GARCH Model with Day of Week Dummies

[TomDoan]

The model is wrong. Look at how the three FRMLS are set up in the revised Baillie-Bollerslev code. The only change you should make to that is to add the AR parameter to the MEANF.

[jack]

Thank you very much. I edited it, I think we should edit this part compute gstart=%regstart()+1,gend=%regend() to compute gstart=%regstart()+2,gend=%regend() too because of Ar(1) component.

Code: Select all

*
* Replication file for Baillie and Bollerslev, "The Message in Daily
* Exchange Rates: A Conditional Variance Tale", JBES 1989, vol 7, pp
* 297-305
*
open data tom1.xlsx
data(format=xlsx,org=columns) 1 3254 sto dow
*
labels sto
# "Stock"
log sto

*
* Unit root tests
* Table 1
*
source ppunit.src
report(action=define)
dofor s = sto
   report(col=new,atrow=1) %l(s)
   @ppunit(lags=22,det=trend) s
   report(col=current,atrow=2) %cdstat
   @ppunit(lags=22) s
   report(col=current,atrow=3) %cdstat
end dofor
report(action=format,picture="*.###")
report(action=show)
*
* Table 2 estimates
*
report(action=define)
dofor s = sto
   report(col=new,atrow=1) %l(s)
   set dx = 100.0*(s{0}-s{1})
   linreg(noprint) dx
   # constant dx{1}
   report(col=curr,atrow=2) %beta(1)
   report(col=curr,atrow=3,special=parens) %stderrs(1)
   report(col=curr,atrow=4) %sigmasq
   report(col=curr,atrow=5) %logl
   set ustd = %resids/sqrt(%seesq)
   corr(print,qstats,number=15,method=yule) ustd
   report(col=curr,atrow=6) %qstat
   set usqr = ustd^2
   corr(print,qstats,number=15,method=yule) usqr
   report(col=curr,atrow=7) %qstat
   stats(noprint) %resids
   report(col=curr,atrow=8) %skewness
   report(col=curr,atrow=9) %kurtosis
end dofor s
report(action=format,picture="*.###",align=decimal)
report(action=show)
*
* Table 3
*
* The DOW series is 1 for Saturday,...,5 for Wednesday. Create dummies for
* each day of the week.
*
dec vect[series] dd(5)
do i=1,5
   set dd(i) = dow==(i)
end do i
*
*
* SKIP is a dummy variable which will be 1 if and only if there is a
* skipped weekday.
*
set(first=1.0) skip = .not.(dow{1}+1==dow.or.(dow{1}==5.and.dow==1))
*
nonlin(parmset=meanparms) b0 b1 b2 b3 b4 b5=-(b1+b2+b3+b4)
nonlin(parmset=garchshifts) w0 w1 w2 w3 w4 w5=-(w1+w2+w3+w4) w6 w7
nonlin(parmset=garchparms) alpha1 beta1 rnu
declare series uu h u
frml hmeanf = w0+w1*dd(1)+w2*dd(2)+w3*dd(3)+w4*dd(4)+w5*dd(5)+w6*skip+w7*skip{1}
frml varf = alpha1*uu{1}+beta1*h{1}+hmeanf-(alpha1+beta1)*hmeanf{1}
frml meanf = b0*dx{1}+b1*dd(1)+b2*dd(2)+b3*dd(3)+b4*dd(4)+b5*dd(5)
frml logl = (u=dx-meanf),(uu(t)=u^2),(h(t)=varf(t)),%logtdensity(h,u,1./rnu)

* Create two parallel reports, one for the model estimates, one for the summary statistics
*
report(action=define,use=garchestimates)
report(action=define,use=garchsummary,hlabels=||"","Stock"||)
report(use=garchsummary,atrow=1,atcol=1,fillby=columns) "$Log L$" "$Q(15)$" "$Q^2(15)$" "$m_3$" "$m_4$" $
   "$3(\hat \nu  - 2)(\hat \nu  - 4)^{ - 1}$"

dofor s = sto
   set dx = 100.0*(s{0}-s{1})
   *
   * * Get preliminary guess values for the mean parameters estimating just the mean model

   *
   nlls(parmset=meanparms,frml=meanf) dx
   * Use the variance of the estimation to initialize the uu and h series
   compute hinit=%seesq
   *
   set uu = %resids^2
   set h  = hinit
   set u  = %resids
   *
   * * Estimation of the full GARCH model will use 1 less data point than
   * the mean model.
   compute gstart=%regstart()+2,gend=%regend()
   *
   * Use base guess values for the variance mean function to avoid
   * problems with negative variances.
   *
   compute w0=hinit,w1=w2=w3=w4=w5=w6=w7=0.0
   *
   * Set initial guess values for the reciprocal degrees of freedom, and
   * the GARCH parameters.
   compute rnu=.10
   compute alpha1=.1,beta1=.8
   *
   maximize(parmset=meanparms+garchshifts+garchparms,pmethod=simplex,piter=10,method=bfgs,reject=(alpha1+beta1)>1.02) logl gstart gend
   report(use=garchestimates,regressors,extra=stderrs)
   report(use=garchsummary,col=new,atrow=1) %funcval
   stats u gstart gend
   set ustd gstart gend = u/sqrt(h)
   corr(print,qstats,number=15,method=yule) ustd gstart gend
   report(use=garchsummary,col=current,atrow=2) %qstat
   set usqr gstart gend = ustd^2
   corr(print,qstats,number=15,method=yule) usqr gstart gend
   report(use=garchsummary,col=current,atrow=3) %qstat
   stats(noprint) ustd
   report(use=garchsummary,col=current,atrow=4) %skewness
   report(use=garchsummary,col=current,atrow=5) %kurtosis
   report(use=garchsummary,col=current,atrow=6) 3.0*(1.0/rnu-2.0)/(1.0/rnu-4.0)
end dofor
report(action=define,use=Table3,title="Table 3 Daily GARCH Models",$
    hlabels=||"","Stock"||)
*

[TomDoan]

You are missing the overall intercept in the MEANF. You need to keep the original B0 (because the B1 to B5 are constrained to add to 0) and add an extra term for the AR(1).

[jack]

I added an extra term for AR(1) [b6]. But it doesn't converge now.

Code: Select all

*
* Replication file for Baillie and Bollerslev, "The Message in Daily
* Exchange Rates: A Conditional Variance Tale", JBES 1989, vol 7, pp
* 297-305
*
open data .xlsx
data(format=xlsx,org=columns) 1 3254 sto dow
*
labels sto
# "Stock"
log sto

*
* Unit root tests
* Table 1
*
source ppunit.src
report(action=define)
dofor s = sto
   report(col=new,atrow=1) %l(s)
   @ppunit(lags=22,det=trend) s
   report(col=current,atrow=2) %cdstat
   @ppunit(lags=22) s
   report(col=current,atrow=3) %cdstat
end dofor
report(action=format,picture="*.###")
report(action=show)
*
* Table 2 estimates
*
report(action=define)
dofor s = sto
   report(col=new,atrow=1) %l(s)
   set dx = 100.0*(s{0}-s{1})
   linreg(noprint) dx
   # constant dx{1}
   report(col=curr,atrow=2) %beta(1)
   report(col=curr,atrow=3,special=parens) %stderrs(1)
   report(col=curr,atrow=4) %sigmasq
   report(col=curr,atrow=5) %logl
   set ustd = %resids/sqrt(%seesq)
   corr(print,qstats,number=15,method=yule) ustd
   report(col=curr,atrow=6) %qstat
   set usqr = ustd^2
   corr(print,qstats,number=15,method=yule) usqr
   report(col=curr,atrow=7) %qstat
   stats(noprint) %resids
   report(col=curr,atrow=8) %skewness
   report(col=curr,atrow=9) %kurtosis
end dofor s
report(action=format,picture="*.###",align=decimal)
report(action=show)
*
* Table 3
*
* The DOW series is 1 for Saturday,...,5 for Wednesday. Create dummies for
* each day of the week.
*
dec vect[series] dd(5)
do i=1,5
   set dd(i) = dow==(i)
end do i
*
*
* SKIP is a dummy variable which will be 1 if and only if there is a
* skipped weekday.
*
set(first=1.0) skip = .not.(dow{1}+1==dow.or.(dow{1}==5.and.dow==1))
*
nonlin(parmset=meanparms) b0 b1 b2 b3 b4 b5=-(b1+b2+b3+b4) b6
nonlin(parmset=garchshifts) w0 w1 w2 w3 w4 w5=-(w1+w2+w3+w4) w6 w7
nonlin(parmset=garchparms) alpha1 beta1 rnu
declare series uu h u
frml hmeanf = w0+w1*dd(1)+w2*dd(2)+w3*dd(3)+w4*dd(4)+w5*dd(5)+w6*skip+w7*skip{1}
frml varf = alpha1*uu{1}+beta1*h{1}+hmeanf-(alpha1+beta1)*hmeanf{1}
frml meanf = b0+b1*dd(1)+b2*dd(2)+b3*dd(3)+b4*dd(4)+b5*dd(5)+b6*dx{1}
frml logl = (u=dx-meanf),(uu(t)=u^2),(h(t)=varf(t)),%logtdensity(h,u,1./rnu)

* Create two parallel reports, one for the model estimates, one for the summary statistics
*
report(action=define,use=garchestimates)
report(action=define,use=garchsummary,hlabels=||"","Stock"||)
report(use=garchsummary,atrow=1,atcol=1,fillby=columns) "$Log L$" "$Q(15)$" "$Q^2(15)$" "$m_3$" "$m_4$" $
   "$3(\hat \nu  - 2)(\hat \nu  - 4)^{ - 1}$"

dofor s = sto
   set dx = 100.0*(s{0}-s{1})
   *
   * * Get preliminary guess values for the mean parameters estimating just the mean model

   *
   nlls(parmset=meanparms,frml=meanf) dx
   * Use the variance of the estimation to initialize the uu and h series
   compute hinit=%seesq
   *
   set uu = %resids^2
   set h  = hinit
   set u  = %resids
   *
   * * Estimation of the full GARCH model will use 1 less data point than
   * the mean model.
   compute gstart=%regstart()+2,gend=%regend()
   *
   * Use base guess values for the variance mean function to avoid
   * problems with negative variances.
   *
   compute w0=hinit,w1=w2=w3=w4=w5=w6=w7=0.0
   *
   * Set initial guess values for the reciprocal degrees of freedom, and
   * the GARCH parameters.
   compute rnu=.10
   compute alpha1=.1,beta1=.8
   *
   maximize(parmset=meanparms+garchshifts+garchparms,pmethod=simplex,piter=10,method=bfgs,reject=(alpha1+beta1)>1.02) logl gstart gend
   report(use=garchestimates,regressors,extra=stderrs)
   report(use=garchsummary,col=new,atrow=1) %funcval
   stats u gstart gend
   set ustd gstart gend = u/sqrt(h)
   corr(print,qstats,number=15,method=yule) ustd gstart gend
   report(use=garchsummary,col=current,atrow=2) %qstat
   set usqr gstart gend = ustd^2
   corr(print,qstats,number=15,method=yule) usqr gstart gend
   report(use=garchsummary,col=current,atrow=3) %qstat
   stats(noprint) ustd
   report(use=garchsummary,col=current,atrow=4) %skewness
   report(use=garchsummary,col=current,atrow=5) %kurtosis
   report(use=garchsummary,col=current,atrow=6) 3.0*(1.0/rnu-2.0)/(1.0/rnu-4.0)
end dofor
report(action=define,use=Table3,title="Table 3 Daily GARCH Models",$
    hlabels=||"","Stock"||)
*

[TomDoan]

You need to shift the W0 out of the HMEANF so it doesn't get clobbered by the fact that alpha+beta sum to almost exactly 1.

frml hmeanf = w1*dd(1)+w2*dd(2)+w3*dd(3)+w4*dd(4)+w5*dd(5)+w6*skip+w7*skip{1}
frml varf = w0+alpha1*uu{1}+beta1*h{1}+hmeanf-(alpha1+beta1)*hmeanf{1}
frml meanf = b0+b1*dd(1)+b2*dd(2)+b3*dd(3)+b4*dd(4)+b5*dd(5)+b6*dx{1}

Your estimates are for a (slightly) drifting IGARCH with rather fat t errors, that is, really noisy data. And this is *after* you've removed outliers?

[jack]

I shifted the W0 out of the HMEANF as you said and run it again. It still doesn't converge even after removing outliers!

Here is original data and program

[TomDoan]

You have w0 in both of these FRML's. You need to take it out of the HMEANF:

frml hmeanf = w0+w1*dd(1)+w2*dd(2)+w3*dd(3)+w4*dd(4)+w5*dd(5)+w6*skip+w7*skip{1}
frml varf = w0+alpha1*uu{1}+beta1*h{1}+hmeanf-(alpha1+beta1)*hmeanf{1}

Change the REJECT option to something like reject=(alpha1+beta1)>1.05. It turns out that you are actually hitting the 1.02 reject limit, which means your model is in the zone where it's stationary but has no unconditional variance.

If this is what you get after removing outliers, you may need to look for a more complicated model than a GARCH. Maybe some type of jump-GARCH.

[jack]

Thanks for your guide. Do you think it is possible to define dummy variables for the days of the week in the proposed model by Chan and Maheu (2002)( "Conditional Jump Dynamics in Stock Market Returns" )?

[TomDoan]

Maheu is still actively doing research. He might know if someone has done that.

[jack]

When I remove outliers the model doesn't converge but when I estimate it without removing the outliers it converges!

When I intend to estimate the model by removing outlier data, I use returns data instead of price index data for estimation from the outset.

Here is the code:

Code: Select all

*
* Replication file for Baillie and Bollerslev, "The Message in Daily
* Exchange Rates: A Conditional Variance Tale", JBES 1989, vol 7, pp
* 297-305
*
open data data.xlsx
data(format=xlsx,org=columns) 1 3232 sto dow
*
labels sto
# "Stock"


*
* Unit root tests
* Table 1
*
source ppunit.src
report(action=define)
dofor s = sto
   report(col=new,atrow=1) %l(s)
   @ppunit(lags=22,det=trend) s
   report(col=current,atrow=2) %cdstat
   @ppunit(lags=22) s
   report(col=current,atrow=3) %cdstat
end dofor
report(action=format,picture="*.###")
report(action=show)
*
* Table 2 estimates
*
report(action=define)
dofor dx = sto
   report(col=new,atrow=1) %l(s)
   linreg(noprint) dx
   # constant dx{1}
   report(col=curr,atrow=2) %beta(1)
   report(col=curr,atrow=3,special=parens) %stderrs(1)
   report(col=curr,atrow=4) %sigmasq
   report(col=curr,atrow=5) %logl
   set ustd = %resids/sqrt(%seesq)
   corr(print,qstats,number=15,method=yule) ustd
   report(col=curr,atrow=6) %qstat
   set usqr = ustd^2
   corr(print,qstats,number=15,method=yule) usqr
   report(col=curr,atrow=7) %qstat
   stats(noprint) %resids
   report(col=curr,atrow=8) %skewness
   report(col=curr,atrow=9) %kurtosis
end dofor dx
report(action=format,picture="*.###",align=decimal)
report(action=show)
*
* Table 3
*
* The DOW series is 1 for Saturday,...,5 for Wednesday. Create dummies for
* each day of the week.
*
dec vect[series] dd(5)
do i=1,5
   set dd(i) = dow==(i)
end do i
*
*
* SKIP is a dummy variable which will be 1 if and only if there is a
* skipped weekday.
*
set(first=1.0) skip = .not.(dow{1}+1==dow.or.(dow{1}==5.and.dow==1))
*
nonlin(parmset=meanparms) b0 b1 b2 b3 b4 b5=-(b1+b2+b3+b4) b6
nonlin(parmset=garchshifts) w0 w1 w2 w3 w4 w5=-(w1+w2+w3+w4) w6 w7
nonlin(parmset=garchparms) alpha1 beta1 rnu
declare series uu h u
frml hmeanf = w1*dd(1)+w2*dd(2)+w3*dd(3)+w4*dd(4)+w5*dd(5)+w6*skip+w7*skip{1}
frml varf = w0+alpha1*uu{1}+beta1*h{1}+hmeanf-(alpha1+beta1)*hmeanf{1}
frml meanf = b0+b1*dd(1)+b2*dd(2)+b3*dd(3)+b4*dd(4)+b5*dd(5)+b6*dx{1}
frml logl = (u=dx-meanf),(uu(t)=u^2),(h(t)=varf(t)),%logtdensity(h,u,1./rnu)

* Create two parallel reports, one for the model estimates, one for the summary statistics
*
report(action=define,use=garchestimates)
report(action=define,use=garchsummary,hlabels=||"","Stock"||)
report(use=garchsummary,atrow=1,atcol=1,fillby=columns) "$Log L$" "$Q(15)$" "$Q^2(15)$" "$m_3$" "$m_4$" $
   "$3(\hat \nu  - 2)(\hat \nu  - 4)^{ - 1}$"

dofor dx = sto

   *
   * * Get preliminary guess values for the mean parameters estimating just the mean model

   *
   nlls(parmset=meanparms,frml=meanf) dx
   * Use the variance of the estimation to initialize the uu and h series
   compute hinit=%seesq
   *
   set uu = %resids^2
   set h  = hinit
   set u  = %resids
   *
   * * Estimation of the full GARCH model will use 1 less data point than
   * the mean model.
   compute gstart=%regstart()+2,gend=%regend()
   *
   * Use base guess values for the variance mean function to avoid
   * problems with negative variances.
   *
   compute w0=hinit,w1=w2=w3=w4=w5=w6=w7=0.0
   *
   * Set initial guess values for the reciprocal degrees of freedom, and
   * the GARCH parameters.
   compute rnu=.10
   compute alpha1=.1,beta1=.8
   *
   maximize(parmset=meanparms+garchshifts+garchparms,pmethod=simplex,piter=10,method=bfgs,reject=(alpha1+beta1)>1.05) logl gstart gend
   report(use=garchestimates,regressors,extra=stderrs)
   report(use=garchsummary,col=new,atrow=1) %funcval
   stats u gstart gend
   set ustd gstart gend = u/sqrt(h)
   corr(print,qstats,number=15,method=yule) ustd gstart gend
   report(use=garchsummary,col=current,atrow=2) %qstat
   set usqr gstart gend = ustd^2
   corr(print,qstats,number=15,method=yule) usqr gstart gend
   report(use=garchsummary,col=current,atrow=3) %qstat
   stats(noprint) ustd
   report(use=garchsummary,col=current,atrow=4) %skewness
   report(use=garchsummary,col=current,atrow=5) %kurtosis
   report(use=garchsummary,col=current,atrow=6) 3.0*(1.0/rnu-2.0)/(1.0/rnu-4.0)
end dofor dx
report(action=define,use=Table3,title="Table 3 Daily GARCH Models",$
    hlabels=||"","Stock"||)
*

Regardless of whether the model converges or not, the software produces the following error, do you think any part of the convergence issue with the model is not related to this error?

The Error Occurred At Location 154, Line 7 of loop/block
## NL6. NONLIN Parameter B0 Has Not Been Initialized. Trying 0
The Error Occurred At Location 154, Line 7 of loop/block
## NL6. NONLIN Parameter B1 Has Not Been Initialized. Trying 0
The Error Occurred At Location 154, Line 7 of loop/block
## NL6. NONLIN Parameter B2 Has Not Been Initialized. Trying 0
The Error Occurred At Location 154, Line 7 of loop/block
## NL6. NONLIN Parameter B3 Has Not Been Initialized. Trying 0
The Error Occurred At Location 154, Line 7 of loop/block
## NL6. NONLIN Parameter B4 Has Not Been Initialized. Trying 0
The Error Occurred At Location 154, Line 7 of loop/block
## NL6. NONLIN Parameter B6 Has Not Been Initialized. Trying 0

[TomDoan]

Those are just warnings. You can silence them by initializing all the B's to zero. (All this is doing is telling you that it's using zero as the guess value, which is fine in this case).

[jack]

Dear Tom,

Here is an estimate of the model without removing the outliers. I have some questions and would appreciate, as always, your guidance on them.

1) Why does the sign of b4 in the first mean model differ from the mean equation of the GARCH model?
2) Why are b3 and b4 not statistically insignificant in the first mean model but significant in the mean equation of the GARCH model?
3) How can I test the statistical significance of b5 and w5?
4) In Baillie and Bollerslev's (1989) study, the author tests the overall effect of different dummy variables in Table 4. How can I conduct similar tests in this context?

Do you recommend that I conclude the work based on these results? Especially considering that the coefficient of variable w3 is negative and, in terms of magnitude, greater than the intercept of variance equation; meaning that the variance on this day is negative!

Code: Select all

Dependent Variable DX
Usable Observations                      3252
Degrees of Freedom                       3246
Skipped/Missing (from 3253)                 1
Centered R^2                        0.1540844
R-Bar^2                             0.1527814
Uncentered R^2                      0.1728856
Mean of Dependent Variable       0.1580648655
Std Error of Dependent Variable  1.0485568659
Standard Error of Estimate       0.9651386850
Sum of Squared Residuals         3023.6252436
Regression F(5,3246)                 118.2525
Significance Level of F             0.0000000
Log Likelihood                     -4495.9931
Durbin-Watson Statistic                1.9650

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  B0                            0.096731495  0.017116960      5.65121  0.00000002
2.  B1                            0.043054439  0.034007896      1.26601  0.20559927
3.  B2                           -0.127004120  0.033770564     -3.76079  0.00017236
4.  B3                            0.052487183  0.033842406      1.55093  0.12101608
5.  B4                           -0.037930066  0.033916115     -1.11835  0.26350056
6.  B6                            0.389512453  0.016179723     24.07411  0.00000000


MAXIMIZE - Estimation by BFGS
Convergence in    28 Iterations. Final criterion was  0.0000071 <=  0.0000100

Usable Observations                      3251
Function Value                     -3395.6225

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  B0                            0.028608146  0.008720096      3.28071  0.00103545
2.  B1                            0.003202237  0.015918821      0.20116  0.84057314
3.  B2                           -0.106792967  0.013465737     -7.93072  0.00000000
4.  B3                            0.034473144  0.013479002      2.55754  0.01054141
5.  B4                            0.031593698  0.014971522      2.11025  0.03483658
6.  B6                            0.391317121  0.015743744     24.85540  0.00000000
7.  W0                            0.005779412  0.001509644      3.82833  0.00012902
8.  W1                            0.071140447  0.015984593      4.45056  0.00000856
9.  W2                           -0.008121147  0.007910554     -1.02662  0.30459858
10. W3                           -0.023370524  0.007395522     -3.16009  0.00157720
11. W4                           -0.014718940  0.008440718     -1.74380  0.08119363
12. W6                           -0.002396808  0.013348094     -0.17956  0.85749659
13. W7                            0.004974879  0.015152529      0.32832  0.74266967
14. ALPHA1                        0.196946799  0.021560844      9.13447  0.00000000
15. BETA1                         0.829015438  0.016030067     51.71628  0.00000000
16. RNU                           0.208133766  0.016793383     12.39380  0.00000000


Statistics on Series U
Observations                  3251
Sample Mean               0.067918      Variance                   0.931953
Standard Error            0.965377      SE of Sample Mean          0.016931
t-Statistic (Mean=0)      4.011382      Signif Level (Mean=0)      0.000062
Skewness                  0.165876      Signif Level (Sk=0)        0.000114
Kurtosis (excess)         4.270679      Signif Level (Ku=0)        0.000000
Jarque-Bera            2485.491787      Signif Level (JB=0)        0.000000

[jack]

Dear Tom

As you know, I've been working on this matter for a while. Now, to bring it to a conclusion, I need your assistance regarding the questions I raised in the previous post.
Furthermore, I haven't received any response despite reaching out to Maheum via email.

[TomDoan]

The revised version of the BBJBES.RPF program posted above includes all the tests in Table 4.

Obviously, it's not my call whether your work is adequate. I would be *very* concerned with the high level of autocorrelation in the data. If it isn't a result of how the data are collected, then why can't someone make a great deal of money by exploiting that? If, however, the underlying prices aren't actually prices at which transactions can be made, then it's not clear what a GARCH model is giving you.