One-Step ARMA-GARCH Forecast Intervals with Skewed Innovations
Summary
The document explains how to turn a one-step ARMA-GARCH point forecast and volatility estimate into a forecast interval when standardized innovations follow a skewed generalized error distribution. The conditional return is modeled as an ARMA mean plus a volatility-scaled innovation, so the innovation keeps the chosen standardized distribution while its location and scale change with the forecast. To construct a central interval, take the lower and upper quantiles of the standardized innovation distribution, multiply each by the forecast volatility, and add the ARMA conditional mean. This gives the requested coverage under the model assumptions.
Because the innovation distribution can be asymmetric, the resulting equal-tail interval need not be the shortest interval with that coverage. The document also distinguishes forecasting returns from estimating uncertainty in the GARCH variance parameter: for the latter, it mentions nonparametric bootstrap methods when the estimator’s distribution is unknown. It provides no worked numerical example or implementation details.
Key ideas
- A one-step conditional innovation has the selected standardized distribution, scaled by forecast volatility and shifted by the ARMA mean.
- Use the standardized innovation’s lower and upper quantiles to form a central forecast interval.
- An equal-tail interval can have correct coverage without being the shortest interval for an asymmetric distribution.
- Bootstrap methods may be useful for uncertainty in an estimated GARCH variance.
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# Confidence Intervals for ARMA+GARCH forecasts
# Confidence Intervals for ARMA+GARCH forecasts
I have fitted an ARMA(1,1)+GARCH(1,1) model to my logreturns series. When it comes to my standarized error's distribution however, I have opted for a Skewed Generalized Error Distribution, because of the much better fit.
My model is therefore:
$$r_t \cdot (1 - \phi_1 \cdot B) = \epsilon_t \cdot (1 + \theta_1 \cdot B)$$
where ${B}$ is the lag-operator and:
$$\epsilon_t = \sigma_t \cdot e_t,$$
$$\sigma_t = \sqrt{\omega + \alpha_1 \cdot \epsilon^2_{t-1} + \beta_1 \cdot \sigma^2_{t-1}}$$
Finally:
$$e_t \stackrel{iid}{\sim} SKED (mean=0, variance =1, skew, shape) $$
At the moment of forecasting (I use R and the rugarch package), I have a point-estimate and a sigma estimate.
Instead of having "one-sigma estimates" I would like to have actual confidence intervals. To calculate this I would of coarse have to know the distribution of ${\epsilon_t}$.
My actual question is: how are these distributed? I suppose that they are also SGED but with other parameters. Could someone confirm this? Or maybe show some convenient function to extract this intervals.
Thanks in advance.
## Answer by Richard Hardy (score 2)
https://quant.stackexchange.com/a/63886
How are these distributed? $\epsilon_{t+1}\sim\text{SGED}(\mu_{t+1},\sigma_{t+1},\text{skew},\text{shape})$.
For a $(1-\alpha)$ level $1$-step-ahead forecast interval that is consistent with the model
- obtain the $\alpha/2$ and $1-\alpha/2$ quantiles of the distribution of the standardized innovation $e$ (regardless of the time index, since $e_t$s are i.i.d.),
- multiply them by $\sigma_{t+1}$ and
- add $\mu_{t+1}$ (due to the ARMA part of the model) to each.
The resulting two points will be the end points of the interval. This may not be the shortest $(1-\alpha)$ level forecast interval if the distribution of $e$ is asymmetric, but it will have the correct coverage.
See also "How to make $h$-step interval forecasts from an ARMA-GARCH model?".
## Answer by Fr1 (score 0)
https://quant.stackexchange.com/a/47179
If you just want a confidence interval for the sigma squared produced by the garch notice that the most popular approach is to assume that the distribution of the estimator is unknown and, as such, use non-parametric methods like bootstrapping. For more info read this answer which is also consistent with other sources like this . On bootstrapping meaning I also suggest this and thisShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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