Seasonal ARIMA

MODEL SARIMA(0,0,4)(0,0,4)^12

\[ y_t = \theta_1 a_{t-1} + \theta_2 a_{t-2} + \theta_3 a_{t-3} + \theta_4 a_{t-4} + \Theta_1 a_{t-12} + \Theta_2 a_{t-24} + \Theta_3 a_{t-36} + \Theta_4 a_{t-48} \]

set.seed(123)
at<- rnorm(1000, mean = 0, sd = 2)
theta<-c(0.4,0.2,0.1,0.1)
Theta<-c(0.3,0.15,0.05,0.05)
s<-12

n <- length(at)
y <- numeric(n)
# Initialize first three values of y (e.g., zero or any arbitrary values)
y[1:max(4,s*4)]<-0
for(t in max(4,s*4+1):n){
  y[t]<-at[t] + theta[1]*at[t-1] + theta[2]*at[t-2] +
  theta[3]*at[t-3] + theta[4]*at[t-4] +
  Theta[1]*at[t-s] + Theta[2]*at[t-2*s] +
  Theta[3]*at[t-3*s] + Theta[4]*at[t-4*s]
  }
y<-y[-(1:100)]

ts.plot(y)

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