5003231060_SARIMA Model Simulation

SARIMA Model Simulation

ARIMA(0,0,0)(0,1,4)^12 \[(1-B^{12})y_t = \Theta_4(B^{12})a_t\] \[y_t - y_{t-12} = (1 - \Theta_1 B^{12} - \Theta_2 B^{24} - \Theta_3 B^{36} - \Theta_4 B^{48})a_t\] \[y_t = y_{t-12} + a_t - \Theta_1 a_{t-12} - \Theta_2 a_{t-24} - \Theta_3 a_{t-36} - \Theta_4 a_{t-48}\]

set.seed(123)
a = rnorm(240, mean = 0, sd = 2)
Theta1 = 0.9
Theta2 = -0.8
Theta3 = 0.6
Theta4 = -0.5

n = length(a)
y = rep(10, n)
for(t in 49:n){
  y[t] = y[t-12] + a[t] - Theta1*a[t-12] - Theta2*a[t-24] - Theta3*a[t-36] - Theta4*a[t-48]
}

ts.plot(y[49:96])

par(mfrow=c(1,2))
acf(y, lag.max = 48)
pacf(y, lag.max = 48)

par(mfrow=c(1,2))
acf(diff(y,12), lag.max = 48)
pacf(diff(y,12), lag.max = 48)