GJR GARCH with dummy variable in variance equation

[emna]

Dear Tom,

I use the following code to estimate a GJR-GARCH(1,1) model with dummy variable in variance eqation
r(t)=beta(1)+beta(2)*r(t-1)
sigma(t)^2=alpha0 + alpha1*u{1}^2+ %if(u{1}<0.0,gamma1*u{1}^2,0.0)+ %if(u{1}<0.0,gamma2*dummy*u{1}^2,0.0)+gammadummy*dummy + beta1*h{1}

Code: Select all

clear all
open data PUT ARCH GARCH.xls
data(format=xls,org=columns)
set x = rendement
linreg x
# constant x{1}

frml(lastreg,vector=beta) meanf
nonlin(parmset=meanparms) beta

set uu = %seesq
set h  = %seesq
set u  = 0.0

nonlin(parmset=garchparms) alpha0 alpha1 beta1 gamma1 gamma2 gammadummy
compute alpha1=gamma1=gamma2=0.005, beta1=0.004,alpha0=%seesq*(1-alpha1-beta1), gammadummy=0.1
   frml varf = alpha0 + beta1*h{1} + alpha1*u{1}^2+ %if(u{1}<0.0,gamma1*u{1}^2,0.0)+ %if(u{1}<0.0,gamma2*dummy*u{1}^2,0.0)+gammadummy*dummy
   frml logl = (h(t)=varf(t)),(u=x-meanf),$
       (uu(t)=u^2),%logdensity(h,u)
maximize(parmset=meanparms+garchparms,pmethod=simplex,piters=100,robust, iters=2500) logl 3 * 

My problem is that model don't converge so I have this message
"NO CONVERGENCE IN 7 ITERATIONS. FINAL NORMED GRADIENT 1.43753
SUBITERATIONS LIMIT EXCEEDED.
ESTIMATION POSSIBLY HAS STALLED OR MACHINE ROUNDOFF IS MAKING FURTHER PROGRESS DIFFICULT
TRY DIFFERENT SETTING FOR EXACTLINE, DERIVES OR ALPHA ON NLPAR
RESTARTING ESTIMATION FROM LAST ESTIMATES OR DIFFERENT INITIAL GUESSES/PMETHOD OPTION MIGHT ALSO WORK

With Heteroscedasticity/Misspecification Adjusted Standard Errors"

I don't know if there is any problem with the code or initial values perhaps or another one... Have you any explanations or comments about my model please ?

Best regards,
Emna

[TomDoan]

In terms of the model, is there a reason that you don't have a dummy x a standard lagged squared residual, just the asymmetric term and the variance constant?

First, that's way too many simplex iterations. Have you actually tried a regular GJR model? You're using guess values which have effectively no GARCH properties. Starting with guess values off a standard GJR with zeros for the dummy coefficients would be the obvious way to start that.

[emna]

Thank you for your response.
1/ In terms of the model I guess that there is a structural break in terms of asymmetry and variance level after a regulation reform that's why I introduce dummy variable for these to variables.

2/ My results for the regular GJR model are as the following :

Code: Select all

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

GARCH Model - Estimation by BFGS
Convergence in    47 Iterations. Final criterion was  0.0000031 <=  0.0000100

Dependent Variable X
Usable Observations                       742
Log Likelihood                     -3366.0123

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Mean(X)                       -8.27906643   0.43129314    -19.19591  0.00000000
2.  C                             73.88707348   6.42176267     11.50573  0.00000000
3.  A                             -0.00750951   0.00060380    -12.43708  0.00000000
4.  B                              0.24756667   0.02306927     10.73145  0.00000000
5.  D                              7.71784140   1.01378913      7.61287  0.00000000

or if I use my code :

Code: Select all

clear all
open data PUT ARCH GARCH.xls
data(format=xls,org=columns)
set x = rendement
linreg x 
# constant x{1}

Linear Regression - Estimation by Least Squares
Dependent Variable X
Usable Observations                       741
Degrees of Freedom                        739
Centered R^2                        0.0163383
R-Bar^2                             0.0150072
Uncentered R^2                      0.0163383
Mean of Dependent Variable        -0.00030205
Std Error of Dependent Variable   55.39263374
Standard Error of Estimate        54.97541708
Sum of Squared Residuals         2233477.1012
Regression F(1,739)                   12.2746
Significance Level of F             0.0004868
Log Likelihood                     -4019.5347
Durbin-Watson Statistic                2.0372

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                     -0.000158954  2.019572410 -7.87066e-05  0.99993722
2.  X{1}                         -0.127822302  0.036484100     -3.50351  0.00048681


frml(lastreg,vector=beta) meanf
nonlin(parmset=meanparms) beta

set uu = %seesq
set h  = %seesq
set u  = 0.0

nonlin(parmset=garchparms) alpha0 alpha1 beta1 gamma1
compute alpha1=5.35, beta1=0.15, alpha0=17.82, gamma1=7
   frml varf = alpha0 + beta1*h{1} + alpha1*u{1}^2+ %if(u{1}<0.0,gamma1*u{1}^2,0.0)
   frml logl = (h(t)=varf(t)),(u=x-meanf),$
       (uu(t)=u^2),%logdensity(h,u)
maximize(parmset=meanparms+garchparms, pmethod=simplex,piters=100,robust, iters=2500) logl 3 *

MAXIMIZE - Estimation by BFGS
Convergence in     1 Iterations. Final criterion was  0.0000032 <=  0.0000100
LOW ITERATION COUNT ON BFGS MAY LEAD TO POOR ESTIMATES FOR STANDARD ERRORS

With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations                       741
Function Value                     -3353.9447

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  BETA(1)                       -8.34554976   0.23799029    -35.06677  0.00000000
2.  BETA(2)                       -0.05954196   0.00974185     -6.11197  0.00000000
3.  ALPHA0                        72.62874708   2.95069357     24.61413  0.00000000
4.  ALPHA1                        -0.00736835   0.00241010     -3.05727  0.00223360
5.  BETA1                          0.24649718   0.00673153     36.61831  0.00000000
6.  GAMMA1                         7.25913378   0.25105248     28.91481  0.00000000

[emna]

And now my model with regular GJR initial values converges but with Heteroscedasticity/Misspecification Adjusted Standard Errors

Code: Select all

data(format=xls,org=columns)
set x = rendement
linreg x 
# constant x{1}

Linear Regression - Estimation by Least Squares
Dependent Variable X
Usable Observations                       741
Degrees of Freedom                        739
Centered R^2                        0.0163383
R-Bar^2                             0.0150072
Uncentered R^2                      0.0163383
Mean of Dependent Variable        -0.00030205
Std Error of Dependent Variable   55.39263374
Standard Error of Estimate        54.97541708
Sum of Squared Residuals         2233477.1012
Regression F(1,739)                   12.2746
Significance Level of F             0.0004868
Log Likelihood                     -4019.5347
Durbin-Watson Statistic                2.0372

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                     -0.000158954  2.019572410 -7.87066e-05  0.99993722
2.  X{1}                         -0.127822302  0.036484100     -3.50351  0.00048681


frml(lastreg,vector=beta) meanf
nonlin(parmset=meanparms) beta

set uu = %seesq
set h  = %seesq
set u  = 0.0

nonlin(parmset=garchparms) alpha0 alpha1 beta1 gamma1 gamma2 gammadummy
compute alpha1=-0.00750951, gamma1=gamma2=7.71784140, beta1=0.24756667,alpha0=73.88707348, gammadummy=0.1
   frml varf = alpha0 + beta1*h{1} + alpha1*u{1}^2+ %if(u{1}<0.0,gamma1*u{1}^2,0.0)+ %if(u{1}<0.0,gamma2*dummy*u{1}^2,0.0)+gammadummy*dummy
 frml logl = (h(t)=varf(t)),(u=x-meanf),$
       (uu(t)=u^2),%logdensity(h,u)

maximize(parmset=meanparms+garchparms, pmethod=simplex,piters=100,robust, iters=2500) logl 3 *

MAXIMIZE - Estimation by BFGS
Convergence in     1 Iterations. Final criterion was  0.0000000 <=  0.0000100
LOW ITERATION COUNT ON BFGS MAY LEAD TO POOR ESTIMATES FOR STANDARD ERRORS

With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations                       741
Function Value                     -3348.7826

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  BETA(1)                       -8.44174085   0.24596736    -34.32057  0.00000000
2.  BETA(2)                       -0.05829411   0.01013334     -5.75270  0.00000001
3.  ALPHA0                        90.58382604   3.52665946     25.68545  0.00000000
4.  ALPHA1                        -0.00775954   0.00231308     -3.35464  0.00079469
5.  BETA1                          0.25307723   0.00688866     36.73826  0.00000000
6.  GAMMA1                         6.29770606   0.25199883     24.99101  0.00000000
7.  GAMMA2                         1.62059489   0.34877321      4.64656  0.00000338
8.  GAMMADUMMY                   -36.43453397   5.06935847     -7.18721  0.00000000


compute alpha1=-0.00750951, gamma1=gamma2=7.71784140, beta1=0.24756667,alpha0=73.88707348, gammadummy=0.1
maximize(parmset=meanparms+garchparms, method= BHHH, pmethod=simplex,piters=100,robust, iters=2500) logl 3 *

MAXIMIZE - Estimation by BHHH
Convergence in     6 Iterations. Final criterion was  0.0000081 <=  0.0000100

With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations                       741
Function Value                     -3348.7826

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  BETA(1)                       -8.44178069   0.44405083    -19.01084  0.00000000
2.  BETA(2)                       -0.05829719   0.02085195     -2.79577  0.00517767
3.  ALPHA0                        90.58637091   5.97952022     15.14944  0.00000000
4.  ALPHA1                        -0.00775944   0.00332409     -2.33431  0.01957973
5.  BETA1                          0.25307885   0.01912558     13.23248  0.00000000
6.  GAMMA1                         6.29763060   0.76974334      8.18147  0.00000000
7.  GAMMA2                         1.62070821   0.58521005      2.76945  0.00561515
8.  GAMMADUMMY                   -36.43666756   7.60316141     -4.79230  0.00000165

What do you think about this please ?

[TomDoan]

Until you fix the fact that you didn't put your chosen mean model into the GARCH instruction (you didn't include the lagged dependent variable---you need to use the REGRESSORS option to put in a non-trivial mean model), I can't be sure, but that looks like you have a major problem with your data. The asymmetry coefficient should not be that big---it really shouldn't be bigger than 1, and usually is more like .1 or .2 at the outside.

[emna]

please find the dataset used to estimate the model in the an attachment

[TomDoan]

OK. Have you looked at that series? What's with those monster spikes, and what do you plan to do about it?

[emna]

I have an database of option prices. As you know, for each trading day the option prices differ according to the exercise price and the number of days remaining until maturity. In order to keep a single price, I have use only ATM options that have the shortest term since they represent the most liquid contracts. The problem here is that the data has peaks each time we have a big difference between the maturities (see example attached

mydata.xls

example

(63 KiB) Downloaded 801 times

).

So I do not know if you have an idea how I should choose the options to avoid this problem

[TomDoan]

That sounds like something that you really need to be discussing with your research advisor. A GARCH model assumes that the DGP for the variance is the same from period to period. It's not clear that anything like that is true for options at a daily level even when the contract isn't changing from day to day, and clearly isn't at all true when the contract shifts.