General Theoretical ACF and PACF of ARIMA Models

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

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 Moving Average Model or MA(1)

The modelZ = + a – a , where = 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 modelZ = + a – a – a , where = 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 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 modelZ = + Z + a , where = (1- )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 modelZ = + Z + Z + a , where = (1 )Zt = + 1 Zt-1 + 2 Zt-2 + at, where = (112)

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]

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

The modelZ = + Z + a a , where = (1 )Zt = + 1 Zt-1 + at 1 at-1 , where = (11)

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

Theoretically of ACF Theoretically of PACF

Dies Down (in fashion 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]