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GARCH literature often lacks consensus regarding the exact definition of any particular class of GARCH model. Software vendors, researchers, and references often disagree about the exact functional form and/or parameter constraints of almost all GARCH models. The following information may help reconcile some of these discrepancies.
Although the functional form of a GARCH(P,Q) model, as described in Equation 2-4, is standard, alternative positivity constraints exist. However, these alternatives involve additional nonlinear inequalities that are difficult to impose in practice. They also do not affect the GARCH(1,1) model, which is by far the most common model. In contrast, the standard linear positivity constraints imposed by the GARCH Toolbox™ software are commonly used, and are straightforward to implement.
Many references and software vendors refer to the GJR(P,Q) model, as described in Equation 2-5, as a TGARCH, or Threshold GARCH, model. However, others make a clear distinction between GJR(P,Q) and TGARCH(P,Q) models: a GJR(P,Q) model is a recursive equation for the conditional variance, and a TGARCH(P,Q) model is the identical recursive equation for the conditional standard deviation (see, for example, Hamilton [22] page 669, Bollerslev, et. al. [8] page 2970). Furthermore, additional discrepancies exist regarding whether to allow both negative and positive innovations to affect the conditional variance process. The GJR(P,Q) model included in the GARCH Toolbox software is relatively standard.
The GARCH Toolbox software parameterizes GARCH(P,Q) and GJR(P,Q) models, as described in Equation 2-4 and Equation 2-5, including constraints, in a way that allows you to interpret a GJR(P,Q) model as an extension of a GARCH(P,Q) model. You can also interpret a GARCH(P,Q) model as a restricted GJR(P,Q) model with zero leverage terms. This latter interpretation is useful for estimation and hypothesis testing via likelihood ratios.
For GARCH(P,Q) and GJR(P,Q) models, the lag lengths P and Q, and the magnitudes of the coefficients Gi and Aj, determine the extent to which disturbances persist. These values then determine the minimum amount of presample data needed to initiate the simulation and estimation processes. The Gi terms capture persistence in EGARCH models.
Although the functional form of an EGARCH(P,Q) model (Equation 2-6) is relatively standard, it is not the same as Nelson's original (see Nelson [28]). Many forms of EGARCH(P,Q) models exist. Another form is
This EGARCH(P,Q) model form appears to offer an advantage. It does not explicitly make assumptions about the conditional probability distribution. That is, it does not assume that the distribution of zt = (εt/σt) is Gaussian or Student's t. However, this is not entirely true. Though the EGARCH(P,Q) model does not explicitly assume a distribution in this equation, such an assumption is required for forecasting and Monte Carlo simulation in the absence of user-specified presample data. In fact, you can easily rearrange this equation to highlight the probability distribution.
The GARCH Toolbox software implements the form of the EGARCH(P,Q) model described by Equation 2-6 because this model closely resembles Nelson's original form.
Although EGARCH(P,Q) models require no parameter constraints to ensure positive conditional variances, stationarity constraints are necessary. The GARCH Toolbox software treats EGARCH(P,Q) models as ARMA(P,Q) models for the logarithm of the conditional variance. Therefore, this toolbox imposes nonlinear constraints on the Gi coefficients to ensure that the eigenvalues of the characteristic polynomial are all inside the unit circle. (See, for example, page 2969 of Bollerslev, Engle, and Nelson [8], and page 12 of Bollerslev, Chou, and Kroner [7].)
Consider the EGARCH(P,Q) and GJR(P,Q) models, as described in Equation 2-6 and Equation 2-5. These asymmetric models capture the leverage effect, or negative correlation, between asset returns and volatility. Both models include leverage terms that explicitly take into account the sign and magnitude of the innovation noise term. Although both models are designed to capture the leverage effect, the way in which they do so differs.
For EGARCH(P,Q) models, the leverage coefficients Li apply to the actual innovations εt-1. For GJR(P,Q) models, the leverage coefficients enter the model through a Boolean indicator, or dummy, variable. Therefore, if the leverage effect does indeed hold, the leverage coefficientsLi should be negative for EGARCH(P,Q) models and positive for GJR(P,Q) models. This is in contrast to GARCH(P,Q) models, which ignore the sign of the innovation.
Although GARCH(P,Q) and GJR(P,Q) models include terms
related to the model innovations,
, EGARCH(P,Q) models include terms
related to the standardized innovations, zt = (εt/σt), such that zt acts as the forcing variable for both the conditional variance and
the error. In this respect, EGARCH(P,Q) models are fundamentally unique.
Generally, there are no asymmetries in foreign-exchange rates. Therefore, asymmetric EGARCH(P,Q) and GJR(P,Q) conditional variance models are often inappropriate for modeling such return series.
This general ARMAX(R,M,Nx) model for the conditional mean
 | (2-2) |
applies to all variance models with autoregressive coefficients {Φi}, moving average coefficients {Φj}, innovations {εt}, and returns {yt}.
X is an explanatory regression matrix in which each column is a time series. X(t, k) denotes the tth row and κth column of this matrix.
The eigenvalues {λi} associated with the characteristic AR polynomial
must lie inside the unit circle to ensure stationarity. Similarly, the eigenvalues associated with the characteristic MA polynomial
must lie inside the unit circle to ensure invertibility.
The conditional variance of the innovations,
, is by definition
 | (2-3) |
The key insight of GARCH lies in the distinction between conditional and unconditional variances of the innovations process {εt}. The term conditional implies explicit dependence on a past sequence of observations. The term unconditional applies more to long-term behavior of a time series, and assumes no explicit knowledge of the past.
The various GARCH models characterize the conditional distribution ofεt by imposing alternative parameterizations to capture serial dependence on the conditional variance of the innovations. About Conditional Mean and Variance Models further defines the conditional variance models.
The general GARCH(P,Q) model for the conditional variance of innovations is
 | (2-4) |
with constraints
κ > 0
Gi ≥ 0 i = 1,2, ..., P
Aj ≥ 0 j = 1,2, ..., Q
The basic GARCH(P,Q) model is a symmetric variance process, in that it ignores the sign of the disturbance.
The general GJR(P,Q) model for the conditional variance of the innovations with leverage terms is
 | (2-5) |
where
St-j = 1 if εt-j < 0
St-j = 0 otherwise,
and
κ > 0
Gi ≥ 0 i = 1,2, ..., P
Aj ≥ 0 j = 1,2, ..., Q
Aj + Lj ≥ 0 j = 1,2, ..., Q
The general EGARCH(P,Q) model for the conditional variance of the innovations, with leverage terms and an explicit probability distribution assumption, is
 | (2-6) |
where
for the Gaussian distribution, and
for the Student's t distribution, with degrees of freedom ν > 2.
The GARCH Toolbox software treats EGARCH(P,Q) models as
ARMA(P,Q) models for log
. Thus, it includes the stationarity constraint for EGARCH(P,Q) models
by ensuring that the eigenvalues of the characteristic polynomial
are inside the unit circle.
EGARCH models are fundamentally different from GARCH and GJR models in that the standardized innovation, zt, serves as the forcing variable for both the conditional variance and the error. GARCH and GJR models allow for volatility clustering (persistence) via a combination of the Gi and Aj terms. The Gi terms capture persistence in EGARCH models.
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