Transfer Function Equation

Persamaan Model

y_t=1.3177+\left(\frac{-0.4379 B+0.5978 B^2}{1-1.2071 B+0.7067 B^2}\right) x_{t-2}+\frac{1}{1-0.4706 B+0.3266 B^2} a_t

Misalkan

• \mu = 1.3177
• \omega_1 = -0.4379
• \omega_2 = 0.5978
• \delta_1 = 1.2071
• \delta_2 = -0.7067
• \phi_1 = 0.4706
• \phi_2 = -0.3266

mu      =  1.3177
omega_1 = -0.4379
omega_2 =  0.5978
delta_1 =  1.2071
delta_2 = -0.7067
phi_1   =  0.4706
phi_2   = -0.3266

Maka, persamaan y_t menjadi

y_t= \mu + \left(\frac{\omega_1 B+ \omega_2 B^2}{1-\delta_1 B - \delta_2 B^2}\right) x_{t-2}+\frac{1}{1- \phi_1 B - \phi_2 B^2} a_t

Misalkan \dot{y}_t = y_t - \mu, maka diperoleh

\dot{y}_t = \left(\frac{\omega_1 B+ \omega_2 B^2}{1-\delta_1 B - \delta_2 B^2}\right) x_{t-2}+\frac{1}{1- \phi_1 B - \phi_2 B^2} a_t

Samakan penyebut di ruas kanan, diperoleh

\dot{y}_t = \left( \frac{(1- \phi_1 B - \phi_2 B^2) (\omega_1 B+ \omega_2 B^2) x_{t-2} + (1-\delta_1 B - \delta_2 B^2) a_t }{(1-\delta_1 B - \delta_2 B^2) (1- \phi_1 B - \phi_2 B^2)} \right)

Penyebut di ruas kanan dipindahkan ke ruas kiri, diperoleh

\dot{y}_t (1-\delta_1 B - \delta_2 B^2) (1- \phi_1 B - \phi_2 B^2) = (1- \phi_1 B - \phi_2 B^2) (\omega_1 B+ \omega_2 B^2) x_{t-2} + (1-\delta_1 B - \delta_2 B^2) a_t

1 ) Komponen ruas kiri:

\dot{y}_t \left[ 1 - (\delta_1 + \phi_1)B + (\delta_1 \phi_1 - \delta_2 - \phi_2)B^2 + (\delta_1 \phi_2 + \delta_2 \phi_1)B^3 + \delta_2 \phi_2 B^4 \right] Kalikan y_t dengan backshift operator B menjadi

\dot{y}_t - (\delta_1 + \phi_1) \dot{y}_{t-1} + (\delta_1 \phi_1 - \delta_2 - \phi_2) \dot{y}_{t-2} + (\delta_1 \phi_2 + \delta_2 \phi_1) \dot{y}_{t-3} + \delta_2 \phi_2 \dot{y}_{t-4}

Hitung nilai koefisien

-(delta_1 + phi_1)

[1] -1.6777

delta_1 * phi_1 - delta_2 - phi_2

[1] 1.601361

delta_1 * phi_2 + delta_2 * phi_1

[1] -0.7268119

delta_2 * phi_2

[1] 0.2308082

Sehingga menjadi

\dot{y}_t - 1.677 \dot{y}_{t-1} + 1.601 \dot{y}_{t-2} -0.726 \dot{y}_{t-3} + 0.230 \dot{y}_{t-4}

Substitusikan kembali \dot{y}_t = y_t - \mu, sehingga diperoleh

(y_t - \mu) -1.677 (y_{t-1} - \mu) + 1.601 (y_{t-2} - \mu) -0.726 (y_{t-3} - \mu) + 0.230 (y_{t-4} - \mu)

y_t -1.677 y_{t-1} + 1.601 y_{t-2} -0.726 y_{t-3} + 0.230 y_{t-4} - \mu (1-0.568 +1.601 -0.726 +0.230)

mu * (1-1.677 +1.601 -0.726 +0.230)

[1] 0.5639756

y_t -1.677 y_{t-1} + 1.601 y_{t-2} -0.726 y_{t-3} + 0.230 y_{t-4} - 0.563

2 ) Komponen ruas kanan bagian 1:

[\omega_1 B + (\omega_2 - \phi_1 \omega_1)B^2 - (\phi_1 \omega_2 + \phi_2 \omega_1)B^3 - \phi_2 \omega_2 B^4] x_{t-2} Kalikan x_t dengan backshift operator B menjadi

\omega_1 x_{t-3} + (\omega_2 - \phi_1 \omega_1)x_{t-4} - (\phi_1 \omega_2 + \phi_2 \omega_1)x_{t-5} - \phi_2 \omega_2 x_{t-6}

Hitung nilai koefisien

omega_1

[1] -0.4379

omega_2 - phi_1 * omega_1

[1] 0.8038757

-(phi_1 * omega_2 + phi_2 * omega_1)

[1] -0.4243428

-(phi_2 * omega_2)

[1] 0.1952415

Sehingga menjadi

-0.437 x_{t-3} +0.803 x_{t-4} -0.424 x_{t-5} + 0.195 x_{t-6}

3 ) Komponen ruas kanan bagian 2:

(1-\delta_1 B - \delta_2 B^2) a_t

Kalikan a_t dengan backshift operator B menjadi

a_t -\delta_1 a_{t-1} - \delta_2 a_{t-2}

Hitung nilai koefisien

delta_1

[1] 1.2071

delta_2

[1] -0.7067

a_t -1.207 a_{t-1} + 0.706 a_{t-2}

4 ) Persamaan akhir menjadi:

\begin{aligned} y_t= & 0.563 \\ & +1.677 y_{t-1}-1.601 y_{t-2}+0.726 y_{t-3}-0.230 y_{t-4} \\ & -0.437 x_{t-3} +0.803 x_{t-4} -0.424 x_{t-5} + 0.195 x_{t-6} \\ & +a_t-1.207 a_{t-1}+0.706 a_{t-2} \end{aligned}