasonal and seasonal level
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Simulation example of ACF and PACF of The
Multiplicative Moving Average Model or MA(1)(1)12
Graphics
illustration
Has spike only at lag 1 (cuts off)
Has spike only at lag 12 (cuts off)
Dies down at non seasonal lags
Dies down at seasonal lags
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Theoretically of ACF and PACF of The
Multiplicative Autore-gressive Model or
ARIMA(1,0,0)(1,0,0)12 or AR(1)(1)12
The model
Zt ?
?1 Zt-1 ?1 Zt-12 ? ?1.?1 Zt-13 at ?
Stationarity condition ?1 lt 1 and ?1 lt 1
Theoretically of ACF
Theoretically of PACF
Cuts off at the lag 1 nonseasonal and lag 12
seasonal level
Dies Down at the nonseasonal and seasonal level
8
Simulation example of ACF and PACF of The
Multiplicative Moving Average Model or AR(1)(1)12
Graphics
illustration
Dies down at non seasonal lags
Dies down at seasonal lags
Has spike only at lag 1 (cuts off)
Has spike only at lag 12 (cuts off)