GARCHUVFLEX.RPF is an example of estimation of a univariate GARCH model with standard variance recursions, but non-standard densities. This uses specialized versions of procedures for the skew-t and skew-GED distributions (%LOGSKEWTGARCH and %LOGSKEWGEDGARCH).
The data used are log returns in the foreign exchange market, using USD/DM. As recommended, these are scaled up by 100.
open data garch.asc
data(format=free,org=columns) 1 1867 bp cd dm jy sf
*
set dlogdm = 100*log(dm/dm{1})
This first estimates a GARCH(1,1) model with AR(1) and t-distributed errors. (Output).
garch(p=1,q=1,reg,distrib=t) / dlogdm
# constant dlogdm{1}
And then the same with GED errors (Output)
garch(p=1,q=1,reg,distrib=ged) / dlogdm
# constant dlogdm{1}
For the non-standard densities, GARCH with the DENSITY and PARMS options is needed. The DENSITY option provides a function which takes the time t variance and residual (both computed using the standard GARCH recursion) and returns the log density conditional on the current values of the parameters in the PARMSET provided by the PARMS option. Those parameters need to be hard coded into the density function, so you need to use those specific names in your PARMSET.
The skew-t example uses the %LOGSKEWTGARCH function, which uses %SkewT_eta (degrees of freedom) and %SkewT_lambda (skew parameter) as its shape parameters.
nonlin(parmset=skewtparms) %SkewT_eta %SkewT_lambda
compute %SkewT_eta=4.0,%SkewT_lambda=0.0
*
garch(p=1,q=1,reg,density=%logskewtgarch,parms=skewtparms,title="GARCH with Skew-t Errors") / dlogdm
# constant dlogdm{1}
(%LOGSKEWTGARCH differs from a standard skew-t by taking the variance of the distribution as an input, rather than the variance of the underlying Normal).
You need to provide guess values for the added parameters. 0 is an obvious guess for the skew parameter. For the degrees of freedom, you can either use the shape parameter from a standard GARCH with t errors (roughly 8), or a generic number such as 4.0 which starts with relatively fat tails. (Output).
The added parameters follow the standard GARCH parameters. Here, we see a modestly better fit than the standard t, but probably not enough to merit adding the skew parameter.
The skew-GED is done with:
nonlin(parmset=skewgedparms) %SkewGED_k %SkewGED_lambda
compute %SkewGED_k=2.0,%SkewGED_lambda=0.0
garch(p=1,q=1,reg,density=%logskewgedgarch,parms=skewgedparms,title="GARCH with Skew-GED Errors") / dlogdm
# constant dlogdm{1}
Again, you specifically need the %SkewGED_k (shape) and %SkewGED_lambda parameters in the PARMSET. (Output). Again, the improvement by adding the skew effect is fairly modest.
Full Program
*
* GARCHUVFLEX.RPF
* Examples of estimates of univariate GARCH models using GARCH with the
* DENSITY and PARMSET options for non-standard densities applied to
* standard GARCH recursions for the variance.
*
open data garch.asc
data(format=free,org=columns) 1 1867 bp cd dm jy sf
*
set dlogdm = 100*log(dm/dm{1})
*
* Estimate a GARCH(1,1) with an AR(1) mean model and t errors
*
garch(p=1,q=1,reg,distrib=t) / dlogdm
# constant dlogdm{1}
*
* Same model with GED errors
*
garch(p=1,q=1,reg,distrib=ged) / dlogdm
# constant dlogdm{1}
*
* Estimate with skew-t
*
source logskewtgarch.src
nonlin(parmset=skewtparms) %SkewT_eta %SkewT_lambda
compute %SkewT_eta=4.0,%SkewT_lambda=0.0
*
garch(p=1,q=1,reg,density=%logskewtgarch,parms=skewtparms) / dlogdm
# constant dlogdm{1}
*
* Estimate with skew-GED
*
source logskewgedgarch.src
nonlin(parmset=skewgedparms) %SkewGED_k %SkewGED_lambda
compute %SkewGED_k=2.0,%SkewGED_lambda=0.0
garch(p=1,q=1,reg,density=%logskewgedgarch,parms=skewgedparms) / dlogdm
# constant dlogdm{1}
Output
GARCH Model - Estimation by BFGS
Convergence in 24 Iterations. Final criterion was 0.0000035 <= 0.0000100
Dependent Variable DLOGDM
Usable Observations 1865
Log Likelihood -2042.0340
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Constant -0.030838422 0.015499805 -1.98960 0.04663495
2. DLOGDM{1} -0.071768172 0.021682467 -3.30996 0.00093308
3. C 0.015039017 0.005071571 2.96536 0.00302332
4. A 0.104822891 0.017649504 5.93914 0.00000000
5. B 0.875418320 0.020043002 43.67701 0.00000000
6. Shape(t degrees) 8.694472198 1.639620186 5.30274 0.00000011
GARCH Model - Estimation by BFGS
Convergence in 31 Iterations. Final criterion was 0.0000000 <= 0.0000100
Dependent Variable DLOGDM
Usable Observations 1865
Log Likelihood -2041.3236
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Constant -0.032612702 0.014268205 -2.28569 0.02227237
2. DLOGDM{1} -0.072179640 0.024172664 -2.98600 0.00282650
3. C 0.015380830 0.005237999 2.93639 0.00332052
4. A 0.107888070 0.017506453 6.16276 0.00000000
5. B 0.871665316 0.019856893 43.89737 0.00000000
6. Shape(GED) 1.358239303 0.058488346 23.22239 0.00000000
GARCH with Skew-t Errors - Estimation by BFGS
Convergence in 21 Iterations. Final criterion was 0.0000001 <= 0.0000100
Dependent Variable DLOGDM
Usable Observations 1865
Log Likelihood -2040.3287
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Constant -0.021970386 0.015820809 -1.38870 0.16492345
2. DLOGDM{1} -0.069843518 0.023472531 -2.97554 0.00292471
3. C 0.016291992 0.005599375 2.90961 0.00361881
4. A 0.103875429 0.017664079 5.88060 0.00000000
5. B 0.873521972 0.020944632 41.70625 0.00000000
6. %SKEWT_ETA 8.782309670 1.712682912 5.12781 0.00000029
7. %SKEWT_LAMBDA 0.059554960 0.029848668 1.99523 0.04601779
GARCH with Skew-GED Errors - Estimation by BFGS
Convergence in 27 Iterations. Final criterion was 0.0000053 <= 0.0000100
Dependent Variable DLOGDM
Usable Observations 1865
Log Likelihood -2038.4773
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Constant -0.081188743 0.026649807 -3.04650 0.00231519
2. DLOGDM{1} -0.073545149 0.023741642 -3.09773 0.00195010
3. C 0.016946203 0.005881049 2.88149 0.00395795
4. A 0.105784824 0.018003154 5.87591 0.00000000
5. B 0.869585279 0.021511551 40.42411 0.00000000
6. %SKEWGED_K 1.474493958 0.068851789 21.41548 0.00000000
7. %SKEWGED_LAMBDA 0.060025188 0.025790375 2.32743 0.01994260