Talk:Autoregressive conditional heteroskedasticity

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 137.222.243.9 (talk) 18:45, August 28, 2007 (UTC)

How is GARCH pronounced aloud? Perhaps someone could add it to the page. —Preceding unsigned comment added by 164.76.125.101 (talk) 14:48, 7 April 2008 (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.

mean reverting?
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
 * $$\sigma_t^2 = \alpha_0 + \sum_{i = 1}^p \alpha_i \epsilon_{t-i}^2 + \sum_{j = 1}^q \beta_j \sigma_{t-j}^2\,$$
 * (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)

will someone please for the love of god tell me how to pronounce ARCH and GARCH —Preceding unsigned comment added by 138.16.18.177 (talk) 01:31, 29 April 2009 (UTC)

Notation
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.  —Preceding unsigned comment added by Aruzinsky (talk • contribs) 16:05, 12 June 2010 (UTC)

"Reads more like a review"
Yes please, thanks! 132.194.3.162 (talk) 23:42, 1 October 2012 (UTC)

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.

Thanks. Sorry for the rant. — Preceding unsigned comment added by 2607:4000:200:13:3CBC:50C5:D36C:5FE (talk) 23:43, 18 January 2013 (UTC)
 * I agree with this it tells NOTHING about the alpha value or the process to select the q lags. Nor does it have links to an explanation. Nor does it tell why you should care if there is an ARCH effect or what it is. -Cs california (talk) 11:29, 14 January 2014 (UTC)

Dr. Bauwens's comment on this article
Dr. Bauwens has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"General comment The explanations are not enough precise and there is a lack of integration. The words conditional variance (or standard deviation) appears for the first time in the TGARCH section, whereas it is a key concept of all ARCH models and should be introduced from the beginning. Estimation is brieflly mentioned An ARCH(q) model can be estimated using ordinary least squares Estimation by (quasi-)maximum likelihood is not mentioned though it is the most popular method.

Some symbols are not defined, e.g. \psi _{t} in the GARCH section

Some corrections: where {\displaystyle T'} T' is the number of equations REPLACE equations BY observations symbol T in T'=T-q is not defined

{\displaystyle \rho (i)} should be the symbol on the left of the = sign in item 2 (instead of {\displaystyle \rho}. of the GARCH(p,q) model specification section"

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Bauwens has published scholarly research which seems to be relevant to this Wikipedia article:


 * Reference 1: Luc, BAUWENS & Arie, PREMINGER & Jeroen, ROMBOUTS, 2007. "Theory and inference for a Markov switching GARCH model," Discussion Papers (ECON - Departement des Sciences Economiques) 2007033, Universite catholique de Louvain, Departement des Sciences Economiques.


 * Reference 2: BAUWENS, Luc & GRIGORYEVA, Lyudmila & ORTEGA, Juan-Pablo, 2014. "Estimation and empirical performance of non-scalar dynamic conditional correlation models," CORE Discussion Papers 2014012, Universite catholique de Louvain, Center for Operations Research and Econometrics (CORE).

ExpertIdeasBot (talk) 18:48, 27 June 2016 (UTC)

Dr. Caporin's comment on this article
Dr. Caporin has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"ARCH(q) model specification First sentence: drop ", i.e. the series terms"

I do not agree with the use of the Ljung-Box test for lag selection in GARCH. This approach would lead to an overparametrization of the model and should not be used.

In EGARCH drop the sentence "E-GARCH model mean Exponential General Autoregressive Conditional Hetroskedacity." as it repeats a concept stated in the first line of the section.

In GARCH-M use the same notation of the top of the article to define innovations epsilon

In GJR the notation is incorrect as with I(t-1) we do not define the 0-1 variable but is should be defined as a function I(epsilon(t-1))=1 if epsilon(t-1)<0 and the opposite for the positive case

I would avoid putting all the terms after TARCH and simply put a reference to the work of Tim Bollerslev "Glossary to GARCH""

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Caporin has expertise on the topic of this article, since he has published relevant scholarly research:


 * Reference 1: Massimiliano Caporin & Eduardo Rossi & Paolo Santucci De Magistris, 2014. "Chasing Volatility. A Persistent Multiplicative Error Model With Jumps," "Marco Fanno" Working Papers 0186, Dipartimento di Scienze Economiche "Marco Fanno".


 * Reference 2: Caporin, Massimiliano & Jimenez-Martin, Juan-Angel & Gonzalez-Serrano, Lydia, 2013. "Currency hedging strategies, strategic benchmarks and the Global and Euro Sovereign financial crises," MPRA Paper 50940, University Library of Munich, Germany, revised 23 Oct 2013.

ExpertIdeasBot (talk) 16:09, 24 August 2016 (UTC)

Dr. Otranto's comment on this article
Dr. Otranto has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

""These {\displaystyle ~\epsilon _{t}~} ~\epsilon _{t}~ are split into a stochastic piece {\displaystyle z_{t}} z_{t} and a time-dependent standard deviation {\displaystyle \sigma _{t}} \sigma _{t}"

Comment: It is more correct to say "time-varying conditional standard deviation"

"A methodology to test for the lag length of ARCH errors using the Lagrange multiplier test was proposed by Engle (1982)".

Comment: this test is used to verify the presence of ARCH-type heteroskedasticity in the series, not to establish the lag length

"Generally, when testing for heteroscedasticity in econometric models, the best test is the White test. However, when dealing with time series data, this means to test for ARCH and GARCH errors."

Comment: I would erase this statement; the test for heteroskedasticity was commented in the previous section

I think that you can add the following text at the end of the description: "There are several other extensions of the GARCH model. A review is present in Bollerslev (2010)." Bollerslev, T. (2010): Glossary to ARCH (GARCH). In Bollerslev, T., Russell J., Watson M. (Eds.): Volatility and Time Series Econometrics, Oxford University Press.

Also I would add: "Another extension of GARCH models is the modelization of realized volatility, developing the Multiplicative Error Models (MEM's) (see Engle, 2002, Engle and Gallo, 2006, Gallo and Otranto, 2015)" Engle, R.F. (2002). New frontiers for ARCH models. Journal of Applied Econometrics 17, 425–446. Engle, R.F., and Gallo, G.M. (2006). A multiple indicators model for volatility using intra-daily data. Journal of Econometrics 131, 3–27. Gallo, G.M. and Otranto, E. (2015). Forecasting realized volatility with changing average volatility levels. International Journal of Forecasting, 31, 620–634. |undefined"

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Otranto has expertise on the topic of this article, since he has published relevant scholarly research:


 * Reference : Edoardo Otranto, 2004. "Classifying the Markets Volatility with ARMA Distance Measures," Econometrics 0402009, EconWPA, revised 05 Mar 2004.

ExpertIdeasBot (talk) 20:08, 24 September 2016 (UTC)

Dr. Terasvirta's comment on this article
Dr. Terasvirta has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"1. The ARCH model is usually NOT estimated using ordinary least squares because the OLS estimates are biased. 1a. What is strong white noise, should be defined? Sometimes i.i.d. is mentioned instead, why change from one sentence to the next? Besides, z_t is assumed to have variance equal to one, important,must not be omitted. 2. It is a bit naive to talk about chi-square table values. It suffices to say that the test statistic has an asymptotic chi-square distribution when the null hypothesis is valid. 3. I do not understand the definition of T'. T' is the number of observations minus the number of parameters in the null hypothesis. By the way, most often one uses T, not T'. 4. The sentence beginning with the words "Generally, when testing ..." is nonsense. It should be deleted. 5. The standard deviation is 1/sqrt(T) only when the observations are iid, that is, they do not contain any conditional (or unconditiona) heteroskedasticity. The text in §3 about GARCH(p,q) model specification is very difficult to understand. The last sentence is impossible to understand. 6. There should be some clarifying discussion as to why one generalises GARCH and has all these variants introduced in this entry. By the way, why is there no reference to the article introducing NGARCH? 6a. It is stated that EGARCH model is by Nelson (1991), which is correct, but later on it is claimed that it is introduced by Nelson & Cao (1991). Sloppy work! By the way, the text following "Nelson and Cao (1991)" is not English, and it is practically impossible to understand what it means. 7. That the GARCH parameters (excluding the intercept) sun up to zero does NOT mean "importing a unit root". The IGARCH process is still strongly, if not weakly, stationary. Why is there no reference to IGARCH (Bollerslev and Engle, 1986)? 8. GARCH-in-mean is more general than what is presented under that heading. Why such a model is introduced must be explained. The description of the model is poor, e_t is not a residual, it is an error term, sigma_t and z_t are not defined, nor are y_t and x_t. 9. Why the change of symbols when presenting GJR-GARCH? For example, beta becomes delta. The intercept changes from omega to K. Sloppy work! 10. fGARCH is actually not a model. Henschel defines a family of GARCH models that, as is stated in the text, are nested in the general framework. There are other families as well: Duan (1996?), He and Teräsvirta (1999), etc. 11. It would be useful to explain why these people have defined a continuous-time GARCH model (of order one). The intercept now becomes alpha_0! 12. Multivariate GARCH is completely ignored, which is a big drawback! Surveys exist: Bauwens et al. (2006), Silvennoinen and Teräsvirta (2009). 13. A list of references should contain review articles because they are very helpful to readers who want to know more. Bera and Higgins (1993?), Bollerslev et al. (1992), Bollerslev and Engle (1994), Palm (1996), Teräsvirta (2009), etc., the book by Francq and Zakoian (2011, very important!), books by Gouriéroux (1996), Tsay (2002), etc. The list of references should contain the articles mentioned in the text. Some of the references are veryodd, not what you would normally expect in an entry written for an encyclopedia."

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Terasvirta has expertise on the topic of this article, since he has published relevant scholarly research:


 * Reference 1: Paul Catani & Timo Terasvirta & Meiqun Yin, 2014. "A Lagrange Multiplier Test for Testing the Adequacy of the Constant Conditional Correlation GARCH Model," CREATES Research Papers 2014-03, School of Economics and Management, University of Aarhus.


 * Reference 2: Annastiina Silvennoinen & Timo Terasvirta, 2012. "Modelling conditional correlations of asset returns: A smooth transition approach," CREATES Research Papers 2012-09, School of Economics and Management, University of Aarhus.


 * Reference 3: Silvennoinen, Annastiina & Terasvirta, Timo, 2007. "Modelling Multivariate Autoregressive Conditional Heteroskedasticity with the Double Smooth Transition Conditional Correlation GARCH model," SSE/EFI Working Paper Series in Economics and Finance 0652, Stockholm School of Economics.

ExpertIdeasBot (talk) 00:32, 30 September 2016 (UTC)