If I wish to run a FHS for VaR model, first I estimate the GARCH model on the historical returns $r_{t}$, then I obtain the historical innovation time series as $z_{t}=\frac{r_{t}}{\sigma_{t}}$, where $\sigma_{t}$ is the volatility estimated by GARCH.

Setting then $\sigma_{0}$ and $r_{0}$ as the volatility and log return of the last day of my historical sample, I estimate the GARCH daily variance on day 1 of risk horizon as
$\sigma_1^2=\gamma+\alpha r_0^2+ \beta\sigma_0^2$ and $r_1=z_1 \sigma_0^2$,

where $z_1$ is simulated through statistical bootstrap from the previous historical innovation time series.. Then the simulated log return over a risk horizon of h days is the sum $r_1 +..+r_h$: I do it thousand of times, resembling a distribution on which computing the VaR.

My question are:

  1. If the original historical return $r_t$ present autocorrelation, should I first model them through AR model and THEN run the GARCH model redoing the above mentioned steps?
  2. Is it improper to call the innovation residuals?
  3. If the innovations are not i.i.d, should I model them with AR in mean equation of GARCH?

1 Answer 1

  1. You can fit directly any ARMA(p,q)-GARCH(h,s) before applying Filtered Historical Simulations by Maximum Likelihood.
  2. You should call the $Z_t$ “innovations” and the estimated $\hat{Z}_t$ standardized residuals.
  3. The innovations must be iid, otherwise FHS cannot be applied. You should check the ACF and PACF of the standardized residuals and their squared values. If they are not iid, the conditional model is not correctly specified and it must be modified.

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