GARCHMVMAX.RPF

GARCHMVMAX.RPF is an example of estimation of a multivariate GARCH model using MAXIMIZE (for model types that can't be done with GARCH). This is described in detailed in "MV GARCH Extensions".

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})

* Estimation using MAXIMIZE

* The initial few lines of this set the estimation range, which needs to

* be done explicitly, and the number of variables. Then, vectors for the

* dependent variables, residuals and residuals formulas are set up. The

* SET instructions copy the dependent variables over into the slots in

* the vector of series.

compute gstart=2,gend=6237

compute n=3

dec vect[series] y(n) u(n)

dec vect[frml] resid(n)

set y(1) = xjpn

set y(2) = xfra

set y(3) = xsui

* This is specific to a mean-only model. It sets up the formulas (the &i

* are needed in the formula definitions when the FRML is defined in a

* loop), and estimates them using NLSYSTEM. This both initializes the

* mean parameters, and computes the unconditional covariance matrix. If

* you want more general mean equations, the simplest way to do that

* would be to define each FRML separately.

dec vect mu(n)

nonlin(parmset=meanparms) mu

do i=1,n

frml resid(i) = (y(&i)-mu(&i))

stats y(i)

compute mu(i)=%mean

end do i

nlsystem(parmset=meanparms,resids=u) gstart gend resid

compute rr=%sigma

* The paths of the covariance matrices and uu' are saved in the

* SERIES[SYMM] names H and UU. UX and HX are used to pull in residuals

* and H matrices.

declare series[symm] h uu

* ux is used when extracting a u vector

declare symm hx(n,n)

declare vect ux(n)

* These are used to initialize pre-sample variances.

gset h * gend = rr

gset uu * gend = rr

* This is a standard (normal) log likelihood formula for any

* multivariate GARCH model. The difference among these will be in the

* definitions of HF and RESID. The function %XT pulls information out of

* a matrix of SERIES.

declare frml[symm] hf

frml logl = $

hx = hf(t) , $

%do(i,1,n,u(i)=resid(i)) , $

ux = %xt(u,t), $

h(t)=hx, uu(t)=%outerxx(ux), $

%logdensity(hx,ux)

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

* Standard GARCH(1,1)

dec symm vcs(n,n) vas(n,n) vbs(n,n)

compute vcs=rr*(1-.05-.80),vas=%const(0.05),vbs=%const(0.80)

nonlin(parmset=garchparms) vcs vas vbs

frml hf = vcs+vas.*uu{1}+vbs.*h{1}

maximize(parmset=meanparms+garchparms,pmethod=simplex,piters=10,$

iters=400) logl gstart gend

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

* CC

* The correlations are parameterized using an (n-1)x(n-1) matrix for the

* subdiagonal. The (i,j) element of this will actually be the

* correlation between i+1 and j.

dec symm qc(n-1,n-1)

dec vect vcv(n) vav(n) vbv(n)

function hfcccgarch time

type symm hfcccgarch

type integer time

do i=1,n

compute hx(i,i)=vcv(i)+$

vbv(i)*h(time-1)(i,i)+vav(i)*uu(time-1)(i,i)

end do i

compute hfcccgarch=%corrtocv(qc,%xdiag(hx))

end

frml hf = hfcccgarch(t)

nonlin(parmset=garchparms) vcv vav vbv qc

compute vcv=%xdiag(rr*(1-.05-.80)),vav=%const(0.05),$

vbv=%const(0.80),qc=%const(0.0)

maximize(parmset=meanparms+garchparms,pmethod=simplex,piters=10) $

logl gstart gend

Output

Statistics on Series Y(1)

Observations 6236

Sample Mean 0.014351 Variance 0.398893

Standard Error 0.631580 SE of Sample Mean 0.007998

t-Statistic (Mean=0) 1.794334 Signif Level (Mean=0) 0.072808

Skewness 0.784016 Signif Level (Sk=0) 0.000000

Kurtosis (excess) 13.265486 Signif Level (Ku=0) 0.000000

Jarque-Bera 46362.538224 Signif Level (JB=0) 0.000000

Statistics on Series Y(2)

Observations 6236

Sample Mean -0.001885 Variance 0.426428

Standard Error 0.653015 SE of Sample Mean 0.008269

t-Statistic (Mean=0) -0.227972 Signif Level (Mean=0) 0.819675

Skewness -0.073929 Signif Level (Sk=0) 0.017182

Kurtosis (excess) 6.554159 Signif Level (Ku=0) 0.000000

Jarque-Bera 11167.339771 Signif Level (JB=0) 0.000000

Statistics on Series Y(3)

Observations 6236

Sample Mean 0.015834 Variance 0.587975

Standard Error 0.766795 SE of Sample Mean 0.009710

t-Statistic (Mean=0) 1.630665 Signif Level (Mean=0) 0.103012

Skewness -0.014788 Signif Level (Sk=0) 0.633634

Kurtosis (excess) 3.447021 Signif Level (Ku=0) 0.000000

Jarque-Bera 3087.554594 Signif Level (JB=0) 0.000000

Non-Linear System Estimation

Convergence in 4 Iterations. Final criterion was 0.0000000 <= 0.0000100

Usable Observations 6236

Log Likelihood -14518.1757

Standard Error of Estimate 0.6315293032

Sum of Squared Residuals 2487.0992707

Durbin-Watson Statistic 1.9120

Standard Error of Estimate 0.6529622576

Sum of Squared Residuals 2658.7791504

Durbin-Watson Statistic 1.9313

Standard Error of Estimate 0.7667335311

Sum of Squared Residuals 3666.0215993

Durbin-Watson Statistic 1.9271

Variable Coeff Std Error T-Stat Signif

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

1. MU(1) 0.014350879 0.007997246 1.79448 0.07273698

2. MU(2) -0.001885176 0.008268658 -0.22799 0.81965356

3. MU(3) 0.015834012 0.009709378 1.63080 0.10293343

MAXIMIZE - Estimation by BFGS

Convergence in 94 Iterations. Final criterion was 0.0000000 <= 0.0000100

Usable Observations 6236

Function Value -11835.6553

Variable Coeff Std Error T-Stat Signif

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

1. MU(1) 0.004645759 0.006713523 0.69200 0.48893720

2. MU(2) -0.003489248 0.006478797 -0.53856 0.59018757

3. MU(3) -0.002354764 0.007610826 -0.30940 0.75701981

4. VCS(1,1) 0.009018591 0.001215970 7.41679 0.00000000

5. VCS(2,1) 0.005698278 0.000759998 7.49776 0.00000000

6. VCS(2,2) 0.011511699 0.001388933 8.28816 0.00000000

7. VCS(3,1) 0.006015431 0.000794083 7.57532 0.00000000

8. VCS(3,2) 0.009938182 0.001266596 7.84637 0.00000000

9. VCS(3,3) 0.012774291 0.001612142 7.92380 0.00000000

10. VAS(1,1) 0.105849650 0.007721223 13.70892 0.00000000

11. VAS(2,1) 0.093970112 0.006254861 15.02353 0.00000000

12. VAS(2,2) 0.128202766 0.007396975 17.33178 0.00000000

13. VAS(3,1) 0.088843871 0.005550942 16.00519 0.00000000

14. VAS(3,2) 0.113665791 0.006069273 18.72807 0.00000000

15. VAS(3,3) 0.111560471 0.006034501 18.48711 0.00000000

16. VBS(1,1) 0.883932095 0.007840007 112.74634 0.00000000

17. VBS(2,1) 0.890954842 0.006562869 135.75692 0.00000000

18. VBS(2,2) 0.860825075 0.007291479 118.05905 0.00000000

19. VBS(3,1) 0.897614729 0.005767301 155.63860 0.00000000

20. VBS(3,2) 0.874828215 0.006242926 140.13112 0.00000000

21. VBS(3,3) 0.877439261 0.005991219 146.45422 0.00000000

MAXIMIZE - Estimation by BFGS

Convergence in 53 Iterations. Final criterion was 0.0000000 <= 0.0000100

Usable Observations 6236

Function Value -12817.3747

Variable Coeff Std Error T-Stat Signif

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

1. MU(1) -0.000774394 0.006339526 -0.12215 0.90277757

2. MU(2) -0.004265435 0.006088158 -0.70061 0.48354537

3. MU(3) 0.003650038 0.007187500 0.50783 0.61157161

4. VCV(1) 0.016829240 0.002136267 7.87787 0.00000000

5. VCV(2) 0.028388105 0.002501625 11.34787 0.00000000

6. VCV(3) 0.032309332 0.002924935 11.04617 0.00000000

7. VAV(1) 0.164136541 0.011604019 14.14480 0.00000000

8. VAV(2) 0.133212681 0.008151332 16.34244 0.00000000

9. VAV(3) 0.112704988 0.007014455 16.06753 0.00000000

10. VBV(1) 0.812626023 0.012303587 66.04789 0.00000000

11. VBV(2) 0.804332251 0.011084932 72.56087 0.00000000

12. VBV(3) 0.831308142 0.010000443 83.12713 0.00000000

13. QC(1,1) 0.564318758 0.008808561 64.06481 0.00000000

14. QC(2,1) 0.579294828 0.008496648 68.17922 0.00000000

15. QC(2,2) 0.828697094 0.003757082 220.56934 0.00000000

Note that the QC's in this are offset by 1 in the row, so QC(1,1) is actually the correlation between 2 and 1.