Autoregressive conditional heteroskedasticity (ARCH) is the condition that one or more data points in a series for which the variance of the current error term or innovation is 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. In econometrics, ARCH models are used to characterize and model time series. A variety of other acronyms are applied to particular structures that have a similar basis.
Contents
- ARCHq model specification
- GARCH
- GARCHp q model specification
- NGARCH
- IGARCH
- EGARCH
- GARCH M
- QGARCH
- GJR GARCH
- TGARCH model
- fGARCH
- COGARCH
- References
ARCH models are commonly employed in modeling financial time series that exhibit time-varying volatility clustering, 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 models, although this is strictly incorrect since at time t the volatility is completely pre-determined (deterministic) given previous values.
ARCH(q) model specification
To model a time series using an ARCH process, let
The random variable
where
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:
- Estimate the best fitting autoregressive model AR(q)
y t = a 0 + a 1 y t − 1 + ⋯ + a q y t − q + ϵ t = a 0 + ∑ i = 1 q a i y t − i + ϵ t - Obtain the squares of the error
ϵ ^ 2 ϵ ^ t 2 = α ^ 0 + ∑ i = 1 q α ^ i ϵ ^ t − i 2 - The null hypothesis is that, in the absence of ARCH components, we have
α i = 0 for alli = 1 , ⋯ , q . The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimatedα i χ 2 T ′ T ′ = 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.
GARCH
If an autoregressive moving average model (ARMA model) is assumed for the error variance, the model is a generalized autoregressive conditional heteroscedasticity(GARCH) model.
In that case, the GARCH (p, q) model (where p is the order of the GARCH terms
Generally, when testing for heteroscedasticity 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.
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
y t = a 0 + a 1 y t − 1 + ⋯ + a q y t − q + ϵ t = a 0 + ∑ i = 1 q a i y t − i + ϵ t - Compute and plot the autocorrelations of
ϵ 2 ρ = ∑ t = i + 1 T ( ϵ ^ t 2 − σ ^ t 2 ) ( ϵ ^ t − 1 2 − σ ^ t − 1 2 ) ∑ t = 1 T ( ϵ ^ t 2 − σ ^ t 2 ) 2 - The asymptotic, that is for large samples, standard deviation of
ρ ( i ) is1 / T χ 2 ϵ t 2
NGARCH
Nonlinear GARCH (NGARCH) is also known as Nonlinear Asymmetric GARCH(1,1) (NAGARCH).
For stock returns, parameter
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 heteroscedasticity I GARCH 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. The condition for this is
EGARCH
The exponential generalized autoregressive conditional heteroskedastic (EGARCH) model by Nelson (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):
where
Since
GARCH-M
The GARCH-in-mean (GARCH-M) model adds a heteroscedasticity term into the mean equation. It has the specification:
The residual
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
where
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
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:
where
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, 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
is the purely discontinuous part of the quadratic variation process of
where the positive parameters