Autoregressive conditional heteroskedasticity
Encyclopedia
In econometrics
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...
, 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 terms will have a characteristic size, or variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
. In particular ARCH models assume the variance of the current error term
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
or innovation
Innovation (signal processing)
In time series analysis — as conducted in statistics, signal processing, and many other fields — the innovation is the difference between the observed value of a variable at time t and the optimal forecast of that value based on information available prior to time t...
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
Mathematical finance
Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
time series
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
that exhibit time-varying volatility
Volatility (finance)
In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...
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
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
where
is a random variable drawn from a Gaussian distribution centered at 0 with standard deviation equal to 1. (i.e. 
) and wherethe series
are modeled by
and where
and 
.An ARCH(q) model can be estimated using ordinary least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
. A methodology to test for the lag length of ARCH errors using the Lagrange multiplier test was proposed by Engle
Robert F. Engle
Robert Fry Engle III is an American economist and the winner of the 2003 Nobel Memorial Prize in Economic Sciences, sharing the award with Clive Granger, "for methods of analyzing economic time series with time-varying volatility ".-Biography:Engle was born in Syracuse, New York and went on to...
(1982). This procedure is as follows:
- Estimate the best fitting autoregressive modelAutoregressive modelIn statistics and signal processing, an autoregressive model is a type of random process which is often used to model and predict various types of natural phenomena...
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 hypothesisNull hypothesisThe practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
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 TR² follows 
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 modelAutoregressive moving average modelIn statistics and signal processing, autoregressive–moving-average models, sometimes called Box–Jenkins models after the iterative Box–Jenkins methodology usually used to estimate them, are typically applied to autocorrelated time series data.Given a time series of data Xt, the ARMA model is a...
. If TR² is smaller than the Chi-square table value, we do not reject the null hypothesis.
GARCH
If an autoregressive moving average modelAutoregressive moving average model
In statistics and signal processing, autoregressive–moving-average models, sometimes called Box–Jenkins models after the iterative Box–Jenkins methodology usually used to estimate them, are typically applied to autocorrelated time series data.Given a time series of data Xt, the ARMA model is a...
(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
and q is the order of the ARCH terms 
) is given by
Generally, when testing for heteroskedasticity
Heteroskedasticity
In statistics, a collection of random variables is heteroscedastic, or heteroskedastic, if there are sub-populations that have different variabilities than others. Here "variability" could be quantified by the variance or any other measure of statistical dispersion...
in econometric models, the best test is the White test
White test
In statistics, the White test is a statistical test that establishes whether the residual variance of a variable in a regression model is constant: that is for homoscedasticity....
. However, when dealing with time series
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
data, this means to test for ARCH errors (as described above) and GARCH errors (below).
Prior to GARCH there was EWMA which has now been superseded by GARCH, although some people utilise both.
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 testLjung-Box testThe Ljung–Box test is a type of statistical test of whether any of a group of autocorrelations of a time series are different from zero...
until the value of these are less than, say, 10% significant. The Ljung-Box Q-statisticQ-statisticThe 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...
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 there are existing such errors in the conditional varianceConditional varianceIn probability theory and statistics, a conditional variance is the variance of a conditional probability distribution. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function...
.
NGARCH
Nonlinear GARCH (NGARCH) also known as Nonlinear Asymmetric GARCH(1,1) (NAGARCH) was introduced by Engle and Ng in 1993.
.For stock returns, parameter
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.
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 rootUnit root
In time series models in econometrics , a unit root is a feature of processes that evolve through time that can cause problems in statistical inference if it is not adequately dealt with....
in the GARCH process. The condition for this is
.EGARCH
The exponential general autoregressive conditional heteroskedastic (EGARCH) model by Nelson (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):
where
, 
is the conditional varianceConditional variance
In probability theory and statistics, a conditional variance is the variance of a conditional probability distribution. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function...
,
, 
, 
, 
and 
are coefficients, and 
may be a standard normal variable or come from a generalized error distribution. 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 (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:
The residual
is defined as
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
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, and the specification is one on conditional standard deviation instead of conditional varianceConditional variance
In probability theory and statistics, a conditional variance is the variance of a conditional probability distribution. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function...
:
where
if 
, and 
if 
. Likewise, 
if 
, and 
if 
.