#install.packages("fGarch")
library(fGarch)


# GARCH Data Generation

# Simple GARCH(1,1) Generation and ACF and PACF
set.seed(2)
a0 <- 0.2
a1 <- 0.5
b1 <- 0.3
w <- rnorm(10000)
e <- rep(0, 10000)
h <- rep(0, 10000)
for (i in 2:10000) {
  h[i] <- a0 + a1 * (e[i-1]^2) + b1 * h[i-1]
  e[i] <- w[i]*sqrt(h[i])
}
par(mfrow = c(2, 1))  
acf(e)
pacf(e^2)


# GARCH simulation and Log Likelihood Function 
set.seed(1234)
garch11<-garchSpec(model = list(omega = 1e-6, alpha = 0.2, beta = 0.6))
y<-garchSim(garch11, n = 1000)

n<-length(y)
sigma2<- vector(length = n)


#estimating the parameters manually by mle assuming normal distribution 
ll <- function(theta) 
{
  sigma2[1]<-theta[1]/(1-theta[2]-theta[3])
  #sigma2[1]<-var(y)
  
  for(i in 1:n)
  {
    sigma2[i+1]<-theta[1]+theta[2]*y[i]^2+theta[3]*sigma2[i]
  }
  
  R=dnorm(y[1:n], 0, sqrt(sigma2[1:n]), log = TRUE)
  L=-sum(R)
  return(L)
}

fit1 <- optim(theta<-c(0.0001,0.01,0.1),ll)
fit1
fit2 <- garchFit(~garch(1,1), y, trace = FALSE)
fit2
plot(fit2)