Ineconometrics, theautoregressive conditional heteroskedasticity (ARCH) model is astatistical model fortime series data that describes thevariance of the currenterror term orinnovation as a function of the actual sizes of the previous time periods' error terms;^[1] often the variance is related to the squares of the previous innovations. The ARCH model is appropriate when the error variance in a time series follows anautoregressive (AR) model; if anautoregressive moving average (ARMA) model is assumed for the error variance, the model is ageneralized autoregressive conditional heteroskedasticity (GARCH) model.^[2]

ARCH models are commonly employed in modelingfinancialtime series that exhibit time-varyingvolatility andvolatility clustering, i.e. periods of swings interspersed with periods of relative calm (this is, when the time series exhibits heteroskedasticity). ARCH-type models are sometimes considered to be in the family ofstochastic volatility models, although this is strictly incorrect since at timet the volatility is completely predetermined (deterministic) given previous values.^[3]

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

${\displaystyle {ϵ}_t={\sigma }_t{z}_t}$

The random variable${\displaystyle z_t}$ is a strongwhite noise process. The series${\displaystyle {\sigma }_t^2}$ is modeled by

${\displaystyle {\sigma }_t^2={\alpha }_0+{\alpha }_1{ϵ}_{t-1}^2+\cdots +{\alpha }_q{ϵ}_{t-q}^2={\alpha }_0+\sum _{i=1}^q{\alpha }_i{ϵ}_{t-i}^2}$,

where${\displaystyle {\alpha }_0>0}$ and${\displaystyle {\alpha }_i\ge 0,i>0}$.

An ARCH(q) model can be estimated usingordinary least squares. A method for testing whether the residuals${\displaystyle {ϵ}_t}$ exhibit time-varying heteroskedasticity using theLagrange multiplier test was proposed byEngle (1982). This procedure is as follows:

1. Estimate the best fittingautoregressive model AR(q)${\displaystyle y_t=a_0+a_1{y}_{t-1}+\cdots +a_q{y}_{t-q}+{ϵ}_t=a_0+\sum _{i=1}^q{a}_i{y}_{t-i}+{ϵ}_t}$.
2. Obtain the squares of the error${\displaystyle {\widehat{ϵ}}^2}$ and regress them on a constant andq lagged values:

   ${\displaystyle {\widehat{ϵ}}_t^2={\alpha }_0+\sum _{i=1}^q{\alpha }_i{\widehat{ϵ}}_{t-i}^2}$

   whereq is the length of ARCH lags.

3. Thenull hypothesis is that, in the absence of ARCH components, we have${\displaystyle {\alpha }_i=0}$ for all${\displaystyle i=1,\cdots ,q}$. The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated${\displaystyle {\alpha }_i}$ coefficients must be significant. In a sample ofT residuals under the null hypothesis of no ARCH errors, the test statisticT'R² follows${\displaystyle {\chi }^2}$ distribution withq degrees of freedom, where${\displaystyle T^{\prime }}$ is the number of equations in the model which fits the residuals vs the lags (i.e.${\displaystyle T^{\prime }=T-q}$). IfT'R² is greater than the Chi-square table value, wereject the null hypothesis and conclude there is an ARCH effect in theARMA model. IfT'R² is smaller than the Chi-square table value, we do not reject the null hypothesis.

If anautoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.^[2]

In that case, the GARCH (p,q) model (wherep is the order of the GARCH terms${\displaystyle {\sigma }^2}$ andq is the order of the ARCH terms${\displaystyle {ϵ}^2}$ ), following the notation of the original paper, is given by

${\displaystyle y_t=x_t^{\prime }b+{ϵ}_t}$

${\displaystyle {ϵ}_t|y_{t-1}\sim \mathcal{N}(0,{\sigma }_t^2)}$

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

Generally, when testing for heteroskedasticity in econometric models, the best test is theWhite test. However, when dealing withtime series data, this means to test for ARCH and GARCH errors.

Exponentially weightedmoving 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 lengthp of a GARCH(p,q) process is established in three steps:

1. Estimate the best fitting AR(q) model

   ${\displaystyle y_t=a_0+a_1{y}_{t-1}+\cdots +a_q{y}_{t-q}+{ϵ}_t=a_0+\sum _{i=1}^q{a}_i{y}_{t-i}+{ϵ}_t}$.

2. Compute and plot the autocorrelations of${\displaystyle {ϵ}^2}$ by

   ${\displaystyle \rho (i)=\frac{\sum _{t=i+1}^T({\widehat{ϵ}}_t^2-{\hat{\sigma }}_t^2)({\widehat{ϵ}}_{t-i}^2-{\hat{\sigma }}_{t-i}^2)}{\sum _{t=1}^T({\widehat{ϵ}}_t^2-{\hat{\sigma }}_t^2{)}^2}}$

3. The asymptotic, that is for large samples, standard deviation of${\displaystyle \rho (i)}$ is${\displaystyle 1/\sqrt{T}}$. Individual values that are larger than this indicate GARCH errors. To estimate the total number of lags, use theLjung–Box test until the value of these are less than, say, 10% significant. The Ljung–BoxQ-statistic follows${\displaystyle {\chi }^2}$ distribution withn degrees of freedom if the squared residuals${\displaystyle {ϵ}_t^2}$ are uncorrelated. It is recommended to consider up to T/4 values ofn. The null hypothesis states that there are no ARCH or GARCH errors. Rejecting the null thus means that such errors exist in theconditional variance.

This sectionneeds expansion with:[4][5]. You can help byadding to it.(October 2017)

Nonlinear Asymmetric GARCH(1,1) (NAGARCH) is a model with the specification:^[6][7]

${\displaystyle {\sigma }_t^2=\omega +\alpha ({ϵ}_{t-1}-\theta {\sigma }_{t-1}{)}^2+\beta {\sigma }_{t-1}^2}$,

where${\displaystyle \alpha \ge 0,\beta \ge 0,\omega >0}$ and, which ensures the non-negativity and stationarity of the variance process.

For stock returns, parameter${\displaystyle \theta }$ 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.^[6][7]

This model should not be confused with the NARCH model, together with the NGARCH extension, introduced by Higgins and Bera in 1992.^[8]

Integrated Generalized Autoregressive Conditional heteroskedasticity (IGARCH) is a restricted version of the GARCH model, where the persistent parameters sum up to one, and imports aunit root in the GARCH process.^[9] The condition for this is

${\displaystyle \sum _{i=1}^p{\beta }_i+\sum _{i=1}^q{\alpha }_i=1}$.

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

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

where${\displaystyle g(Z_t)=\theta Z_t+\lambda (|Z_t|-E(|Z_t|))}$,${\displaystyle {\sigma }_t^2}$ is theconditional variance,${\displaystyle \omega }$,${\displaystyle \beta }$,${\displaystyle \alpha }$,${\displaystyle \theta }$ and${\displaystyle \lambda }$ are coefficients.${\displaystyle Z_t}$ may be astandard normal variable or come from ageneralized error distribution. The formulation for${\displaystyle g(Z_t)}$ allows the sign and the magnitude of${\displaystyle Z_t}$ to have separate effects on the volatility. This is particularly useful in an asset pricing context.^[10][11]

Since${\displaystyle \log {\sigma }_t^2}$ 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:

${\displaystyle y_t=\beta x_t+\lambda {\sigma }_t+{ϵ}_t}$

The residual${\displaystyle {ϵ}_t}$ is defined as:

${\displaystyle {ϵ}_t={\sigma }_t\times z_t}$

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${\displaystyle {\sigma }_t}$ is

${\displaystyle {ϵ}_t={\sigma }_t{z}_t}$

where${\displaystyle z_t}$ is i.i.d. and

${\displaystyle {\sigma }_t^2=K+\alpha {ϵ}_{t-1}^2+\beta {\sigma }_{t-1}^2+\varphi {ϵ}_{t-1}}$

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${\displaystyle {ϵ}_t={\sigma }_t{z}_t}$ where${\displaystyle z_t}$ is i.i.d., and

${\displaystyle {\sigma }_t^2=K+\delta {\sigma }_{t-1}^2+\alpha {ϵ}_{t-1}^2+\varphi {ϵ}_{t-1}^2{I}_{t-1}}$

where${\displaystyle I_{t-1}=0}$ if${\displaystyle {ϵ}_{t-1}\ge 0}$, and${\displaystyle I_{t-1}=1}$ if.

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

${\displaystyle {\sigma }_t=K+\delta {\sigma }_{t-1}+{\alpha }_1^{+}{ϵ}_{t-1}^{+}+{\alpha }_1^{-}{ϵ}_{t-1}^{-}}$

where${\displaystyle {ϵ}_{t-1}^{+}={ϵ}_{t-1}}$ if${\displaystyle {ϵ}_{t-1}>0}$, and${\displaystyle {ϵ}_{t-1}^{+}=0}$ if${\displaystyle {ϵ}_{t-1}\le 0}$. Likewise,${\displaystyle {ϵ}_{t-1}^{-}={ϵ}_{t-1}}$ if${\displaystyle {ϵ}_{t-1}\le 0}$, and${\displaystyle {ϵ}_{t-1}^{-}=0}$ if${\displaystyle {ϵ}_{t-1}>0}$.

Hentschel'sfGARCH model,^[12] also known asFamily 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

${\displaystyle {ϵ}_t={\sigma }_t{z}_t,}$

${\displaystyle {\sigma }_t^2={\alpha }_0+{\alpha }_1{ϵ}_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2={\alpha }_0+{\alpha }_1{\sigma }_{t-1}^2{z}_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2,}$

and then to replace the strong white noise process${\displaystyle z_t}$ by the infinitesimal increments${\displaystyle \mathrm{d}{L}_t}$ of aLévy process${\displaystyle (L_t{)}_{t\ge 0}}$, and the squared noise process${\displaystyle z_t^2}$ by the increments${\displaystyle \mathrm{d}[L,L{]}_t^{\mathrm{d}}}$, where

${\displaystyle [L,L{]}_t^{\mathrm{d}}=\sum _{s\in [0,t]}(\mathrm{\Delta }{L}_t{)}^2,t\ge 0,}$

is the purely discontinuous part of thequadratic variation process of${\displaystyle L}$. The result is the following system ofstochastic differential equations:

${\displaystyle \mathrm{d}{G}_t={\sigma }_{t-}\mathrm{d}{L}_t,}$

${\displaystyle \mathrm{d}{\sigma }_t^2=(\beta -\eta {\sigma }_t^2)\mathrm{d}t+\phi {\sigma }_{t-}^2\mathrm{d}[L,L{]}_t^{\mathrm{d}},}$

where the positive parameters${\displaystyle \beta }$,${\displaystyle \eta }$ and${\displaystyle \phi }$ are determined by${\displaystyle {\alpha }_0}$,${\displaystyle {\alpha }_1}$ and${\displaystyle {\beta }_1}$. Now given some initial condition${\displaystyle (G_0,{\sigma }_0^2)}$, the system above has a pathwise unique solution${\displaystyle (G_t,{\sigma }_t^2{)}_{t\ge 0}}$ which is then called the continuous-time GARCH (COGARCH) model.^[13]

An ARCH model without intercept was proposed by Hafner and Preminger (2015),^[14] who set the intercept term to zero (${\displaystyle \omega =0}$), in the first order ARCH model${\displaystyle {ϵ}_t={\sigma }_t{z}_t}$, where${\displaystyle z_t}$ is i.i.d., and the conditional variance is:

${\displaystyle {\sigma }_t^2={\alpha }_1{ϵ}_{t-1}^2.}$

This model was extended by Li, Zhang, Zhu and Ling (2018)^[15] which consider the Zero-Drift GARCH (ZD-GARCH) with the specification:

${\displaystyle {\sigma }_t^2={\alpha }_1{ϵ}_{t-1}^2+{\beta }_1{\sigma }_{t-1}^2.}$

The ZD-GARCH model does not require${\displaystyle {\alpha }_1+{\beta }_1=1}$, and hence it nests theExponentially weighted moving average (EWMA) model in "RiskMetrics". Since${\displaystyle \omega =0}$, 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${\displaystyle {\alpha }_1}$ and${\displaystyle {\beta }_1}$ can be estimated by the generalizedQMLE method.

Spatial GARCH processes by Otto, Schmid and Garthoff (2018)^[16] are considered as the spatial equivalent to the temporal generalized autoregressive conditional heteroscedasticity (GARCH) models.^[17] 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 contemporaneous dependence between neighboring spatial locations. The spatial model is given by${\displaystyle ϵ(s_i)=\sigma (s_i)z(s_i)}$ and

${\displaystyle \sigma (s_i{)}^2={\alpha }_i+\sum _{v=1}^n\rho w_{iv}ϵ(s_v{)}^2,}$

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

In spatiotemporal extensions, the conditional variance is modelled as a joint function of spatially lagged past squared observations and temporally lagged volatilities, allowing for both cross-sectional and serial dependence. These models have been applied in fields such as environmental statistics, regional economics, and financial econometrics, where shocks can propagate over space and time. Recent reviews summarise methodological developments, estimation techniques, and applications across disciplines.^[17]

In a different vein, the machine learning community has proposed the use of Gaussian process regression models to obtain a GARCH scheme.^[18] 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.^{[citation needed]}

• Bollerslev, Tim; Russell, Jeffrey; Watson, Mark (May 2010)."Chapter 8: Glossary to ARCH (GARCH)"(PDF).Volatility and Time Series Econometrics: Essays in Honor of Robert Engle (1st ed.). Oxford: Oxford University Press. pp. 137–163.ISBN 9780199549498.
• Enders, W. (2004). "Modelling Volatility".Applied Econometrics Time Series (Second ed.). John-Wiley & Sons. pp. 108–155.ISBN 978-0-471-45173-0.
• Engle, Robert F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation".Econometrica.50 (4):987–1008.doi:10.2307/1912773.JSTOR 1912773.S2CID 18673159.(the paper which sparked the general interest in ARCH models)
• Engle, Robert F. (1995).ARCH: selected readings. Oxford University Press.ISBN 978-0-19-877432-7.
• Engle, Robert F. (2001)."GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics".Journal of Economic Perspectives.15 (4):157–168.doi:10.1257/jep.15.4.157.JSTOR 2696523.(a short, readable introduction)
• Gujarati, D. N. (2003).Basic Econometrics. pp. 856–862.
• Hacker, R. S.; Hatemi-J, A. (2005)."A Test for Multivariate ARCH Effects".Applied Economics Letters.12 (7):411–417.doi:10.1080/13504850500092129.S2CID 218639533.
• Nelson, D. B. (1991). "Conditional Heteroskedasticity in Asset Returns: A New Approach".Econometrica.59 (2):347–370.doi:10.2307/2938260.JSTOR 2938260.
• Otto, P.; Dogan, O.; Taspinar, S.; Schmid, W.; Bera, A. K. (2025)."Spatial and Spatiotemporal Volatility Models: A Review".Journal of Economic Surveys.39 (3).doi:10.1111/joes.12643.

• Nonlinear time series analysis
• Autocorrelation

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