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)

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)

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

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

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

Theoretically of ACF and PACF of The Second-order Moving Average Model or MA(2) The model Zt =  + at – 1 at-1 – 2 at-2 , 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)

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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