In
econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
, the autoregressive conditional heteroskedasticity (ARCH) model is a
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
for
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
data that describes the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of the current
error term In mathematics and statistics, an error term is an additive type of error. Common examples include:
* errors and residuals in statistics, e.g. in linear regression
* the error term in numerical integration
In analysis, numerical integration ...
or
innovation
Innovation is the practical implementation of ideas that result in the introduction of new goods or services or improvement in offering goods or services. ISO TC 279 in the standard ISO 56000:2020 defines innovation as "a new or changed entity ...
as a function of the actual sizes of the previous time periods' error terms; often the variance is related to the squares of the previous
innovation
Innovation is the practical implementation of ideas that result in the introduction of new goods or services or improvement in offering goods or services. ISO TC 279 in the standard ISO 56000:2020 defines innovation as "a new or changed entity ...
s. The ARCH model is appropriate when the error variance in a time series follows an
autoregressive
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
(AR) model; if an
autoregressive moving average
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
(ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.
ARCH models are commonly employed in modeling
financial
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
that exhibit time-varying
volatility and
volatility clustering In finance, volatility clustering refers to the observation, first noted by Mandelbrot (1963), that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes." A quantitative manifes ...
, i.e. periods of swings interspersed with periods of relative calm. ARCH-type models are sometimes considered to be in the family of
stochastic volatility
In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name ...
models, although this is strictly incorrect since at time ''t'' the volatility is completely pre-determined (deterministic) given previous values.
Model specification
To model a time series using an ARCH process, let
denote the error terms (return residuals, with respect to a mean process), i.e. the series terms. These
are split into a stochastic piece
and a time-dependent standard deviation
characterizing the typical size of the terms so that
:
The random variable
is a strong
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
process. The series
is modeled by
:
,
:where
and
.
An ARCH(''q'') model can be estimated using
ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the prin ...
. A method for testing whether the residuals
exhibit time-varying heteroskedasticity using the
Lagrange multiplier test
In statistics, the score test assesses constraints on statistical parameters based on the gradient of the likelihood function—known as the '' score''—evaluated at the hypothesized parameter value under the null hypothesis. Intuitively, if th ...
was proposed by
Engle (1982). This procedure is as follows:
# Estimate the best fitting
autoregressive model
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
AR(''q'')
.
# Obtain the squares of the error
and regress them on a constant and ''q'' lagged values:
#:
#: where ''q'' is the length of ARCH lags.
#The
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
is that, in the absence of ARCH components, we have
for all
. The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated
coefficients must be significant. In a sample of ''T'' residuals under the null hypothesis of no ARCH errors, the test statistic ''T'R²'' follows
distribution with ''q'' degrees of freedom, where
is the number of equations in the model which fits the residuals vs the lags (i.e.
). 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.
GARCH
If an
autoregressive moving average model
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
(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
and ''q'' is the order of the ARCH terms
), following the notation of the original paper, is given by
Generally, when testing for heteroskedasticity in econometric models, the best test is the
White test
In statistics, the White test is a statistical test that establishes whether the variance of the errors in a regression model is constant: that is for homoskedasticity.
This test, and an estimator for heteroscedasticity-consistent standard erro ...
. However, when dealing with
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
data, this means to test for ARCH and GARCH errors.
Exponentially weighted
moving average
In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
(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.
GARCH(''p'', ''q'') model specification
The lag length ''p'' of a GARCH(''p'', ''q'') process is established in three steps:
# Estimate the best fitting AR(''q'') model
#:
.
# Compute and plot the autocorrelations of
by
#:
# The asymptotic, that is for large samples, standard deviation of
is
. Individual values that are larger than this indicate GARCH errors. To estimate the total number of lags, use the
Ljung–Box test
The Ljung–Box test (named for Greta M. Ljung and George E. P. Box) is a type of statistical test of whether any of a group of autocorrelations of a time series are different from zero. Instead of testing randomness at each distinct lag, it tes ...
until the value of these are less than, say, 10% significant. The Ljung–Box
Q-statistic
The Q-statistic is a test statistic output by either the Box-Pierce test or, in a modified version which provides better small sample properties, by the Ljung-Box test. It follows the chi-squared distribution. See also Portmanteau test.
The q ...
follows
distribution with ''n'' degrees of freedom if the squared residuals
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 In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables.
Particularly in econometrics, the conditional variance is also known as the scedastic function or ...
.
NGARCH
NAGARCH
Nonlinear Asymmetric GARCH(1,1) (NAGARCH) is a model with the specification:
:
,
:where
and
, which ensures the non-negativity and stationarity of the variance process.
For stock returns, parameter
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.
This model should not be confused with the NARCH model, together with the NGARCH extension, introduced by Higgins and Bera in 1992.
IGARCH
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 probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is ...
in the GARCH process. The condition for this is
.
EGARCH
The exponential generalized autoregressive conditional heteroskedastic (EGARCH) model by Nelson & Cao (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):
where
,
is the
conditional variance In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables.
Particularly in econometrics, the conditional variance is also known as the scedastic function or ...
,
,
,
,
and
are coefficients.
may be a
standard normal variable
A standard normal deviate is a normally distributed deviate. It is a realization of a standard normal random variable, defined as a random variable with expected value 0 and variance 1.Dodge, Y. (2003) The Oxford Dictionary of Statisti ...
or come from a
generalized error distribution
The generalized normal distribution or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. To dis ...
. The formulation for
allows the sign and the magnitude of
to have separate effects on the volatility. This is particularly useful in an asset pricing context.
Since
may be negative, there are no sign restrictions for the parameters.
GARCH-M
The GARCH-in-mean (GARCH-M) model adds a heteroskedasticity term into the mean equation. It has the specification:
The residual
is defined as:
QGARCH
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
is
where
is i.i.d. and
GJR-GARCH
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
where
is i.i.d., and
where
if
, and
if
.
TGARCH model
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 In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables.
Particularly in econometrics, the conditional variance is also known as the scedastic function or ...
:
where
if
, and
if
. Likewise,
if
, and
if
.
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.
COGARCH
In 2004,
Claudia Klüppelberg
Claudia Klüppelberg is a German mathematical statistician and applied probability theorist, known for her work in risk assessment and statistical finance. She is a professor emerita of mathematical statistics at the Technical University of Mu ...
, 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
:
:
and then to replace the strong white noise process
by the infinitesimal increments
of a
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...
, and the squared noise process
by the increments
, where
:
is the purely discontinuous part of the
quadratic variation In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.
Definition
Suppose that X_t is a real-valued sto ...
process of
. The result is the following system of
stochastic differential equations
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock pr ...
:
:
:
where the positive parameters
,
and
are determined by
,
and
. Now given some initial condition
, the system above has a pathwise unique solution
which is then called the continuous-time GARCH (COGARCH) model.
ZD-GARCH
Unlike GARCH model, the Zero-Drift GARCH (ZD-GARCH) model by Li, Zhang, Zhu and Ling (2018)
lets the drift term
in the first order GARCH model. The ZD-GARCH model is to model
, where
is i.i.d., and
The ZD-GARCH model does not require
, and hence it nests the
Exponentially weighted moving average
In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
(EWMA) model in "
RiskMetrics
The RiskMetrics variance model (also known as exponential smoother) was first established in 1989, when Sir Dennis Weatherstone, the new chairman of J.P. Morgan, asked for a daily report measuring and explaining the risks of his firm. Nearly f ...
". Since the drift term
, 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
and
can be estimated by the generalized
QMLE In statistics a quasi-maximum likelihood estimate (QMLE), also known as a pseudo-likelihood estimate or a composite likelihood estimate, is an estimate of a parameter ''θ'' in a statistical model that is formed by maximizing a function that is rel ...
method.
Spatial GARCH
Spatial GARCH processes by Otto, Schmid and Garthoff (2018)
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
and
:
where
denotes the
-th spatial location and
refers to the
-th entry of a spatial weight matrix and
for
. The spatial weight matrix defines which locations are considered to be adjacent.
Gaussian process-driven GARCH
In a different vein, the machine learning community has proposed the use of Gaussian process regression models to obtain a GARCH scheme.
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.
References
Further reading
*
*
* ''(the paper which sparked the general interest in ARCH models)''
*
* ''(a short, readable introduction)''
*
*
*
{{DEFAULTSORT:Autoregressive Conditional Heteroskedasticity
Nonlinear time series analysis
Autocorrelation