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UV GARCH Asymmetry

It has long been recognized that equity returns exhibit asymmetrical conditional variance behavior, that is, that positive values of the residuals have a different effect than negative ones. That won’t be captured by any of the models so far, since the residual always enters the variance as a square or absolute value. The EGARCH model from Nelson (1991) adds an extra term to provide for this; an EGARCH(1,1) would have the variance evolving according to

\begin{equation} \log h_t = c_0 + a_1 \frac{{\left| {u_{t - 1} } \right|}}{{\sqrt {h_{t - 1} } }} + b_1 \log h_{t - 1} + d_1 \frac{{u_{t - 1} }}{{\sqrt {h_{t - 1} } }} \end{equation}

There are many equivalent ways to introduce the asymmetric effect into the model—we choose this one to maintain a similar form to that used in other models you can estimate with GARCH. Note that, with this parameterization, a negative value of \(d_1\) means that negative residuals tend to produce higher variances in the immediate future. (This sign convention is used only with the EGARCH).

An analogous change to the standard GARCH model was proposed by Glosten, et. al. (1993) and is known as GJR, after the originators. Again, looking at a GARCH(1,1), the GJR variance model is

\begin{equation} h_t = c_0 + a_1 u_{t - 1}^2 + b_1 h_{t - 1} + d_1 u_{t - 1}^2 I_{u \le 0} \left( {u_{t - 1} } \right) \end{equation}

where \(I\) is an indicator function, in this case, for \(u<0\). With this formulation, a positive value of \(d_1\) means negative residuals tend to increase the variance more than positive ones.

To estimate with asymmetric effects, just add the ASYMMETRIC option:

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

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

Output

Asymmetry adds a D coefficient to the output, or D{1},...,D{q} if you have chosen a higher value for the number of "ARCH" terms. Again, note that the EGARCH and standard GARCH are using different sign conventions (keeping those from the original papers).

EGARCH Model - Estimation by BFGS

Convergence in 47 Iterations. Final criterion was 0.0000014 <= 0.0000100

Dependent Variable DLOGDM

Usable Observations 1866

Log Likelihood -2065.1214

Variable Coeff Std Error T-Stat Signif

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

1. Mean(DLOGDM) -0.027982045 0.016275648 -1.71926 0.08556732

2. C -0.184511441 0.024287836 -7.59687 0.00000000

3. A 0.215085921 0.028043634 7.66969 0.00000000

4. B 0.967687031 0.009281676 104.25779 0.00000000

5. D -0.017257427 0.012664170 -1.36270 0.17297801

GARCH Model - Estimation by BFGS

Convergence in 24 Iterations. Final criterion was 0.0000004 <= 0.0000100

Dependent Variable DLOGDM

Usable Observations 1866

Log Likelihood -2066.7407

Variable Coeff Std Error T-Stat Signif

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

1. Mean(DLOGDM) -0.027168020 0.018387564 -1.47752 0.13953592

2. C 0.013681319 0.004828922 2.83320 0.00460840

3. A 0.089768070 0.019190241 4.67780 0.00000290

4. B 0.877557723 0.019279382 45.51794 0.00000000

5. D 0.032974948 0.020725016 1.59107 0.11159386