Hi, I am trying to write the Realized GARCH model with order (1,1)
The model can be describe bellow:
r_t = sqrt( h_t) * z_t
logh_t = w + b*logh_(t-1) + r*logx_(t-1)
logx_t = c + q*logh_t + t1*z_t +t2*(z_t ^2 -1) + u_t
and z follow N(0,1) , u follow N(0, sigma.u^2)
But I'm troubled with the simulation check for my code.
After I simulate data from the model and estimate the data,
I can't get precise estimation for my setting parameters.
This is my simulation code:
=======================
sim<-function(theta)
{
omega = theta[1]
bet = theta[2]
gam = theta[3]
xi = theta[4]
phi = theta[5]
tau1 = theta[6]
tau2 = theta[7]
sigma.u = theta[8]
n = theta[9]
n.warm = 500
n = n+n.warm
z = rnorm(n+1)
u = rnorm((n+1),0,sigma.u)
logh.pre = 1
logx.pre = xi+phi*logh.pre+tau1*z[1]+tau2*(z[1]^2-1)+u[1]
logh = c(logh.pre,rep(0,n))
r = c(rep(0,(n+1)))
logx = c(logx.pre,rep(0,n))
for(i in 2:(n+1))
{
logh[i] = omega+bet*logh[(i-1)]+gam*logx[(i-1)]
logx[i] = xi+phi*logh[i]+tau1*z[i]+tau2*(z[i]^2-1)+u[i]
r[i] = sqrt(exp(logh[i]))*z[i]
}
gdata = rbind(r,exp(logx))
ans = gdata[,-(1:(n.warm+1))]
ans
}
===========================
This is my estimation code:
============================
LLH<-function(data,theta)
{
r = data[1,]
x = data[2,]
n = dim(data)[2]
logx = log(x)
logh = rep(0,n)
u = rep(0,n)
garchDist = function(r, hh) { dnorm(x = r/hh)/hh }
measureDist = function(u,sigma){dnorm(x = u/sigma)/sigma}
omega = theta[1]
bet = theta[2]
gam = theta[3]
xi = theta[4]
phi = theta[5]
tau1 = theta[6]
tau2 = theta[7]
sigma.u = theta[8]
logh.pre = 0.1
logx.pre = 0.1
logh[1]<-omega+bet*logh.pre+gam*logx.pre
u[1]<-logx[1]-xi-phi*logh[1]-tau1*(r[1]/sqrt(exp(logh[1])))-tau2*((r[1]/sqrt(exp(logh[1])))^2-1)
for(i in 2:n)
{
logh[i] = omega+bet*logh[(i-1)]+gam*logx[(i-1)]
u[i] =
logx[i]-xi-phi*logh[i]-tau1*(r[i]/sqrt(exp(logh[i])))-tau2*((r[i]/sqrt(exp(logh[i])))^2-1)
}
h = exp(logh)
hh = sqrt(h)
llh = -sum(log(garchDist(r, hh)))-sum(log(measureDist(u,sigma.u)))
llh
}
fitting<-function(data)
{
LLH1<-function(theta) {LLH(data,theta)}
ini = c(0.1,0.6,0.3,0.07,1,-0.03,0.1,0.4)
for(i in 1:10)
{
fit = optim(ini,LLH1,control=list(trace=2),hessian=T)
ini = fit$par
}
hessian = fit$hessian
std = sqrt(diag(solve(hessian)))
A = rbind(fit$par,std)
A
}
=====================
Does anyone can point out that where I am wrong? Thank you very much!
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