Autoregressive conditional heteroskedasticity#GARCH

To model a time series using an ARCH process, let $~\backslash epsilon\_t~$denote the error terms (return residuals, with respect to a mean process), i.e. the series terms. These $~\backslash epsilon\_t~$ are split into a stochastic piece $z\_t$ and a time-dependent standard deviation $\backslash sigma\_t$ characterizing the typical size of the terms so that

:$~\backslash epsilon\_t=\backslash sigma\_t\; z\_t\; ~$

The random variable $z\_t$ is a strong white noise process. The series $\backslash sigma\_t^2$ is modeled by

:$\backslash sigma\_t^2=\backslash alpha\_0+\backslash alpha\_1\; \backslash epsilon\_\{t-1\}^2+\backslash cdots+\backslash alpha\_q\; \backslash epsilon\_\{t-q\}^2\; =\; \backslash alpha\_0\; +\; \backslash sum\_\{i=1\}^q\; \backslash alpha\_\{i\}\; \backslash epsilon\_\{t-i\}^2$,

:where $~\backslash alpha\_0>0~$ and $\backslash alpha\_i\backslash ge\; 0,~i>0$.

An ARCH(q) model can be estimated using ordinary least squares. A method for testing whether the residuals $\backslash epsilon\_t$ exhibit time-varying heteroskedasticity using the Lagrange multiplier test was proposed by Engle (1982). This procedure is as follows:

1. Estimate the best fitting autoregressive model AR(q) $y\_t\; =\; a\_0\; +\; a\_1\; y\_\{t-1\}\; +\; \backslash cdots\; +\; a\_q\; y\_\{t-q\}\; +\; \backslash epsilon\_t\; =\; a\_0\; +\; \backslash sum\_\{i=1\}^q\; a\_i\; y\_\{t-i\}\; +\; \backslash epsilon\_t$.
2. Obtain the squares of the error $\backslash hat\; \backslash epsilon^2$ and regress them on a constant and q lagged values:
3. : $\backslash hat\; \backslash epsilon\_t^2\; =\; \backslash alpha\_0\; +\; \backslash sum\_\{i=1\}^\{q\}\; \backslash alpha\_i\; \backslash hat\; \backslash epsilon\_\{t-i\}^2$
4. : where q is the length of ARCH lags.
5. The null hypothesis is that, in the absence of ARCH components, we have $\backslash alpha\_i\; =\; 0$ for all $i\; =\; 1,\; \backslash cdots,\; q$. The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated $\backslash alpha\_i$ coefficients must be significant. In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic T'R² follows $\backslash chi^2$ distribution with q degrees of freedom, where $T\text{'}$ is the number of equations in the model which fits the residuals vs the lags (i.e. $T\text{'}=T-q$). If T'R² 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 T'R² is smaller than the Chi-square table value, we do not reject the null hypothesis.

If an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.

In that case, the GARCH (p, q) model (where p is the order of the GARCH terms $~\backslash sigma^2$ and q is the order of the ARCH terms $~\backslash epsilon^2$ ), following the notation of the original paper, is given by

$y\_t=x\text{'}\_t\; b\; +\backslash epsilon\_t$

$\backslash epsilon\_t|\; \backslash psi\_\{t-1\}\; \backslash sim\backslash mathcal\{N\}(0,\; \backslash sigma^2\_t)$

$\backslash sigma\_t^2=\backslash omega\; +\; \backslash alpha\_1\; \backslash epsilon\_\{t-1\}^2\; +\; \backslash cdots\; +\; \backslash alpha\_q\; \backslash epsilon\_\{t-q\}^2\; +\; \backslash beta\_1\; \backslash sigma\_\{t-1\}^2\; +\; \backslash cdots\; +\; \backslash beta\_p\backslash sigma\_\{t-p\}^2\; =\; \backslash omega\; +\; \backslash sum\_\{i=1\}^q\; \backslash alpha\_i\; \backslash epsilon\_\{t-i\}^2\; +\; \backslash sum\_\{i=1\}^p\; \backslash beta\_i\; \backslash sigma\_\{t-i\}^2$

Generally, when testing for heteroskedasticity 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.

Exponentially weighted moving average (EWMA) is an alternative model in a separate class of exponential smoothing models. As an alternative to GARCH modelling it has some attractive properties such as a greater weight upon more recent observations, but also drawbacks such as an arbitrary decay factor that introduces subjectivity into the estimation.

The lag length p of a GARCH(p, q) process is established in three steps:

1. Estimate the best fitting AR(q) model
2. : $y\_t\; =\; a\_0\; +\; a\_1\; y\_\{t-1\}\; +\; \backslash cdots\; +\; a\_q\; y\_\{t-q\}\; +\; \backslash epsilon\_t\; =\; a\_0\; +\; \backslash sum\_\{i=1\}^q\; a\_i\; y\_\{t-i\}\; +\; \backslash epsilon\_t$.
3. Compute and plot the autocorrelations of $\backslash epsilon^2$ by
4. : $\backslash rho(i)\; =\; \{\{\backslash sum^T\_\{t=i+1\}\; (\backslash hat\; \backslash epsilon^2\_t\; -\; \backslash hat\; \backslash sigma^2\_t)\; (\backslash hat\; \backslash epsilon^2\_\{t-i\}\; -\; \backslash hat\; \backslash sigma^2\_\{t-i\})\}\; \backslash over\; \{\backslash sum^T\_\{t=1\}\; (\backslash hat\; \backslash epsilon^2\_t\; -\; \backslash hat\; \backslash sigma^2\_t)^2\}\}$
5. The asymptotic, that is for large samples, standard deviation of $\backslash rho\; (i)$ is $1/\backslash sqrt\{T\}$. Individual values that are larger than this indicate GARCH errors. To estimate the total number of lags, use the Ljung–Box test until the value of these are less than, say, 10% significant. The Ljung–Box Q-statistic follows $\backslash chi^2$ distribution with n degrees of freedom if the squared residuals $\backslash epsilon^2\_t$ are uncorrelated. It is recommended to consider up to T/4 values of n. The null hypothesis states that there are no ARCH or GARCH errors. Rejecting the null thus means that such errors exist in the conditional variance.

{{expand section|{{cite journal|last1=Lanne|first1=Markku|last2=Saikkonen|first2=Pentti|title=Non-linear GARCH models for highly persistent volatility|journal=The Econometrics Journal|date=July 2005|volume=8|issue=2|pages=251–276|doi=10.1111/j.1368-423X.2005.00163.x|jstor=23113641|hdl=10419/65348 |s2cid=15252964|url=https://edoc.hu-berlin.de/bitstream/18452/4120/1/20.pdf}}{{cite book | last1=Bollerslev | first1=Tim | last2=Russell | first2=Jeffrey | last3=Watson | first3=Mark | title=Volatility and Time Series Econometrics: Essays in Honor of Robert Engle | date=May 2010 | publisher=Oxford University Press | location=Oxford | isbn=9780199549498 | pages=137–163 | edition=1st | chapter=Chapter 8: Glossary to ARCH (GARCH) | chapter-url=http://public.econ.duke.edu/~boller/Published_Papers/glossary_10.pdf | access-date=27 October 2017 }}|date=October 2017}}

Nonlinear Asymmetric GARCH(1,1) (NAGARCH) is a model with the specification:

:$~\backslash sigma\_\{t\}^2=\; ~\backslash omega\; +\; ~\backslash alpha\; (~\backslash epsilon\_\{t-1\}\; -\; ~\backslash theta~\backslash sigma\_\{t-1\})^2\; +\; ~\backslash beta\; ~\backslash sigma\_\{t-1\}^2$,

:where $~\backslash alpha\backslash geq\; 0\; ,\; ~\backslash beta\; \backslash geq\; 0\; ,\; ~\backslash omega\; >\; 0$ and , which ensures the non-negativity and stationarity of the variance process.

For stock returns, parameter $~\; \backslash theta$ is usually estimated to be positive; in this case, it reflects a phenomenon commonly referred to as the "leverage effect", signifying that negative returns increase future volatility by a larger amount than positive returns of the same magnitude.{{cite journal|last1=Engle|first1=Robert F.|last2=Ng|first2=Victor K.|title=Measuring and testing the impact of news on volatility|journal=Journal of Finance|year=1993|volume=48|issue=5|pages=1749–1778|ssrn=262096|doi=10.1111/j.1540-6261.1993.tb05127.x|url=http://www.finance.martinsewell.com/stylized-facts/volatility/EngleNg1993.pdf|quote=It is not yet clear in the finance literature that the asymmetric properties of variances are due to changing leverage. The name "leverage effect" is used simply because it is popular among researchers when referring to such a phenomenon.|doi-access=free}}{{cite journal|last=Posedel|first=Petra|year=2006|title=Analysis Of The Exchange Rate And Pricing Foreign Currency Options On The Croatian Market: The Ngarch Model As An Alternative To The Black Scholes Model|journal=Financial Theory and Practice|volume=30|issue=4|pages=347–368|url=http://www.ijf.hr/eng/FTP/2006/4/posedel.pdf | quote = Special attention to the model is given by the parameter of asymmetry [theta (θ)] which describes the correlation between returns and variance.⁶ ...
⁶ In the case of analyzing stock returns, the positive value of [theta] reflects the empirically well known leverage effect indicating that a downward movement in the price of a stock causes more of an increase in variance more than a same value downward movement in the price of a stock, meaning that returns and variance are negatively correlated}}

This model should not be confused with the NARCH model, together with the NGARCH extension, introduced by Higgins and Bera in 1992.{{cite journal|last1=Higgins|first1=M.L|last2=Bera|first2=A.K|title=A Class of Nonlinear Arch Models|journal=International Economic Review|date=1992|volume=33|issue=1|pages=137–158|jstor=2526988|doi=10.2307/2526988}}

Integrated Generalized Autoregressive Conditional heteroskedasticity (IGARCH) is a restricted version of the GARCH model, where the persistent parameters sum up to one, and imports a unit root in the GARCH process.{{Cite journal |last1=Caporale |first1=Guglielmo Maria |last2=Pittis |first2=Nikitas |last3=Spagnolo |first3=Nicola |date=October 2003 |title=IGARCH models and structural breaks |url=http://www.tandfonline.com/doi/abs/10.1080/1350485032000138403 |journal=Applied Economics Letters |language=en |volume=10 |issue=12 |pages=765–768 |doi=10.1080/1350485032000138403 |issn=1350-4851}} The condition for this is

\sum^p_{i=1} ~\beta_{i} +\sum_{i=1}^q~\alpha_{i} = 1

.

The exponential generalized autoregressive conditional heteroskedastic (EGARCH) model by Nelson & Cao (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):

$\backslash log\backslash sigma\_\{t\}^2=\backslash omega+\backslash sum\_\{k=1\}^\{q\}\backslash beta\_\{k\}g(Z\_\{t-k\})+\backslash sum\_\{k=1\}^\{p\}\backslash alpha\_\{k\}\backslash log\backslash sigma\_\{t-k\}^\{2\}$

where $g(Z\_\{t\})=\backslash theta\; Z\_\{t\}+\backslash lambda(|Z\_\{t\}|-E(|Z\_\{t\}|))$, $\backslash sigma\_\{t\}^\{2\}$ is the conditional variance, $\backslash omega$, $\backslash beta$, $\backslash alpha$, $\backslash theta$ and $\backslash lambda$ are coefficients. $Z\_\{t\}$ may be a standard normal variable or come from a generalized error distribution. The formulation for $g(Z\_\{t\})$ allows the sign and the magnitude of $Z\_\{t\}$ to have separate effects on the volatility. This is particularly useful in an asset pricing context.{{cite journal |last=St. Pierre |first=Eilleen F. |year=1998 |title=Estimating EGARCH-M Models: Science or Art |journal=The Quarterly Review of Economics and Finance |volume=38 |issue=2 |pages=167–180 |doi=10.1016/S1062-9769(99)80110-0 }}{{cite journal |last1=Chatterjee |first1=Swarn |last2=Hubble | first2=Amy | year=2016 | title=Day-Of-The-Whieek Effect In Us Biotechnology Stocks—Do Policy Changes And Economic Cycles Matter? | journal=Annals of Financial Economics |volume=11 |issue=2 |pages=1–17 |doi=10.1142/S2010495216500081 }}

Since $\backslash log\backslash sigma\_\{t\}^\{2\}$ may be negative, there are no sign restrictions for the parameters.

The GARCH-in-mean (GARCH-M) model adds a heteroskedasticity term into the mean equation. It has the specification:

y_t = ~\beta x_t + ~\lambda ~\sigma_t + ~\epsilon_t

The residual $~\backslash epsilon\_t$ is defined as:

~\epsilon_t = ~\sigma_t ~\times z_t

The Quadratic GARCH (QGARCH) model by Sentana (1995) is used to model asymmetric effects of positive and negative shocks.

In the example of a GARCH(1,1) model, the residual process $~\backslash sigma\_t$ is

~\epsilon_t = ~\sigma_t z_t

where $z\_t$ is i.i.d. and

~\sigma_t^2 = K + ~\alpha ~\epsilon_{t-1}^2 + ~\beta ~\sigma_{t-1}^2 + ~\phi ~\epsilon_{t-1}

Similar to QGARCH, the Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model by Glosten, Jagannathan and Runkle (1993) also models asymmetry in the ARCH process. The suggestion is to model $~\backslash epsilon\_t\; =\; ~\backslash sigma\_t\; z\_t$ where $z\_t$ is i.i.d., and

~\sigma_t^2 = K + ~\delta ~\sigma_{t-1}^2 + ~\alpha ~\epsilon_{t-1}^2 + ~\phi ~\epsilon_{t-1}^2 I_{t-1}

where $I\_\{t-1\}\; =\; 0$ if $~\backslash epsilon\_\{t-1\}\; \backslash ge\; 0$, and $I\_\{t-1\}\; =\; 1$ if .

The Threshold GARCH (TGARCH) model by Zakoian (1994) is similar to GJR GARCH. The specification is one on conditional standard deviation instead of conditional variance:

~\sigma_t = K + ~\delta ~\sigma_{t-1} + ~\alpha_1^{+} ~\epsilon_{t-1}^{+} + ~\alpha_1^{-} ~\epsilon_{t-1}^{-}

where $~\backslash epsilon\_\{t-1\}^\{+\}\; =\; ~\backslash epsilon\_\{t-1\}$ if $~\backslash epsilon\_\{t-1\}\; >\; 0$, and $~\backslash epsilon\_\{t-1\}^\{+\}\; =\; 0$ if $~\backslash epsilon\_\{t-1\}\; \backslash le\; 0$. Likewise, $~\backslash epsilon\_\{t-1\}^\{-\}\; =\; ~\backslash epsilon\_\{t-1\}$ if $~\backslash epsilon\_\{t-1\}\; \backslash le\; 0$, and $~\backslash epsilon\_\{t-1\}^\{-\}\; =\; 0$ if $~\backslash epsilon\_\{t-1\}\; >\; 0$.

Hentschel's fGARCH model,{{cite journal |last=Hentschel |first=Ludger |year=1995 |title=All in the family Nesting symmetric and asymmetric GARCH models |journal=Journal of Financial Economics |volume=39 |issue=1 |pages=71–104 |doi=10.1016/0304-405X(94)00821-H |citeseerx=10.1.1.557.8941 }} also known as Family GARCH, is an omnibus model that nests a variety of other popular symmetric and asymmetric GARCH models including APARCH, GJR, AVGARCH, NGARCH, etc.

In 2004, Claudia Klüppelberg, Alexander Lindner and Ross Maller proposed a continuous-time generalization of the discrete-time GARCH(1,1) process. The idea is to start with the GARCH(1,1) model equations

:$\backslash epsilon\_t\; =\; \backslash sigma\_t\; z\_t,$

:$\backslash sigma\_t^2\; =\; \backslash alpha\_0\; +\; \backslash alpha\_1\; \backslash epsilon^2\_\{t-1\}\; +\; \backslash beta\_1\; \backslash sigma^2\_\{t-1\}\; =\; \backslash alpha\_0\; +\; \backslash alpha\_1\; \backslash sigma\_\{t-1\}^2\; z\_\{t-1\}^2\; +\; \backslash beta\_1\; \backslash sigma^2\_\{t-1\},$

and then to replace the strong white noise process $z\_t$ by the infinitesimal increments $\backslash mathrm\{d\}L\_t$ of a Lévy process $(L\_t)\_\{t\backslash geq0\}$, and the squared noise process $z^2\_t$ by the increments $\backslash mathrm\{d\}[L,L]^\backslash mathrm\{d\}\_t$, where

:$[L,L]^\backslash mathrm\{d\}\_t\; =\; \backslash sum\_\{s\backslash in[0,t]\}\; (\backslash Delta\; L\_t)^2,\backslash quad\; t\backslash geq0,$

is the purely discontinuous part of the quadratic variation process of $L$. The result is the following system of stochastic differential equations:

:$\backslash mathrm\{d\}G\_t\; =\; \backslash sigma\_\{t-\}\; \backslash ,\backslash mathrm\{d\}L\_t,$

:$\backslash mathrm\{d\}\backslash sigma\_t^2\; =\; (\backslash beta\; -\; \backslash eta\; \backslash sigma^2\_t)\backslash ,\backslash mathrm\{d\}t\; +\; \backslash varphi\; \backslash sigma\_\{t-\}^2\; \backslash ,\backslash mathrm\{d\}[L,L]^\backslash mathrm\{d\}\_t,$

where the positive parameters $\backslash beta$, $\backslash eta$ and $\backslash varphi$ are determined by $\backslash alpha\_0$, $\backslash alpha\_1$ and $\backslash beta\_1$. Now given some initial condition $(G\_0,\backslash sigma^2\_0)$, the system above has a pathwise unique solution $(G\_t,\backslash sigma^2\_t)\_\{t\backslash geq0\}$ which is then called the continuous-time GARCH (COGARCH) model.{{cite journal |last1=Klüppelberg |first1=C.|author1-link= Claudia Klüppelberg |last2=Lindner |first2=A. |last3=Maller |first3=R. |year=2004 |title=A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour |journal=Journal of Applied Probability |volume=41 |issue=3 |pages=601–622 |doi=10.1239/jap/1091543413 |s2cid=17943198|url=http://nbn-resolving.de/urn:nbn:de:bvb:19-epub-1794-9|hdl=10419/31047 |hdl-access=free }}

Unlike GARCH model, the Zero-Drift GARCH (ZD-GARCH) model by Li, Zhang, Zhu and Ling (2018) {{cite journal | last1=Li|first1=D.|last2=Zhang|first2=X.|last3=Zhu|first3=K. |last4=Ling|first4=S. |year=2018 |title=The ZD-GARCH model: A new way to study heteroscedasticity |journal=Journal of Econometrics |volume=202 |issue=1 |pages=1–17|doi=10.1016/j.jeconom.2017.09.003 |url=https://mpra.ub.uni-muenchen.de/68621/1/MPRA_paper_68621.pdf}} lets the drift term $~\backslash omega=\; 0$ in the first order GARCH model. The ZD-GARCH model is to model $~\backslash epsilon\_t\; =\; ~\backslash sigma\_t\; z\_t$, where $z\_t$ is i.i.d., and

~\sigma_t^2 = ~\alpha_{1} ~\epsilon_{t-1}^2 + ~\beta_{1} ~\sigma_{t-1}^2.

The ZD-GARCH model does not require $~\backslash alpha\_\{1\}\; +\; ~\backslash beta\_\{1\}=\; 1$, and hence it nests the Exponentially weighted moving average (EWMA) model in "RiskMetrics". Since the drift term $~\backslash omega=\; 0$, the ZD-GARCH model is always non-stationary, and its statistical inference methods are quite different from those for the classical GARCH model. Based on the historical data, the parameters $~\backslash alpha\_\{1\}$ and $~\backslash beta\_\{1\}$ can be estimated by the generalized QMLE method.

Spatial GARCH processes by Otto, Schmid and Garthoff (2018) {{cite journal | last1=Otto|first1=P.|last2=Schmid|first2=W.|last3=Garthoff|first3=R. |year=2018 |title=Generalised spatial and spatiotemporal autoregressive conditional heteroscedasticity |journal=Spatial Statistics |volume=26 |issue=1 |pages=125–145|doi=10.1016/j.spasta.2018.07.005 |s2cid=88521485|arxiv=1609.00711|bibcode=2018SpaSt..26..125O }} are considered as the spatial equivalent to the temporal generalized autoregressive conditional heteroscedasticity (GARCH) models. In contrast to the temporal ARCH model, in which the distribution is known given the full information set for the prior periods, the distribution is not straightforward in the spatial and spatiotemporal setting due to the interdependence between neighboring spatial locations. The spatial model is given by $~\backslash epsilon(s\_i)\; =\; ~\backslash sigma(s\_i)\; z(s\_i)$ and

:$$

~\sigma(s_i)^2 = ~\alpha_i + \sum_{v=1}^{n} \rho w_{iv} \epsilon(s_v)^2,

where $~s\_i$ denotes the $i$-th spatial location and $~w\_\{iv\}$ refers to the $iv$-th entry of a spatial weight matrix and $w\_\{ii\}=0$ for $~i\; =\; 1,\; ...,\; n$. The spatial weight matrix defines which locations are considered to be adjacent.

In a different vein, the machine learning community has proposed the use of Gaussian process regression models to obtain a GARCH scheme.{{cite journal | last1=Platanios|first1=E.|last2=Chatzis|first2=S. |year=2014 |title=Gaussian process-mixture conditional heteroscedasticity |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |volume=36 |issue=5 |pages=889–900|doi=10.1109/TPAMI.2013.183|pmid=26353224|arxiv=1211.4410|s2cid=10424638}} This results in a nonparametric modelling scheme, which allows for: (i) advanced robustness to overfitting, since the model marginalises over its parameters to perform inference, under a Bayesian inference rationale; and (ii) capturing highly-nonlinear dependencies without increasing model complexity.{{cn|date=September 2021}}

{{Reflist}}

• {{cite book | last1=Bollerslev | first1=Tim | last2=Russell | first2=Jeffrey | last3=Watson | first3=Mark | title=Volatility and Time Series Econometrics: Essays in Honor of Robert Engle | date=May 2010 | publisher=Oxford University Press | location=Oxford | isbn=9780199549498 | pages=137–163 | edition=1st | chapter=Chapter 8: Glossary to ARCH (GARCH) | chapter-url=http://public.econ.duke.edu/~boller/Published_Papers/glossary_10.pdf }}
• {{cite book |last=Enders |first=W. |year=2004 |chapter=Modelling Volatility |title=Applied Econometrics Time Series |publisher=John-Wiley & Sons |edition=Second |pages=108–155 |isbn=978-0-471-45173-0 }}
• {{cite journal |author-link=Robert F. Engle |last=Engle |first=Robert F. |year=1982 |title=Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation |journal=Econometrica |volume=50 |issue=4 |pages=987–1008 |jstor=1912773 |doi=10.2307/1912773 |s2cid=18673159 }} (the paper which sparked the general interest in ARCH models)
• {{cite book |last=Engle |first=Robert F. |year=1995 |title=ARCH: selected readings |publisher=Oxford University Press |isbn=978-0-19-877432-7 }}
• {{cite journal |last=Engle |first=Robert F. |year=2001 |title=GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics |journal=Journal of Economic Perspectives |volume=15 |issue=4 |pages=157–168 |jstor=2696523 |doi=10.1257/jep.15.4.157|url=http://www.aeaweb.org/articles.php?doi=10.1257/jep.15.4.157 }} (a short, readable introduction)
• {{cite book |last=Gujarati |first=D. N. |year=2003 |title=Basic Econometrics |pages=856–862 }}
• {{cite journal |last1=Hacker |first1=R. S. |last2=Hatemi-J |first2=A. |year=2005 |title=A Test for Multivariate ARCH Effects |journal=Applied Economics Letters |volume=12 |issue=7 |pages=411–417 |url=https://ideas.repec.org/a/taf/apeclt/v12y2005i7p411-417.html |doi=10.1080/13504850500092129 |s2cid=218639533 }}
• {{cite journal |last=Nelson |first=D. B. |year=1991 |title=Conditional Heteroskedasticity in Asset Returns: A New Approach |journal=Econometrica |volume=59 |issue=2 |pages=347–370 |jstor=2938260 |doi=10.2307/2938260 }}

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