SARIMA Model Simulation

ARIMA(0,1,3)(3,1,3)^12 Model

\[ (1 - B)(1 - B^{12})y_t = \theta_3(B)\Theta_3(B^{12})a_t \] \[ (1 - B)(1 - B^{12})y_t = (1 + \theta_1 B + \theta_2 B^2 + \theta_3 B^3)(1 + \Theta_1 B^{12} + \Theta_2 B^{24} + \Theta_3 B^{36})a_t \] \[ y_t - y_{t-1} - y_{t-12} + y_{t-13} = a_t + \theta_1 a_{t-1} + \theta_2 a_{t-2} + \theta_3 a_{t-3} + \Theta_1 a_{t-12} + \Theta_2 a_{t-24} + \Theta_3 a_{t-36} \] \[ + \theta_1 \Theta_1 a_{t-13} + \theta_2 \Theta_1 a_{t-14} + \theta_3 \Theta_1 a_{t-15} \\ + \theta_1 \Theta_2 a_{t-25} + \theta_2 \Theta_2 a_{t-26} + \theta_3 \Theta_2 a_{t-27} \\ + \theta_1 \Theta_3 a_{t-37} + \theta_2 \Theta_3 a_{t-38} + \theta_3 \Theta_3 a_{t-39} \]

# Simulasi proses SARIMA(0,1,3)(3,1,3)[12]
set.seed(123)
at <- rnorm(1039, mean = 0, sd = 2)

theta <- c(0.5, -0.3, 0.2)  # Koefisien MA(3)
Theta <- c(0.4, -0.25, 0.15)  # Koefisien SMA(3)

n <- length(at)
y <- numeric(n)

# Inisialisasi nilai awal
y[1:39] <- 0

for (t in 40:n) {
  y[t] <- y[t-1] + y[t-12] - y[t-13] +
    at[t] +
    theta[1]*at[t-1] + theta[2]*at[t-2] + theta[3]*at[t-3] +
    Theta[1]*at[t-12] + Theta[2]*at[t-24] + Theta[3]*at[t-36] +
    theta[1]*Theta[1]*at[t-13] + theta[2]*Theta[1]*at[t-14] + theta[3]*Theta[1]*at[t-15] +
    theta[1]*Theta[2]*at[t-25] + theta[2]*Theta[2]*at[t-26] + theta[3]*Theta[2]*at[t-27] +
    theta[1]*Theta[3]*at[t-37] + theta[2]*Theta[3]*at[t-38] + theta[3]*Theta[3]*at[t-39]
}

y <- y[-(1:100)]