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In econometrics, AutoRegressive Conditional Heteroskedasticity (ARCH) models are used to characterize and model observed time series. They are used whenever there is reason to believe that, at any point in a series, the error terms will have a characteristic size, or variance. In particular ARCH models assume the variance of the current error term or innovation to be a function of the actual sizes of the previous time periods' error terms: often the variance is related to the squares of the previous innovations.

Such models are often called ARCH models (Engle, 1982), although a variety of other acronyms are applied to particular structures of model which have a similar basis. ARCH models are employed commonly in modeling financial time series that exhibit time-varying volatility clustering, i.e. periods of swings followed by periods of relative calm.

ARCH(q) model Specification

Suppose one wishes to model a time series using an ARCH process. Let $~\epsilon _{t}~$ denote the error terms (return residuals, with respect to a mean process) i.e. the series terms. These $~\epsilon _{t}~$ are split into a stochastic piece $z_{t}$ and a time-dependent standard deviation $\sigma _{t}$ characterizing the typical size of the terms so that

$~\epsilon _{t}=\sigma _{t}z_{t}~$

The random variable $z_{t}$ is a strong White noise process. The series $\sigma _{t}^{2}$ is modelled by

$\sigma _{t}^{2}=\alpha _{0}+\alpha _{1}\epsilon _{t-1}^{2}+\cdots +\alpha _{q}\epsilon _{t-q}^{2}=\alpha _{0}+\sum _{i=1}^{q}\alpha _{i}\epsilon _{t-i}^{2}$

where $~\alpha _{0}>0~$ and $\alpha _{i}\geq 0,~i>0$.

An ARCH(q) model can be estimated using ordinary least squares. A methodology to test for the lag length of ARCH errors using the Lagrange multiplier test was proposed by Engle (1982). This procedure is as follows:

1. Estimate the best fitting autoregressive model AR(q) $y_{t}=a_{0}+a_{1}y_{t-1}+\cdots +a_{q}y_{t-q}+\epsilon _{t}=a_{0}+\sum _{i=1}^{q}a_{i}y_{t-i}+\epsilon _{t}$.
2. Obtain the squares of the error ${\hat {\epsilon }}^{2}$ and regress them on a constant and q lagged values:

   ${\hat {\epsilon }}_{t}^{2}={\hat {\alpha }}_{0}+\sum _{i=1}^{q}{\hat {\alpha }}_{i}{\hat {\epsilon }}_{t-i}^{2}$

   where q is the length of ARCH lags.

3. The null hypothesis is that, in the absence of ARCH components, we have $\alpha _{i}=0$ for all $i=1,\cdots ,q$. The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated $\alpha _{i}$ coefficients must be significant. In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic TR² follows $\chi ^{2}$ 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.

GARCH

If an autoregressive moving average model (ARMA model) is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH, Bollerslev (1986)) model.

In that case, the GARCH (p, q) model (where p is the order of the GARCH terms $~\sigma ^{2}$ and q is the order of the ARCH terms $~\epsilon ^{2}$ ) is given by

$\sigma _{t}^{2}=\alpha _{0}+\alpha _{1}\epsilon _{t-1}^{2}+\cdots +\alpha _{q}\epsilon _{t-q}^{2}+\beta _{1}\sigma _{t-1}^{2}+\cdots +\beta _{p}\sigma _{t-p}^{2}=\alpha _{0}+\sum _{i=1}^{q}\alpha _{i}\epsilon _{t-i}^{2}+\sum _{i=1}^{p}\beta _{i}\sigma _{t-i}^{2}$

Generally, when testing for heteroskedasticity in econometric models, the best test is the White test. However, when dealing with time series data, this means to test for ARCH errors (as described above) and GARCH errors (below).

EWMA is an alternative model in a separate class of exponential smoothing models. It can be an alternative to GARCH modelling as it has some attractive properties such as a greater weight upon more recent observations but also some drawbacks such as an arbitrary decay factor that introduce subjectivity into the estimation.

GARCH(p, q) model specification

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

1. Estimate the best fitting AR(q) model

   $y_{t}=a_{0}+a_{1}y_{t-1}+\cdots +a_{q}y_{t-q}+\epsilon _{t}=a_{0}+\sum _{i=1}^{q}a_{i}y_{t-i}+\epsilon _{t}$.

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

   $\rho ={{\sum _{t=i+1}^{T}({\hat {\epsilon }}_{t}^{2}-{\hat {\sigma }}_{t}^{2})({\hat {\epsilon }}_{t-1}^{2}-{\hat {\sigma }}_{t-1}^{2})} \over {\sum _{t=1}^{T}({\hat {\epsilon }}_{t}^{2}-{\hat {\sigma }}_{t}^{2})^{2}}}$

3. The asymptotic, that is for large samples, standard deviation of $\rho (i)$ is $1/{\sqrt {T}}$. 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 $\chi ^{2}$ distribution with n degrees of freedom if the squared residuals $\epsilon _{t}^{2}$ 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.

NGARCH

Nonlinear GARCH (NGARCH) also known as Nonlinear Asymmetric GARCH(1,1) (NAGARCH) was introduced by Engle and Ng in 1993.
$~\sigma _{t}^{2}=~\omega +~\alpha (~\epsilon _{t-1}-~\theta ~\sigma _{t-1})^{2}+~\beta ~\sigma _{t-1}^{2}$

$~\alpha ,~\beta \geq 0;~\omega >0$.
For stock returns, parameter $~\theta$ is usually estimated to be positive; in this case, it reflects the leverage effect, signifying that negative returns increase future volatility by a larger amount than positive returns of the same magnitude.

This model shouldn't be confused with the NARCH model, together with the NGARCH extension, introduced by Higgins and Bera in 1992.Template:Clarify

IGARCH

Integrated Generalized Autoregressive Conditional Heteroskedasticity IGARCH is a restricted version of the GARCH model, where the persistent parameters sum up to one, and therefore there is a unit root in the GARCH process. The condition for this is

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

EGARCH

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

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

where $g(Z_{t})=\theta Z_{t}+\lambda (|Z_{t}|-E(|Z_{t}|))$, $\sigma _{t}^{2}$ is the conditional variance, $\omega$, $\beta$, $\alpha$, $\theta$ and $\lambda$ are coefficients, and $Z_{t}$ 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.

Since $\log \sigma _{t}^{2}$ may be negative there are no (fewer) restrictions on the parameters.

GARCH-M

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 $~\epsilon _{t}$ is defined as

$~\epsilon _{t}=~\sigma _{t}~\times z_{t}$

QGARCH

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

In the example of a GARCH(1,1) model, the residual process $~\sigma _{t}$ is

$~\epsilon _{t}=~\sigma _{t}z_{t}$

where $z_{t}$ is i.i.d. and

$~\sigma _{t}^{2}=K+~\alpha ~\epsilon _{t-1}^{2}+~\beta ~\sigma _{t-1}^{2}+~\phi ~\epsilon _{t-1}$

GJR-GARCH

Similar to QGARCH, The Glosten-Chris Hughton-Runkle GARCH (GJR-GARCH) model by Glosten, Jagannathan and Runkle (1993) also models asymmetry in the ARCH process. The suggestion is to model $~\epsilon _{t}=~\sigma _{t}z_{t}$ 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 $~\epsilon _{t-1}\geq 0$, and $I_{t-1}=1$ if $~\epsilon _{t-1}<0$.

TGARCH model

The Threshold GARCH (TGARCH) model by Nikolai (2013) is similar to GJR GARCH, and 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 $~\epsilon _{t-1}^{+}=~\epsilon _{t-1}$ if $~\epsilon _{t-1}>0$, and $~\epsilon _{t-1}^{+}=0$ if $~\epsilon _{t-1}\leq 0$. Likewise, $~\epsilon _{t-1}^{-}=~\epsilon _{t-1}$ if $~\epsilon _{t-1}\leq 0$, and $~\epsilon _{t-1}^{-}=0$ if $~\epsilon _{t-1}>0$.

fGARCH

Hentschel's fGARCH model, 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.

References

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Further reading

• Bollerslev, Tim (1986). "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, 31:307-327
• Bollerslev, Tim (2008). Glossary to ARCH (GARCH), working paper
• Enders, W. (1995). Applied Econometrics Time Series, John-Wiley & Sons, 139-149, ISBN 0-471-11163-5
• Engle, Robert F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation", Econometrica 50:987-1008. (the paper which sparked the general interest in ARCH models)
• Engle, Robert F. (2001). "GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics", Journal of Economic Perspectives 15(4):157-168. (a short, readable introduction) Preprint
• Engle, R.F. (1995) ARCH: selected readings. Oxford University Press. ISBN 0-19-877432-X
• Gujarati, D. N. (2003) Basic Econometrics, 856-862
• Hacker, R. S. and Hatemi-J, A. (2005). A Test for Multivariate ARCH Effects, Applied Economics Letters, 12(7), 411–417.
• Nelson, D. B. (1991). "Conditional heteroskedasticity in asset returns: A new approach", Econometrica 59: 347-370.