value at risk estimation using fGarch package in R

Question

I am trying to make a similar analysis to McNeil & Frey in their paper 'Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach' but I am stuck with a problem when implementing the models.

The approach is to fit a AR(1)-GARCH(1,1) model in order to estimate the the one-day ahead forecast of the VaR using a window of 1000 observations.

I have simulated data that should work fine with my model, and I assume that if I would be doing this correct, the observed coverage rate should be close to the theoretical one. However it is always below the theoretical coverage rate, and I don´t know why.

I beleive that this is how the calculation of the estimated VaR is done

VaR_hat = mu_hat + sigma_hat * qnorm(alpha)

, but I might be wrong. I have tried to find related questions here at stack but I have not found any.

How I approach this can be summarized in three steps.

1. Simulate 2000 AR(1)-GARCH(1,1) observations and fit a corresponding model and extract the one day prediction of the conditional mean and standard deviation using a window of 1000 observations.(Thereby making 1000 predictions)

2. Use the predicted values and the normal quantile to calculate the VaR for the wanted confidence level.

3. Check if the coverage rate is close to the theoretical one.

If someone could help me I would be extremely thankful, and if I'm unclear in my formalation please just tell me and I'll try to come up with a better explanation to the problem.

The code I'm using is attached below. Thank you in advance

library(fGarch)

nObs <- 2000                  # Number of observations.
quantileLevel <- 0.95         # Since we expect 5% exceedances.  
from <- seq(1,1000)           # Lower index vector for observations in model.
to <- seq(1001,2000)          # Upper index vector for observations in model.
VaR_vec <- rep(0,(nObs-1000)) # Empty vector for storage of 1000 VaR estimates.

# Specs for simulated data (including AR(1) component and all components for GARC(1,1)).
spec = garchSpec(model = list(omega = 1e-6, alpha = 0.08, beta = 0.91, ar = 0.10),
             cond.dist = 'norm')
# Simulate 1000 data points.
data_sim <- c(garchSim(spec, n = nObs, n.start = 1000))

for (i in 1:1000){
  # The rolling window of 1000 observations.
  data_insert <- data_sim[from[i]:to[i]]
  # Fitting an AR(1)-GARCH(1,1) model with normal cond.dist.
  fitted_model <- garchFit(~ arma(1,0) + garch(1,1), data_insert,
                       trace = FALSE,
                       cond.dist = "norm")
  # One day ahead forecast of conditional mean and standard deviation.
  predict(fitted_model, n.ahead = 1)
  prediction_model <- predict(fitted_model, n.ahead = 1)
  mu_pred <- prediction_model$meanForecast
  sigma_pred <- prediction_model$standardDeviation
  # Calculate VaR forecast
  VaR_vec[i] <- mu_pred + sigma_pred*qnorm(quantileLevel)

  if (length(to)-i != 0){
    print(c('Countdown, just',(length(to) - i),'iterations left'))
  } else {
    print(c('Done!'))
  }
}

# Exctract only the estiamtes ralated to the forecasts.
compare_data_sim <- data_sim[1001:length(data_sim)]
hit <- rep(0,length(VaR_vec))
# Count the amount of exceedances. 
for (i in 1:length(VaR_vec)){
  hit[i] <- sum(VaR_vec[i] <= compare_data_sim[i])
}


plot(data_sim[1001:2000], type = 'l', 
 ylab = 'Simulated data', main = 'Illustration of one day ahead prediction of 95%-VaR')
lines(VaR_vec, col = 'red')

cover_prop <- sum(hit)/length(hit)
print(sprintf("Diff theoretical level and VaR coverage = %f", (1-quantileLevel) - cover_prop))