Course 9 Arima Box-Jenskin - WordPress.com

ARIMA Box-Jenkins

Model ARIMA Box-Jenkins

Identification of STATIONER TIME SERIES Identification of STATIONER TIME SERIES Estimation of ARIMA model Diagnostic Check of ARIMA model Forecasting

Prepocessing

Identifikasi modelIdentifikasi model

Estimasi model

Diagnostic check

Peramalan

Ya

tidak

Model ACF PACF

MA(q): moving average of order q Cuts off Dies downafter lag q

AR(p): autoregressive of order p Dies down Cuts offafter lag p

General Theoretical ACF and PACF of ARIMA Models

after lag p

ARMA(p,q): mixed autoregressive- Dies down Dies downmoving average of order (p,q)

AR(p) or MA(q) Cuts off Cuts offafter 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 MovingAverage Model or MA(1)

The modelZt = + at – 1 at-1 , where =

Invertibility condition : –1 < 1 < 1

Theoretically of ACF Theoretically of PACFTheoretically of ACF Theoretically of PACF

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

PACFACF

ACF PACF

PACF

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

Theoretically of ACF and PACF of The Second-orderMoving Average Model or MA(2)

The modelZt = + at – 1 at-1 – 2 at-2 , where =

Invertibility condition : 1 + 2 < 1 ; 2 1 < 1 ; |2| < 1

Theoretically of ACF Theoretically of PACFTheoretically of ACF Theoretically of PACF

Dies Down (according to amixture of damped exponentials

and/or damped sine waves)

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

PACFACF PACF

ACF PACF

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

PACFACF 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-orderAutoregressive Model or AR(1)

The modelZt = + 1 Zt-1 + at , where = (1-1)

Stationarity condition : –1 < 1 < 1

Theoretically of ACF Theoretically of PACFTheoretically of ACF Theoretically of PACF

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

PACFACF PACF

ACF PACF

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

Theoretically of ACF and PACF of The Second-orderAutoregressive Model or AR(2)

The modelZt = + 1 Zt-1 + 2 Zt-2 + at, where = (112)

Stationarity condition : 1 + 2 < 1 ; 2 1 < 1 ; |2| < 1

Theoretically of ACF Theoretically of PACFTheoretically of ACF Theoretically of PACF

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

PACFACF PACF

ACF PACF

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

PACFACF PACF

ACF PACF

Simulation example of ACF and PACF of The Second-order

Autoregressive Model or AR(2) … [Graphics illustration]

Theoretically of ACF and PACF of The Mixed Autoregressive-Moving Average Model or ARMA(1,1)

The modelZt = + 1 Zt-1 + at 1 at-1 , where = (11)

Stationarity and Invertibility condition : |1| < 1 and |1| < 1

Theoretically of ACF Theoretically of PACFTheoretically of ACF Theoretically of PACF

Dies Down (in fashiondominated by dampedexponentials decay)

Theoretically of ACF and PACF of The Mixed Autoregressive-

Moving Average Model or ARMA(1,1) … [Graphics illustration] … (1)

ACF PACFACF PACF

ACF PACF

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

PACFACF PACF

ACF PACF

Theoretically of ACF and PACF of The Mixed Autoregressive-

Moving Average Model or ARMA(1,1) … [Graphics illustration] … (3)

PACFACF PACF

ACF PACF

Simulation example of ACF and PACF of The Mixed Autoregressive-

Moving Average Model or ARMA(1,1) … [Graphics illustration]