Nelson (1991) introduced a number of refinements on the GARCH model, which we’ll examine separately, as you can choose them separately on the GARCH instruction. The first was to model the log of the variance, rather than the level. This avoids any problem that could arise because of negative coefficients in standard ARCH and GARCH models, and gives rise to an EGARCH model. The form for the variance that we use in RATS is

\begin{equation} \begin{array}{c}\log h_t = c_0 + a_1 \frac{{\left| {u_{t - 1} } \right|}}{{\sqrt {h_{t - 1} } }} + a_2 \frac{{\left| {u_{t - 2} } \right|}}{{\sqrt {h_{t - 2} } }} + \ldots + a_q \frac{{\left| {u_{t - q} } \right|}}{{\sqrt {h_{t - q} } }} + \\ \,\,\,\,\,\,\,b_1 \log h_{t - 1} + \ldots + b_p \log h_{t - p} \\ \end{array} \label{eq:garch_egarch} \end{equation}

A few things to note about this. First, it doesn’t include “asymmetry” effects, where positive and negative \(u\)’s have different effects. Nelson also used an alternative distribution to the normal (the GED).

 

 

EGARCH models are often parameterized using

\begin{equation} \begin{array}{c}\log h_t = c_0 + a_1 \frac{{\left| {u_{t - 1} } \right|}}{{\sqrt {h_{t - 1} } }} + a_2 \frac{{\left| {u_{t - 2} } \right|}}{{\sqrt {h_{t - 2} } }} + \ldots + a_q \frac{{\left| {u_{t - q} } \right|}}{{\sqrt {h_{t - q} } }} + \\ \,\,\,\,\,\,\,b_1 \log h_{t - 1} + \ldots + b_p \log h_{t - p} \\ \end{array} \end{equation}

which, by definition, has mean 0. However, the expected value can be a complicated function of the distribution, and has no effect on the volatility as it just washes into the the constant term in \eqref{eq:garch_egarch}.

 

To estimate a GARCH(1,1) with an exponential variance model, use

 

garch(p=1,q=1,exp) / dlogdm

 

 

Output

EGARCH Model - Estimation by BFGS

Convergence in    42 Iterations. Final criterion was  0.0000055 <=  0.0000100

 

Dependent Variable DLOGDM

Usable Observations                      1866

Log Likelihood                     -2066.0184

 

    Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  Mean(DLOGDM)                 -0.024192164  0.004368412     -5.53798  0.00000003

2.  C                            -0.191334378  0.021973796     -8.70739  0.00000000

3.  A                             0.220275484  0.025181585      8.74748  0.00000000

4.  B                             0.963515999  0.009443625    102.02819  0.00000000

 

Because the lagged residuals come into the model in “standardized” form, the variance persistence of the EGARCH comes solely through the \(b\) coefficients.

 

There is nothing wrong with a negative C, since the variance recursion is in log form.

 

Copyright © 2025 Thomas A. Doan