Preamble

library("forecast")
library("rugarch")
Loading required package: methods
Loading required package: parallel

Attaching package: 'rugarch'
The following object is masked from 'package:stats':

    sigma

Simulating ARCH(1)

n <- 1100
omega <- 0.1
alpha <- 0.5

e <- rnorm(n)  
x <- numeric(n)
x[1] <- rnorm(1, sd = sqrt(omega/(1.0-alpha[1])))
for(i in 2:n) {
  h <- omega + alpha[1] * x[i-1]^2
  x[i] <- e[i] * sqrt(h)
}
x1 <- x[101:1100] # throw away "burn-in"

Time Series Plot

plot(x1, type="l", col=2)

ACF, PACF of Series

par(mfrow=c(1,2))
Acf(x1)
Pacf(x1)

ACF, PACF of Squares

par(mfrow=c(1,2))
Acf(x1^2)
Pacf(x1^2)