GARCHMVBOOTSTRAP.RPF

GARCHMVBOOTSTRAP.RPF is an example of bootstrapping (for out-of-sample simulations) for MV GARCH models. It does both a DVECH model (any "VECH" family model can be done in a similar fashion) and a DCC model; the DCC model, in particular, requires a very different recursion from the VECH family.

GARCHBOOT.RPF and EGARCHBOOTSTRAP.RPF are simpler examples of bootstrapping univariate GARCH and EGARCH models. As in those simpler cases, you can't reshuffle the residuals from multivariate GARCH models because they have a complicated (second order) dependence pattern. Instead, you have to (jointly) standardize the residuals at time $t$ using the GARCH covariance matrix at $t$, reshuffle the standardized residuals, then use the GARCH model to rebuild a set of residuals with the desired GARCH properties.

Full Program

open data g10xrate.xls
data(format=xls,org=columns) 1 6237 usxjpn usxfra usxsui
*
set xjpn = 100.0*log(usxjpn/usxjpn{1})
set xfra = 100.0*log(usxfra/usxfra{1})
set xsui = 100.0*log(usxsui/usxsui{1})
***********************************************************************
*
* Multivariate GARCH model with DVECH. This same basic idea applies to
* any of the models supported by MVGARCHTOVECH. You have to save the
* residuals (most conveniently done using RVECTORS since we need to keep
* the time periods together) and the H matrices.
*
garch(p=1,q=1,pmethod=simplex,piters=10,rvectors=rv,hmatrices=h) / xjpn xfra xsui
compute gstart=%regstart(),gend=%regend()
*
@MVGARCHtoVECH(mv=standard)
*
* Create the series of outer products of the residuals
*
dec series[symm] uu
gset uu gstart gend = %outerxx(rv)
*
* Create the formula for updating the GARCH variance
*
dec frml[symm] hf
frml hf = %vectosymm(%%vech_c+%%vech_b*%vec(h{1})+%%vech_a*%vec(uu{1}),%nvar)
*
* Transform residuals into standardized form
*
dec series[vect] rvstd
gset rvstd gstart gend = %solve(%decomp(h(t)),rv(t))
*
* Bootstrap model for 10 periods out-of-sample. 10000 repetitions
*
compute span=10
compute ndraws=10000
compute bstart=gend+1,bend=gend+span
*
* Extend UU and H out-of-sample. (Values don't really matter at this
* point).
*
gset uu bstart bend = uu{1}
gset h bstart bend = h{1}
*
dec series[vect] portfolio
gset portfolio bstart bend = %unitv(%nvar,1)
*
set returns 1 ndraws = %na
*
compute [vector] ones=%fill(%nvar,1,1.0)
*
do draw=1,ndraws
   boot entries bstart bend gstart gend
   *
   * Update the variance, post-multiply the shuffled standardized
   * residuals by a square root of the variance, save the outer product
   * and return the re-flated residuals.
   *
   gset rv bstart bend = h=hf,uboot=%decomp(h)*rvstd(entries(t)),uu=%outerxx(uboot),uboot
   *
   * Compute cumulative return on minimum variance portfolio. (This
   * assumes zero mean returns).
   *
   gset portfolio bstart bend = hinv=inv(h),hinv*ones/%qform(hinv,ones)
   sstats bstart bend %dot(portfolio,rv)>>returns(draw)
end do draw
*
* Get quantiles on the performance of the portfolio
*
stats(fractiles) returns 1 ndraws
***********************************************************************
*
* MV GARCH with DCC
*
garch(p=1,q=1,mv=dcc,rvectors=rv,hmatrices=h) / xjpn xfra xsui
compute gstart=%regstart(),gend=%regend()
*
* Transform residuals into standardized form. For a DCC model, there are
* two types of standardizations necessary - one set transforms the
* residuals down to fully uncorrelated with unit variances (RVSTD) and
* one which merely standardizes each variance but leaves correlations
* intact (RVSTDC). Also save the vectors of variances and squared
* residuals.
*
dec series[vect] rvstd rvstdc
dec series[vect] hv uuv
gset rvstd gstart gend = %solve(%decomp(h(t)),rv(t))
gset rvstdc gstart gend = rv(t)./%xdiag(h(t))
gset hv     gstart gend = %xdiag(h(t))
gset uuv    gstart gend = rv(t).*rv(t)
*
* Set up the individual variance recursions
*
dec vect %%g_c(%nvar) %%g_a(%nvar) %%g_b(%nvar)
ewise %%g_c(i)=%beta(%nregmean+0*%nvar+i)
ewise %%g_a(i)=%beta(%nregmean+1*%nvar+i)
ewise %%g_b(i)=%beta(%nregmean+2*%nvar+i)
*
dec frml[vect] hvf
frml hvf = %%g_c+%%g_b.*hv{1}+%%g_a.*uuv{1}
*
* Pull out the DCC recursion coefficients
*
compute %%dcc_a=%beta(%nreg-1),%%dcc_b=%beta(%nreg)
*
* The "constant" term in the DCC recursion is based upon the
* unconditional correlation matrix of the data.
*
vcv(center)
# xjpn xfra xsui
compute qbar=%cvtocorr(%sigma)
*
* Set up the DCC recursion
*
dec series[symm] q
dec frml[symm] qf
frml qf = (1-%%dcc_a-%%dcc_b)*qbar+%%dcc_b*q{1}+%%dcc_a*%outerxx(rvstdc{1})
*
* The Q sequence can't be backed out of the information available from
* GARCH. This re-generates it.
*
gset q gstart   gstart = qbar
gset q gstart+1 gend   = qf(t)
*
* Bootstrap model for 10 periods out-of-sample. 10000 repetitions
*
compute span=10
compute ndraws=10000
compute bstart=gend+1,bend=gend+span
*
* Extend various objects out-of-sample. (Values don't really matter at
* this point).
*
gset uuv    bstart bend = uuv{1}
gset hv     bstart bend = hv{1}
gset h      bstart bend = h{1}
gset q      bstart bend = q{1}
gset rvstd bstart bend = rvstd{1}
gset rvstdc bstart bend = rvstdc{1}
*
dec series[vect] portfolio
gset portfolio bstart bend = %unitv(%nvar,1)
*
set returns 1 ndraws = %na
*
compute [vector] ones=%fill(%nvar,1,1.0)
*
do draw=1,ndraws
   boot entries bstart bend gstart gend
   *
   * Do the DCC recursion and the recursion for the univariate variances,
   * convert the pair into the covariance matrix, post-multiply the
   * shuffled standardized residuals by a square root of the variance,
   * save the squares for the next GARCH recusion, save the standardized
   * residuals for the next DCC recursion, and return the re-flated
   * residuals.
   *
   * To do the bootstrap, update Q(t) and HV(t). Convert Q to a
   * correlation matrix and create H(t) from the correlations and the
   * variances in HV. Use that to reflate the shuffled fully
   * standardized residuals.
   *
   gset rv bstart bend = q=qf,hv=hvf,h=%corrtocv(%cvtocorr(q),hv),$
     uboot=%decomp(h)*rvstd(entries(t)),uuv=uboot.*uboot,rvstdc=uboot./hv,uboot
   *
   * Compute cumulative return on minimum variance portfolio. (This
   * assumes zero mean returns).
   *
   gset portfolio bstart bend = hinv=inv(h),hinv*ones/%qform(hinv,ones)
   sstats bstart bend %dot(portfolio,rv)>>returns(draw)
end do draw
*
* Get quantiles on the performance of the portfolio
*
stats(fractiles) returns 1 ndraws

Output

The statistics on the simulated returns depend upon random numbers and so will not be exactly reproducible.

MV-GARCH - Estimation by BFGS

Convergence in 180 Iterations. Final criterion was 0.0000031 <= 0.0000100

Usable Observations 6236

Log Likelihood -11835.6454

Variable Coeff Std Error T-Stat Signif

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

1. Mean(XJPN) 0.004674636 0.006903238 0.67717 0.49830081

2. Mean(XFRA) -0.003474442 0.006786384 -0.51197 0.60867030

3. Mean(XSUI) -0.002331133 0.008361692 -0.27879 0.78040814

4. C(1,1) 0.008960589 0.001121259 7.99154 0.00000000

5. C(2,1) 0.005646258 0.000730696 7.72723 0.00000000

6. C(2,2) 0.011403662 0.001196925 9.52746 0.00000000

7. C(3,1) 0.005959232 0.000763548 7.80466 0.00000000

8. C(3,2) 0.009833899 0.001076759 9.13287 0.00000000

9. C(3,3) 0.012645698 0.001362519 9.28112 0.00000000

10. A(1,1) 0.105448983 0.007737425 13.62843 0.00000000

11. A(2,1) 0.093505182 0.005844222 15.99960 0.00000000

12. A(2,2) 0.127430154 0.006235690 20.43561 0.00000000

13. A(3,1) 0.088431373 0.005252148 16.83718 0.00000000

14. A(3,2) 0.112992867 0.005405432 20.90358 0.00000000

15. A(3,3) 0.110936860 0.005468652 20.28596 0.00000000

16. B(1,1) 0.884418010 0.007784760 113.60890 0.00000000

17. B(2,1) 0.891576893 0.006302766 141.45804 0.00000000

18. B(2,2) 0.861703430 0.006492651 132.71982 0.00000000

19. B(3,1) 0.898171217 0.005545496 161.96408 0.00000000

20. B(3,2) 0.875632965 0.005654273 154.86216 0.00000000

21. B(3,3) 0.878186944 0.005668731 154.91772 0.00000000

Statistics on Series RETURNS

Observations 10000

Sample Mean -0.057737 Variance 2.500103

Standard Error 1.581171 SE of Sample Mean 0.015812

t-Statistic (Mean=0) -3.651522 Signif Level (Mean=0) 0.000262

Skewness 0.996806 Signif Level (Sk=0) 0.000000

Kurtosis (excess) 13.986229 Signif Level (Ku=0) 0.000000

Jarque-Bera 83162.124491 Signif Level (JB=0) 0.000000

Minimum -11.783097 Maximum 23.893259

01-%ile -3.879249 99-%ile 3.895327

05-%ile -2.491524 95-%ile 2.324091

10-%ile -1.862923 90-%ile 1.699969

25-%ile -0.962258 75-%ile 0.824934

Median -0.061246

MV-DCC GARCH - Estimation by BFGS

Convergence in 40 Iterations. Final criterion was 0.0000080 <= 0.0000100

Usable Observations 6236

Log Likelihood -11814.4403

Variable Coeff Std Error T-Stat Signif

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

1. Mean(XJPN) 0.003987923 0.006009027 0.66366 0.50691089

2. Mean(XFRA) -0.003132340 0.006238983 -0.50206 0.61562576

3. Mean(XSUI) -0.003069840 0.007698257 -0.39877 0.69006207

4. C(1) 0.008499197 0.001141242 7.44732 0.00000000

5. C(2) 0.012485965 0.001377298 9.06555 0.00000000

6. C(3) 0.016566678 0.001823971 9.08276 0.00000000

7. A(1) 0.151663636 0.009063915 16.73268 0.00000000

8. A(2) 0.138383290 0.007750557 17.85463 0.00000000

9. A(3) 0.123692432 0.007406780 16.69989 0.00000000

10. B(1) 0.852004681 0.008025825 106.15790 0.00000000

11. B(2) 0.848526128 0.007936097 106.91982 0.00000000

12. B(3) 0.858001040 0.007867699 109.05362 0.00000000

13. DCC(A) 0.053230483 0.003249015 16.38358 0.00000000

14. DCC(A) 0.939072178 0.003941594 238.24678 0.00000000

Covariance\Correlation Matrix

XJPN XFRA XSUI

XJPN 0.398829261 0.55994 0.56526

XFRA 0.230897587 0.426359710 0.82230

XSUI 0.273707444 0.411684556 0.587880308

Statistics on Series RETURNS

Observations 10000

Sample Mean -0.030108 Variance 2.415239

Standard Error 1.554104 SE of Sample Mean 0.015541

t-Statistic (Mean=0) -1.937328 Signif Level (Mean=0) 0.052733

Skewness 0.393117 Signif Level (Sk=0) 0.000000

Kurtosis (excess) 5.747051 Signif Level (Ku=0) 0.000000

Jarque-Bera 14019.484674 Signif Level (JB=0) 0.000000

Minimum -10.880054 Maximum 17.770789

01-%ile -3.886063 99-%ile 3.811708

05-%ile -2.478113 95-%ile 2.392488

10-%ile -1.856786 90-%ile 1.794328

25-%ile -0.915057 75-%ile 0.856821

Median -0.042741