Robert Engle published a kindly exposition of the GARCH model which he in good humor called Garch 101. --- Journal of Economic Perspectives(Volume 15, Number 4—Fall 2000, p. 157–168)
This article is definitely worth the attention of any Stat 434 or Stat 956 student. The notation of this paper is not exactly the same as the notation we use for compatibility with S-Plus, but one should not have any trouble making the translation.
Engle's GARCH 101 is closely related to his Noble Lecture though the Nobel Lecture has a few parts that will not be transparent to 434 students.
Still, it is perfectly OK to just read the juicy bits.
The obvious value of the ARCH/GARCH models opened a flood gate of variations.
In a way the stream is now a bit polluted. There are so many variations, the modeler cannot help but wonder how to make a reasonable choice.
Moreover, except when one follows a specific branch of this family, the models are non-nested, so the traditional information criteria are not helpful.
Eventually, one needs to say how one expects to use the chosen model. Once this is done, then it becomes possible to judge the models in terms of their "fitness for use."
Since GARCH family models address the notion of volatility, it is reasonable to consider how well individual models do at forecasting out-of-sample volatility. The paper
Hansen, P. R. and Lunde, A. "A Forecast Comparison of Volatility Models: Does Anything Beat a GARCH(1,1)." Journal of Applied Econometrics, March 2004.
takes up this idea and the authors --- bless their hearts --- test 330 alternative models.
In class we will discuss their list of models (beginning on page 31) and the tables of relative performance that follow in the appendix.
Again, you will want to skip over some parts of this article, but with luck the main messages will still flow through.
Volatility prediction is not necessarily the only purpose for a model for the GARCH family, but it certainly in a natural and important use.
For example, we should be encouraged to explore GARCH(1,1) more thoroughly in trading game contexts.
This is where fitness for use is truly well defined --- though of course we face a fair share of other difficulties.
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