Garch模型

首先得看下Garch(1,1)模型长什么样

（1）首先生成一个Garch(1,1)序列

n <- 10000
a <- c(0.1, 0.3, 0.6)  # GARCH(1,1) coefficients

sigma_0 <- sqrt(a[1]/(1 - a[2] - a[3]))
r_0 <- sigma_0 * rnorm(1)

r <- double(n)
sigma <- double(n)
sigma[1] <- sqrt(a[1] + a[2] * r_0^2 + a[3] * sigma_0^2)
r[1] <- sigma[1] * rnorm(1)

for(i in 2:n)  # Generate GARCH(1,1) process
{
  sigma[i] <- sqrt(a[1] + a[2]*r[i-1]^2 + a[3]*sigma[i-1]^2)
  r[i] <- sigma[i] * rnorm(1)
}

  （2）用R的rugarch包来求下参数

library(rugarch)
r.garch <- ugarchspec(mean.model = list(armaOrder = c(0,0),
include.mean = FALSE),variance.model = list(garchOrder = c(1, 1)))  # Fit garch(1,1) 
fit = ugarchfit(data = r, spec = r.garch)                           # Diagnostic tests

 （3）写出似然函数，用optim求得参数

loglik <- function(param){
  a0 <- param[1]; a1 <- param[2] ; a2 <- param[3]
  sigma_0 <- sqrt(a[1]/(1 - a[2] - a[3]))
  r_0 <- sigma_0 * rnorm(1)
  sigma <- double(n)
  sigma[1] <- sqrt(a[1] + a[2] * r_0^2 + a[3] * sigma_0^2)
  d <- 0
  for(i in 2:n)  
  {
    sigma[i] <- sqrt(a[1] + a[2]*r[i-1]^2 + a[3]*sigma[i-1]^2)
  }
  for(i in 1:n){
    d <- d + log(dnorm(r[i],0,sigma[i]))
  }
  return(-d)
}
loglik(c(0.1,0.1,0.3))
optim(c(0.1,0.1,0.3), loglik)