1. Teknik Peramalan: Materi minggu kedelapan
 Model ARIMA Box-Jenkins
 Identification of STATIONER TIME SERIES  Estimation of ARIMA model  Diagnostic Check of ARIMA model  Forecasting
 Studi Kasus : Model ARIMAX (Analisis Intervensi, Fungsi Transfer dan Neural Networks)

2. General Theoretical ACF and PACF of ARIMA Models
Model ACF PACF
MA(q): moving average of order q Cuts off Dies down after lag q
AR(p): autoregressive of order p Dies down Cuts off after lag p
ARMA(p,q): mixed autoregressive- Dies down Dies down moving average of order (p,q)
AR(p) or MA(q) Cuts off Cuts off after lag q after lag p
No order AR or MA No spike No spike (White Noise or Random process)

3. Theoretically of ACF and PACF of The First-order Moving Average Model or MA(1)
The model Zt =  + at – 1 at , where  = 
 Invertibility condition : –1 < 1 < 1
Theoretically of ACF
Theoretically of PACF

4. Theoretically of ACF and PACF of The First-order Moving Average Model or MA(1) … [Graphics illustration]
ACF
PACF
ACF
PACF

5. Simulation example of ACF and PACF of The First-order Moving Average Model or MA(1) … [Graphics illustration]

6. Theoretically of ACF and PACF of The Second-order Moving Average Model or MA(2)
The model Zt =  + at – 1 at-1 – 2 at , where  = 
 Invertibility condition : 1 + 2 < 1 ; 2  1 < 1 ; |2| < 1
Theoretically of ACF
Theoretically of PACF
Dies Down (according to a mixture of damped exponentials and/or damped sine waves)

7. Theoretically of ACF and PACF of The Second-order Moving Average Model or MA(2) … [Graphics illustration] … (1)
ACF
PACF
ACF
PACF

8. Theoretically of ACF and PACF of The Second-order Moving Average Model or MA(2) … [Graphics illustration] … (2)
ACF
PACF
ACF
PACF

9. Simulation example of ACF and PACF of The Second-order Moving Average Model or MA(2) … [Graphics illustration]

10. Theoretically of ACF and PACF of The First-order Autoregressive Model or AR(1)
The model Zt =  + 1 Zt-1 + at , where  =  (1-1)
 Stationarity condition : –1 < 1 < 1
Theoretically of ACF
Theoretically of PACF

11. Theoretically of ACF and PACF of The First-order Autoregressive Model or AR(1) … [Graphics illustration]
ACF
PACF
ACF
PACF

12. Simulation example of ACF and PACF of The First-order Autoregressive Model or AR(1) … [Graphics illustration]

13. Theoretically of ACF and PACF of The Second-order Autoregressive Model or AR(2)
The model Zt =  + 1 Zt-1 + 2 Zt-2 + at, where  = (112)
 Stationarity condition : 1 + 2 < 1 ; 2  1 < 1 ; |2| < 1
Theoretically of ACF
Theoretically of PACF

14. Theoretically of ACF and PACF of The Second-order Autoregressive Model or AR(2) … [Graphics illustration] … (1)
ACF
PACF
ACF
PACF

15. Theoretically of ACF and PACF of The Second-order Autoregressive Model or AR(2) … [Graphics illustration] … (2)
ACF
PACF
ACF
PACF

16. Simulation example of ACF and PACF of The Second-order Autoregressive Model or AR(2) … [Graphics illustration]

17. Dies Down (in fashion dominated by damped exponentials decay)
Theoretically of ACF and PACF of The Mixed Autoregressive-Moving Average Model or ARMA(1,1)
The model Zt =  + 1 Zt-1 + at  1 at-1 , where  =  (11)
 Stationarity and Invertibility condition : |1| < 1 and |1| < 1
Theoretically of ACF
Theoretically of PACF
Dies Down (in fashion dominated by damped exponentials decay)

18. Theoretically of ACF and PACF of The Mixed Autoregressive-Moving Average Model or ARMA(1,1) … [Graphics illustration] … (1)
ACF
PACF
ACF
PACF

19. Theoretically of ACF and PACF of The Mixed Autoregressive-Moving Average Model or ARMA(1,1) … [Graphics illustration] … (2)
ACF
PACF
ACF
PACF

20. Theoretically of ACF and PACF of The Mixed Autoregressive-Moving Average Model or ARMA(1,1) … [Graphics illustration] … (3)
ACF
PACF
ACF
PACF

21. Simulation example of ACF and PACF of The Mixed Autoregressive-Moving Average Model or ARMA(1,1) … [Graphics illustration]