GARCH, IGARCH, EGARCH, and GARCH-M Models

The AUTOREG Procedure

GARCH, IGARCH, EGARCH, and GARCH-M Models

Consider the series y_t, which follows the GARCH process. The conditional distribution of the series Y for time t is written

$y_{t}| {\Psi}_{t-1} {\sim} \rm{N}(0, h_{t})$

where ${\Psi}_{t-1}$ denotes all available information at time t-1. The conditional variance h_t is

$h_{t} = {\omega} + \sum_{i=1}^q{{\alpha}_{i} y_{t-i}^2} + \sum_{j=1}^p{{\gamma}_{j} h_{t-j}}$

where

$p{\ge}0, q\gt$

${\omega}\gt, {\alpha}_{i}{\ge}0, {\gamma}_{j}{\ge}0$

The GARCH(p,q) model reduces to the ARCH(q) process when p=0. At least one of the ARCH parameters must be nonzero (q > 0). The GARCH regression model can be written

$y_{t} = x_{t}' {\beta} + {\epsilon}_{t}$

${\epsilon}_{t} = \sqrt{ h_{t}} e_{t}$

$h_{t} = {\omega} + \sum_{i=1}^q{{\alpha}_{i} {\epsilon}_{t-i}^2} + \sum_{j=1}^p{{\gamma}_{j} h_{t-j}}$

where ${ e_{t} {\sim} \rm{IN}(0, 1)}$ .

In addition, you can consider the model with disturbances following an autoregressive process and with the GARCH errors. The AR(m)-GARCH(p,q) regression model is denoted

$y_{t} = x_{t}' {\beta} + {\nu}_{t}$

${\nu}_{t} = {\epsilon}_{t} - {\varphi}_{1} {\nu}_{t-1} -{ ... }- {\varphi}_{m} {\nu}_{t-m}$

${\epsilon}_{t} = \sqrt{ h_{t}} e_{t}$

$h_{t} = {\omega} + \sum_{i=1}^q{{\alpha}_{i} {\epsilon}_{t-i}^2} + \sum_{j=1}^p{{\gamma}_{j} h_{t-j}}$

GARCH Estimation with Nelson-Cao Inequality Constraints

The GARCH(p,q) model is written in ARCH( ${\infty}$ ) form as

$h_{t} &=& ( 1-\sum_{j=1}^p {\gamma}_{j}B^j )^{-1} [{\ssbeleven {\omega} + \sum... ...cr &=& {\omega}^{*} + \sum_{i=1}^{{\infty}}{{\phi}_{i} {\epsilon}_{t-i}^2}$

where B is a backshift operator. Therefore, ${h_{t}{\ge}0}$ if ${ {\omega}^{*}{\ge}0}$ and ${{\phi}_{i}{\ge}0, {\forall} i}$ . Assume that the roots of the following polynomial equation are inside the unit circle:

$\sum_{j=0}^p{-{\gamma}_{j}Z^{p-j}}$

where ${{\gamma}_{0}=-1}$ and Z is a complex scalar. $\sum_{j=0}^p{-{\gamma}_{j}Z^{p-j}}$ and ${\sum_{i=1}^q{{\alpha}_{i}Z^{q-i}}}$ , do not share common factors. Under these conditions, ${{| {\omega}^{*}|}\lt{\infty}}$ , ${{|{\phi}_{i}|} \lt {\infty}}$ , and these coefficients of the ARCH( ${\infty}$ ) process are well defined.

Define n=max(p,q). The coefficient ${{\phi}_{i}}$ is written

${\phi}_{0} &=& {\alpha}_{1} \cr {\phi}_{1} &=& {\gamma}_{1}{\phi}_{0}+{\alpha}... ... ... }+{\gamma}_{n}{\phi}_{k-n} \hspace*{.15in}\rm{for}\hspace*{.1in}k{\ge}n$

where ${{\alpha}_{i}=0}$ for i>q and ${{\gamma}_{j}=0}$ for j>p.

Nelson and Cao (1992) proposed the finite inequality constraints for GARCH(1,q) and GARCH(2,q) cases. However, it is not straightforward to derive the finite inequality constraints for the general GARCH(p,q) model.

For the GARCH(1,q) model, the nonlinear inequality constraints are

${\omega} &{\ge}& 0 \cr {\gamma}_{1} &{\ge}& 0 \cr {\phi}_{k} &{\ge}& 0\hspace*{.15in}\rm{for}\hspace*{.1in}k=0,1,{ ... },q-1$

For the GARCH(2,q) model, the nonlinear inequality constraints are

${\Delta}_{i} &{\in}& R \hspace*{.15in}\rm{for}\hspace*{.1in}i=1,2 \cr {\omega}^... ...\cr {\phi}_{k} &{\ge}& 0\hspace*{.15in}\rm{for}\hspace*{.1in}k=0,1,{ ... },q$

where ${{\Delta}_{1}}$ and ${{\Delta}_{2}}$ are the roots of ${(Z^2-{\gamma}_{1}Z-{\gamma}_{2})}$ .

For the GARCH(p,q) model with p > 2, only max(q-1,p)+1 nonlinear inequality constraints ( ${{\phi}_{k} {\ge} 0}$ for k=0 to max(q-1,p)) are imposed, together with the in-sample positivity constraints of the conditional variance h_t.

IGARCH and Stationary GARCH Model

The condition ${\sum_{i=1}^q{{\alpha}_{i}}+\sum_{j=1}^p{{\gamma}_{j}} \lt 1}$ implies that

the GARCH process is weakly stationary since the mean, variance, and autocovariance are finite and constant over time. However, this condition is not sufficient for weak stationarity in the presence of autocorrelation. For example, the stationarity condition for an AR(1)-GARCH(p,q) process is

$\frac{1}{1- {\varphi}^2_{1}} \sum_{i=1}^q{{\alpha}_{i}} + \sum_{j=1}^p{{\gamma}_{j}} \lt 1$

When the GARCH process is stationary, the unconditional variance of ${{\epsilon}_{t}}$ is computed as

$V({\epsilon}_{t}) = \frac{{\omega}}{(1- \sum_{i=1}^q{{\alpha}_{i}}-\sum_{j=1}^p{{\gamma}_{j}})}$

where ${ {\epsilon}_{t}=\sqrt{ h_{t}} e_{t}}$ and h_t is the

GARCH(p,q) conditional variance.

Sometimes, the multistep forecasts of the variance do not approach the unconditional variance when the model is integrated in variance; that is, ${\sum_{i=1}^q{{\alpha}_{i}} + \sum_{j=1}^p{{\gamma}_{j}} = 1}$ .

The unconditional variance for the IGARCH model does not exist. However, it is interesting that the IGARCH model can be strongly stationary even though it is not weakly stationary. Refer to Nelson (1990) for details.

EGARCH Model

The EGARCH model was proposed by Nelson (1991). Nelson and Cao (1992) argue that the nonnegativity constraints in the linear GARCH model are too restrictive. The GARCH model imposes the nonnegative constraints on the parameters, ${\alpha}_{i}$ and ${\gamma}_{j}$ , while there are no restrictions on these parameters in the EGARCH model. In the EGARCH model, the conditional variance, h_t, is an asymmetric function of lagged disturbances ${\epsilon}_{t-i}$ :

${\ln}( h_{t}) = {\omega} + \sum_{i=1}^q{{\alpha}_{i}g( z_{t-i})} + \sum_{j=1}^p{{\gamma}_{j}{\ln}( h_{t-j})}$

where

$g( z_{t}) = {\theta} z_{t}+{\gamma}[{| z_{t}|} -E{| z_{t}|}]$

$z_{t} = {\epsilon}_{t}/\sqrt{ h_{t}}$

The coefficient of the second term in g( z_t) is set to be 1 ( ${\gamma}$ =1) in our formulation. Note that ${E{| z_{t}|} = (2/{\pi})^{1/2}}$ if ${ z_{t} {\sim} \rm{N}(0,1)}$ . The properties of the EGARCH model are summarized as follows:

The function g( z_t) is linear in z_t with slope coefficient ${{\theta}+1}$ if z_t is positive while g( z_t) is linear in z_t with slope coefficient ${{\theta}-1}$ if z_t is negative
Suppose that ${{\theta}=0}$ . Large innovations increase the conditional variance if ${{\ssbeleven {| z_{t}|}- E{| z_{t}|}} \gt 0}$ and decrease
the conditional variance if ${{\ssbeleven {| z_{t}|}- E{| z_{t}|}} \lt 0}$ .
Suppose that ${{\theta}\lt 1}$ . The innovation in variance, g( z_t), is positive if the innovations z_t are less than ${(2/{\pi})^{1/2}/({\theta}-1)}$ . Therefore, the negative innovations in returns, ${{\epsilon}_{t}}$ , cause the innovation to the conditional variance to be positive if ${\theta}$ is much less than 1.

GARCH-in-Mean

The GARCH-M model has the added regressor that is the conditional standard deviation:

$y_{t} = x_{t}' {\beta} + {\delta}\sqrt{ h_{t}} + {\epsilon}_{t}$

${\epsilon}_{t} = \sqrt{ h_{t}} e_{t}$

where h_t follows the ARCH or GARCH process.

Maximum Likelihood Estimation

The family of GARCH models are estimated using the maximum likelihood method. The log-likelihood function is computed from the product of all conditional densities of the prediction errors.

When e_t is assumed to have a standard normal distribution ( ${e_{t} {\sim} \rm{N}(0,1)}$ ), the likelihood function is given by

$l = \sum_{t=1}^N{\frac{1}2[-{\ln}(2{\pi}) -{\ln}( h_{t})- {{\epsilon}_{t}^2 \over h_{t}}]}$

where ${ {\epsilon}_{t}= y_{t}- x_{t}'{\beta}}$ and h_t is the conditional variance. When the GARCH(p,q)-M model is estimated, ${ {\epsilon}_{t}= y_{t}- x_{t}'{\beta}- {\delta}\sqrt{ h_{t}}}$ . When there are no regressors, the residuals ${{\epsilon}_{t}}$ are denoted as y_t or ${y_{t} - {\delta}\sqrt{ h_{t}}}$ .

If e_t has the standardized Student's t distribution the log likelihood function for the conditional t distribution is

${\ell} = \sum_{t=1}^{N} \Biggl[ {\log}({\Gamma}(\frac{{\nu}+1}2)) -{\log}({\Gamma}(\frac{{\nu}}2)) -\frac{1}2{\log}(({\nu}-2)h_{t})$

where ${{\Gamma}({\cdot})}$ is the gamma function and ${{\nu}}$ is the degree of freedom ( ${{\nu}\gt 2}$ ). Under the conditional t distribution, the additional parameter $1/{\nu}$ is estimated. The log likelihood function for the conditional t distribution converges to the log likelihood function of the conditional normal GARCH model as ${1/{\nu} {arrow} 0}$ .

The likelihood function is maximized via either the dual quasi-Newton or trust region algorithm. The default is the dual quasi-Newton algorithm. The starting values for the regression parameters ${{\beta}}$ are obtained from the OLS estimates. When there are autoregressive parameters in the model, the initial values are obtained from the Yule-Walker estimates. The starting value 1.0^-6 is used for the GARCH process parameters.

The variance-covariance matrix is computed using the Hessian matrix. The dual quasi-Newton method approximates the Hessian matrix while the quasi-Newton method gets an approximation of the inverse of Hessian. The trust region method uses the Hessian matrix obtained using numerical differentiation. When there are active constraints, that is, ${q({\theta})=0}$ , the variance-covariance matrix is given by

$V(\hat{{\theta}}) = H^{-1}[I - Q' (Q H^{-1}Q')^{-1}Q H^{-1}]$

where ${H=-{\partial}^2l/{\partial}{\theta}{\partial}{\theta}'}$ and ${Q = {\partial}q({\theta})/{\partial}{\theta}'}$ . Therefore, the variance-covariance matrix without active constraints reduces to ${V(\hat{{\theta}}) = H^{-1}}$ .

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