Selecting ARMA-GARCH Models and Computing Distribution-Based Forecast Intervals
Summary
The discussion addresses information-criterion comparisons across ARMA-GARCH specifications, including models with different innovation distributions, and how to form forecast intervals when innovations are nonnormal. It describes AIC and BIC as likelihood-based measures that account for model complexity and sample size. These criteria can compare candidate models, but selecting the best in-sample fit may overfit historical conditions; out-of-sample forecast performance is suggested as an additional selection check.
For interval construction, the response centers bounds on the conditional mean and scales the innovation quantile by the conditional standard deviation. The quantile should come from the fitted innovation distribution, using its estimated shape or degrees-of-freedom parameters, with the distribution parameterized to have zero mean and unit variance. One answer gives a normal-style multiple of conditional standard deviation, while another explains how to use distribution-specific quantiles. Thus fixed normal bounds need not be appropriate for skewed or heavy-tailed innovations, and interval coverage depends on the fitted model and distribution.
Key ideas
- AIC and BIC can compare ARMA-GARCH candidates with different orders and innovation distributions.
- Information criteria balance likelihood fit against model complexity but can favor historical overfit.
- Out-of-sample forecast performance provides another basis for choosing a specification.
- Forecast intervals combine the conditional mean with the conditional standard deviation and an innovation quantile.
- For nonnormal innovations, calculate the quantile from the fitted distribution using its estimated parameters.
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# ARMA-GARCH model, bset model selection and confidence levels calculations
# ARMA-GARCH model, bset model selection and confidence levels calculations
I'm a newbie in GARCH models. I tried to realize ARMA(p, q)-GARCH(u, v) model via fGarch.
So, 2 main questions.
1) Can I use BIC/AIC for selection best model for all (p, q)-(u, v) models? So, is it correct to compare BICs from, for example, ARMA(2, 1)-GARCH(1, 2) and ARMA(1, 3)-GARCH(2, 1)? If not, what should I do in this case (best way)? [Added] And is it correct to compare BICs/AICs for different ARMA-GARCH model, based on different distributions (sged vs t-studentm for example)?
2) I want to use distributions which are not normal. They may be skewed and so on. I know formula for $r_{t}$, that $r_{t} = z_{t}\sigma_{t}$, where $z_{t}$ iid, zero mean and unit variation. For example, I have same model, fitted values for $r_{t}$ and $\sigma_{t}$ by the same ARMA-GARCH model I choose as best. And I choose skewed normal distribution when I fitted model. So, how should I calculate quantiles? Is it correct to get quantiles of $z_{t}$ as a skewed normal with 0 mean and unit variation and calculate conf.levels as $r_{t}^{fitted} +- q_{1,2}*\sigma_{t}^{fitted}$, where $q_{1,2}$ are quantiles, which I get by skewed normal distribution (sure, zero mean and unit variance)?
Thank you.
## Answer by simmy (score 2, accepted)
https://quant.stackexchange.com/a/25026
1) The answer is yes: you can use the AIC/BIC to select the best model.
2) You can have confidence intervals by: $r_t^{fitted} \pm 2* \sigma_t^{fitted}$ so that you have a confidence interval of $\pm 2$ times the conditional standard deviation.
You can see a plot of this by: `plot(your_garchFit_object)` and typing 3 to select the plot of the series with the confidence bounds.
## Answer by Malick (score 2)
https://quant.stackexchange.com/a/25033
- Information Criteria estimate the quality of a model based on the likelihood / the numbers of parameters (or degree of freedom) and the number of observations. It is a measure of goodness-of-fit and so it may suffers of overfitting problems. You can use it to compare any models (even with different errors distributions) however you may risk to select the model which fit the best to your past data (to a specific scenario). An additional way is is to select the "best" model based on its predictive power by using out-of-sample forecast.
- The confidence interval is given by : conditional mean +/- $\sigma_{t} \times q $ where $\sigma$ is the conditional variance at time $t$ and $q$ is the quantile obtained via the inverse CDF based on the parameters you obtained during the estimation and the confidence interval. So if you use a Skew Normal Dist, the Inverse CDF must also be based on the shape parameter ($\alpha$) you obtained. At the end the $z_{t}$ have a zero mean and unit variance distribution but you need to use the specific estimated shape parameter to obtain the quantile. When we use a density different that the normal one, it is always re-parameterize in such a way that the innovation process has zero mean and unit variance. (For instance if you use the student t distribution the quantile is obtained via the $t(0,1,v)$ student density with $v$ the degree of freedom)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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