ARIMA Model Simulation

ARIMA(2,1,1) Model

\[ y_t = y_{t-1} + \phi_1 (y_{t-1} - y_{t-2}) + \phi_2 (y_{t-2} - y_{t-3}) + a_t + \theta_1 a_{t-1} \]

set.seed(123)
at     <- rnorm(1000, mean = 0, sd = 2)
phi1   <- 0.7
phi2   <- -0.8
theta1 <- 0.65

n <- length(at)
y <- numeric(n)
# Initialize first three values of y (e.g., zero or any arbitrary values)
y[1] <- 0
y[2] <- 0
y[3] <- 0

for (t in 4:n) {
  y[t] <- y[t-1] +
    phi1 * (y[t-1] - y[t-2]) +
    phi2 * (y[t-2] - y[t-3]) +
    at[t] +
    theta1 * at[t-1]
}

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

ts.plot(y)

par(mfrow=c(1,2))
acf(y)
pacf(y)

par(mfrow=c(1,2))
acf(diff(y,1))
pacf(diff(y,1))