Talk:Autoregressive conditional heteroskedasticity
|WikiProject Statistics||(Rated C-class, High-importance)|
|WikiProject Economics||(Rated C-class, Low-importance)|
Alessio Farhadi in the external link (ARCH and GARCH models for forecasting volatility, quantnotes.com) does not seem to follow the more conventional notation that a GARCH(p,q) process has p GARCH terms and q ARCH terms, and not viceversa. These links support the notation I have introduced: , , . The application EViews has the command "arch(p,q)" which uses the convention used by Fahardi (for what I would call a GARCH(q,p)), but even the new versions (>5) have a note "Note the order of the arguments in which the ARCH and GARCH terms are entered, which gives precedence to the ARCH term.", presumably because this is not the standard convention, and it is not even the convention used in their own help file describing GARCH models. (It might even be the case that confounding the order of the terms is a common mistake.)
In my opinion the P Q order is not important given that the formula is specified. However links to similar methods (Locally Stationary Wavelets of Nason for example) should be referenced. There is also no example application or detailed explaination. This should not only be linked to economics but also to mathematics (via statistics). I can also see no consideration of the assumed distribution, ML estimation for GARCH models is often improved assuming a students t distribution as apposed to a guassian distribution. D M —Preceding unsigned comment added by 184.108.40.206 (talk) 18:45, August 28, 2007 (UTC)
Generally, when testing for heteroskedasticity in econometric models, the best test is the White test. However, when dealing with time series data, there means to test for ARCH errors (as described above) and GARCH errors (below).
There is something wrong with the second sentence here, and I can't tell what it trying to say.
this the mean reverting model isn't it? i'm gonna add that into the definition. i thought i should state it here because i'm not exactly sure. —Preceding unsigned comment added by ToyotaPanasonic (talk • contribs) 07:20, 16 June 2008 (UTC)
GARCH(p, q) model specification
The notation applied here for GARCH(p,q) differs from the one used in gretl, where a GARCH(p,q) refers to a model
- (weak reference: gretl mailing list), and probably also in R (programming language) (reference: garchFit documentation, but it's not clear what model is used; the model appears in garchSim, and it seems that alpha and beta are switched). Albmont (talk) 11:47, 27 November 2008 (UTC)
- I have just found an interesting source for these models: Parameter Estimation of ARMA Models with GARCH/APARCH Errors - An R and SPlus Software Implementation, by Diethelm Würtz, Yohan Chalabi, and Ladislav Luksan. Albmont (talk) 12:03, 27 November 2008 (UTC)
I think the page should include a "Notation" paragraph. For exemple, in the R documentation, a (complete) model is referenced like "AR(1)-GARCH(1,1)", meaning that the model of the variable is AR(1) and the model of the volatility is GARCH(1,1). See, for example, the documentation of function garchFit: formula object describing the mean and variance equation of the ARMA-GARCH/APARCH model. A pure GARCH(1,1) model is selected when e.g. formula=~garch(1,1). To specify for example an ARMA(2,1)-APARCH(1,1) use formula = ~arma(2,1)+apaarch(1,1). Albmont (talk) 14:34, 22 May 2009 (UTC)
- PS: the GARCH paragraph seems to be in error, with p and q reversed. Albmont (talk) 14:34, 22 May 2009 (UTC)
Regarding the test,
"The null hypothesis is that, in the absence of ARCH components, we have αi = 0 for all . The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated αi coefficients must be significant. In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic TR² follows χ2 distribution with q degrees of freedom. If TR² is greater than the Chi-square table value, we reject the null hypothesis and conclude there is an ARCH effect in the ARMA model. If TR² is smaller than the Chi-square table value, we do not reject the null hypothesis.",
it is wrong in that it does not discriminate between an ARCH process and an AR process with time varying AR parameters with driving noise of constant variance. If one assumes a standard AR model when the actual process is one with time varying AR parameters and constant noise variance, then the noise estimates will appear to have time varying variance and the above test will lead one to wrongly assume an ARCH model. Similarly, for a process with time varying ARMA parameters, one would be led to wrongly assume a GARCH model. The time varying AR parameter model is widely used, e.g., in linear predictive coding of speech, see http://en.wikipedia.org/wiki/Linear_predictive_coding , and should not be ignored.
Steven A. Ruzinsky, Ph.D.
"Reads more like a review"
Seriously, let's say that I want to figure out WHAT an ARCH model is and WHERE TO USE IT. How am I supposed to accomplish this task (that should be easy)? An ARCH is used to model time series. What does that even mean? Model what? Use where? I understand what conditional homeskedasticity is after reading this article but I still have absolutely no idea why these models are useful. No idea what they are doing. I have a faint hint that they might do something related to linear regression but nothing else. Can somebody please extend the intro to include the most basic aspects of them all: WHAT is it and WHY BOTHER WITH IT.