Using FIGARCH Estimates for One-Day-Ahead Value at Risk

Summary

The document describes a practical question that arises after fitting a FIGARCH model: how to obtain the next day’s conditional volatility for an analytic Value at Risk estimate. It contrasts the task with GARCH(1,1), where the one-step forecast follows directly from the last conditional variance, the latest squared residual, and the estimated parameters. In FIGARCH, fractional integration changes the recursion, making that familiar update insufficient on its own.

The author reports using an R interface to estimate a FIGARCH(1,1) model and provides estimated parameters, but the document contains no answer or forecast formula. It therefore identifies the implementation gap rather than teaching a complete VaR calculation. Readers would need the model’s precise parameterization and its infinite ARCH representation or an equivalent recursion, along with the assumed innovation distribution, to turn fitted estimates into a one-day VaR forecast.

Key ideas

• A one-step GARCH volatility forecast uses the latest variance, squared residual, and fitted parameters.
• FIGARCH fractional integration makes the forecasting recursion more involved than the standard GARCH update.
• The document reports fitted FIGARCH estimates but does not derive the next conditional variance.
• Analytic VaR also requires a quantile from the assumed return distribution.

Tags

# FIGARCH estimation in R


# FIGARCH estimation in R












I am trying to estimate a FIGARCH(1,1) model in R for Value-at-Risk purposes. As I understand it, the rugarch package does not support FIGARCH or FIEGARCH. To that end, I used the garchOxFit function (which runs the estimation in Ox, whilst interfacing with R).

It all works and I am left with the the fitted conditional volatility and the parameter estimates. My problem now is to use that to get the analytic VaR estimate for the next day.

For a simple GARCH(1,1) that is fine: take the last estimated conditional volatility of the sample as well as the last squared residual; plug those into the GARCH equation along with the parameter estimates to get the next day's predicted volatility. One would then use with a quantile function based on whatever distribution was assumed to calculate the analytic VaR.

Problem is I am too simple to see how to do get the vol estimate with a FIGARCH model. I have the following maximum likelihood estimates for the FIGARCH parameters:

Cst(V) x 10^4 : 0.076547 #ie. constant in GARCH equation (omega) d-Figarch : 0.584467 ARCH(Phi1) : 0.122547 GARCH(Beta1) : 0.643318

I have looked at Bollerslev's initial paper on FIGARCH, and am still clueless as to how one gets the next recursive volatility estimate given the parameter estimates and previous day's volatility and squared residual. Any ideas? Any help would be very much appreciated.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.