ARIMA Model Simulation

ARIMA(0,0,0)(0,0,4)^12 Model

\[ y_t = a_t + \Theta_1 a_{t-12} + \Theta_2 a_{t-24} + \Theta_3 a_{t-36} + \Theta_4 a_{t-48} \]

set.seed(1234)
n <- 500
at <- rnorm(n + 48, mean = 0, sd = 2)

# Parameter seasonal MA
theta1 <- 0.5
theta2 <- 0.4
theta3 <- 0.3
theta4 <- 0.2

y <- numeric(n + 48)

y[1] <- 0
y[2] <- 0
y[3] <- 0

for (t in 49:(n + 48)) {
  y[t] <- at[t] + 
          theta1 * at[t - 12] +
          theta2 * at[t - 24] +
          theta3 * at[t - 36] +
          theta4 * at[t - 48]
}

y <- y[49:(n + 48)]

ts.plot(y, main = "Simulasi ARIMA(0,0,0)(0,0,4)[12]", ylab = "y", xlab = "Time")

ts.plot(y[1:100], main = "Simulasi ARIMA(0,0,0)(0,0,4)[12] (n = 1–100)", ylab = "y", xlab = "Time")

par(mfrow = c(1, 2))
acf(y, lag.max = 80, main = "ACF - Original Series")
pacf(y, lag.max = 80, main = "PACF - Original Series")