EGARCHBOOTSTRAP.RPF

EGARCHBOOTSTRAP.RPF is an example of bootstrapping an EGARCH model for forecasting variance out-of-sample and/or producing a bootstrapped sample. A companion (and almost identical) program EGARCHSIMULATE.RPF does the same type of analysis, but with random simulations rather than bootstrapping.

This uses a sample data set of (old) exchange rates. The one being analyzed is the US$/Deutsche Mark exchange rate. The log returns are multiplied by 100 for improved performance of non-linear estimation. Note that this means that what you read off from the analysis are returns to 100 DM.

open data garch.asc

data(format=free,org=columns) 1 1867 bp cd dm jy sf

set dlogdm = 100*log(dm/dm{1})

This estimates a univariate GARCH model with asymmetry, saving the estimated variances in the series H. (Output)

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

This pulls the coefficients out of the output.

compute gmu=%beta(1),c=%beta(2),a=%beta(3),b=%beta(4),d=%beta(5)

This pulls out the estimation range (which is also the source range for bootstrapped residuals), and computes the standardized residuals.

compute gstart=%regstart(),gend=%regend()

set stdu = %resids/sqrt(h)

This sets the number of draws and number of out-of-sample steps:

compute ndraws=10000

compute nsteps=50

This is the out-of-sample range over which we will do the simulations.

compute bstart=gend+1,bend=gend+nsteps

We will be saving statistics on draws for H and for the data X over the BSTART to BEND range. For each of those, we need a VECTOR of size NDRAWS at each data point, so HDRAWS and XDRAWS will be type SERIES[VECT]. We also need series for the variance H and simulated values X to do the recursion in each simulation.

set h bstart bend = %na

set x bstart bend = %na

dec series[vect] hdraws xdraws

gset hdraws bstart bend = %zeros(ndraws,1)

gset xdraws bstart bend = %zeros(ndraws,1)

This is the draw loop. EGARCH is a recursion in H and the standardized residuals. This uses BOOT to draw random entry numbers (with replacement) from the GSTART to GEND estimation range, creating the SERIES[INTEGER] named ENTRIES over the bootstrapping range BSTART to BEND. Within the SET instruction, the bootstrapped draw for the standardized residual is put into SU, it then computes the new value of H using the past values of H and STDU, computes a new value of X using the estimated mean plus a rescaled (by the standard deviation) SU, and saves the new draw for the Standard Normal for the next period. The SET instruction generates new values of H and X over the BSTART to BEND range. X is then accumulated into multiple period returns (the original X simulations are just bootstrapped future single period returns) and saves the H and X values into the DRAW location in each of the HDRAWS and XDRAWS.

do draw=1,ndraws

boot entries bstart bend gstart gend

set stdu bstart bend = su=stdu(entries(t)),$

h=exp(c+b*log(h{1})+a*abs(stdu{1})+d*stdu{1}),$

x=gmu+sqrt(h)*su,su

* Replace x by its accumulation

acc x bstart bend

do t=bstart,bend

compute hdraws(t)(draw)=h(t)

compute xdraws(t)(draw)=x(t)

end do t

end do draw

This extracts the .05 and .95 values for the H simulations for each out-of-sample entry, and graphs those.

set lower bstart bend = %fractiles(hdraws(t),.05)(1)

set upper bstart bend = %fractiles(hdraws(t),.95)(1)

set median bstart bend = %fractiles(hdraws(t),.50)(1)

graph(header="Variance Forecasts for EGARCH model using Bootstrapping",min=0.0) 4

# h 1800 1867

# median 1868 1868+nsteps-1

# lower 1868 1868+nsteps-1 3

# upper 1868 1868+nsteps-1 3

And this extracts and graphs the lower 10% and 5% cumulative returns (again, per 100 DM).

set lower10 bstart bend = %fractiles(xdraws(t),.10)(1)

set lower05 bstart bend = %fractiles(xdraws(t),.05)(1)

graph(header="Extreme Cumulative Returns by Bootstrapping") 2

# lower10

# lower05

Full Program

open data garch.asc
data(format=free,org=columns) 1 1867 bp cd dm jy sf
*
set dlogdm = 100*log(dm/dm{1})
garch(p=1,q=1,exp,asymmetric,hseries=h) / dlogdm
*
* Pull out the coefficients for the EGARCH process
*
compute gmu=%beta(1),c=%beta(2),a=%beta(3),b=%beta(4),d=%beta(5)
*
* Estimation range, which is also the source range for bootstrapped residuals
*
compute gstart=%regstart(),gend=%regend()
*
set stdu = %resids/sqrt(h)
*
compute ndraws=10000
compute nsteps=50
*
compute bstart=gend+1,bend=gend+nsteps
set h bstart bend = %na
set x bstart bend = %na
dec series[vect] hdraws xdraws
gset hdraws bstart bend = %zeros(ndraws,1)
gset xdraws bstart bend = %zeros(ndraws,1)
do draw=1,ndraws
   boot entries bstart bend gstart gend
   *
   * This is way the asymmetric EGARCH instruction is parameterized in
   * RATS. Nelson's original formulation is equivalent, but the c
   * coefficient has a different meaning. The recursion for the variance
   * uses the standardized residuals only.
   *
   set stdu bstart bend = su=stdu(entries(t)),$
             h=exp(c+b*log(h{1})+a*abs(stdu{1})+d*stdu{1}),$
              x=gmu+sqrt(h)*su,su
   *
   * Replace x by its accumulation
   *
   acc x bstart bend
   do t=bstart,bend
      compute hdraws(t)(draw)=h(t)
      compute xdraws(t)(draw)=x(t)
   end do t
end do draw
*
set lower bstart bend = %fractiles(hdraws(t),.05)(1)
set upper bstart bend = %fractiles(hdraws(t),.95)(1)
set median bstart bend = %fractiles(hdraws(t),.50)(1)
*
graph(header="Bootstrapped Variance Forecasts for EGARCH model",min=0.0) 4
# h      1800 1867
# median 1868 1868+nsteps-1
# lower 1868 1868+nsteps-1 3
# upper 1868 1868+nsteps-1 3
*
set lower10 bstart bend = %fractiles(xdraws(t),.10)(1)
set lower05 bstart bend = %fractiles(xdraws(t),.05)(1)
graph(header="Extreme Cumulative Returns by Bootstrapping") 2
# lower10
# lower05

Output

GARCH 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 0.016275592 -1.71926 0.08556624

2. C -0.184511441 0.024287988 -7.59682 0.00000000

3. A 0.215085921 0.028043749 7.66966 0.00000000

4. B 0.967687031 0.009281720 104.25729 0.00000000

5. D -0.017257427 0.012664163 -1.36270 0.17297778

Graph