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Examples / GARCHMV.RPF |
GARCHMV.RPF is an illustrative example which includes several variants on multivariate GSARCH models, including "stock" estimates for DVECH, BEKK, CC and DCC models, plus GARCH with non-standard mean models (different explanatory variables in each) and GARCH with several types of M effects.
Note that this illustrates a wide range of GARCH models applied to a single set of data. In practice, you would focus in on one or two model types. Specifying, estimating and testing these types of models forms a large part of the RATS ARCH/GARCH and Volatility Models e-course, and we strongly recommend that you get that if you are planning to focus in this area.
This takes quite a while to run—it's a fairly large data set and it's doing many models. The notes on the instructions are mixed in with the output below.
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})
*
* Examples with the different choices for the MV option
*
garch(p=1,q=1,pmethod=simplex,piters=10) / xjpn xfra xsui
garch(p=1,q=1,mv=bekk,pmethod=simplex,piters=10) / xjpn xfra xsui
*
* Restricted correlation models
*
garch(p=1,q=1,mv=cc) / xjpn xfra xsui
garch(p=1,q=1,mv=dcc) / xjpn xfra xsui
garch(p=1,q=1,mv=cholesky) / xjpn xfra xsui
*
* CC with VARMA variances
* This needs some special treatment to get convergence. Along with extra
* iterations on the GARCH, it uses NLPAR with DERIVE=FOURTH which does
* slower, but more accurate, numerical derivatives. EXACTLINE does
* slower exact line minimizations. These usually aren't necessary, but
* some model types can have some optimization problems that can require
* more careful treatment.
*
nlpar(derive=fourth,exactline)
garch(p=1,q=1,mv=cc,variances=varma,iters=500,pmethod=simplex,piters=10) / $
xjpn xfra xsui
*
* This resets the NLPAR to the standard values
*
nlpar(derive=first,noexactline)
*
* EWMA with t-errors with an estimated degrees of freedom parameter
*
garch(p=1,q=1,mv=ewma,distrib=t) / xjpn xfra xsui
*
* CC-EGARCH with asymmetry
*
garch(p=1,q=1,mv=cc,asymmetric,variances=exp) / xjpn xfra xsui
*
* Estimates with graphs of conditional correlations
*
garch(p=1,q=1,mv=dcc,variances=koutmos,hmatrices=hh,iters=500) / $
xjpn xfra xsui
*
set jpnfra = %cvtocorr(hh(t))(1,2)
set jpnsui = %cvtocorr(hh(t))(1,3)
set frasui = %cvtocorr(hh(t))(2,3)
*
spgraph(vfields=3,footer="Conditional Correlations")
graph(header="Japan with France",min=-1.0,max=1.0)
# jpnfra
graph(header="Japan with Switzerland",min=-1.0,max=1.0)
# jpnsui
graph(header="France with Switzerland",min=-1.0,max=1.0)
# frasui
spgraph(done)
*
* Estimates for a BEKK with t errors, saving the residuals and the
* variances (in the VECT[SERIES] and SYMM[SERIES] forms), and using them
* to compute the empirical probability of a residual (for Japan) being
* in the left .05 tail.
*
garch(p=1,q=1,mv=bekk,pmethod=simplex,piters=10,distrib=t,$
rseries=rs,mvhseries=hhs) / xjpn xfra xsui
*
compute fixt=(%shape-2)/%shape
set trigger = %tcdf(rs(1)/sqrt(hhs(1,1)*fixt),%shape)<.05
sstats(mean) / trigger>>VaRp
disp "Probability of being below .05 level" #.#### VaRp
*
* Univariate AR(1) mean models for each series, DCC model for the variance
*
equation(constant) jpneq xjpn 1
equation(constant) fraeq xfra 1
equation(constant) suieq xsui 1
group ar1 jpneq fraeq suieq
garch(p=1,q=1,model=ar1,mv=dcc,pmethod=simplex,piter=10)
*
* VAR(1) model for the mean, BEKK for the variance
*
system(model=var1)
variables xjpn xfra xsui
lags 1
det constant
end(system)
*
garch(p=1,q=1,model=var1,mv=bekk,pmethod=simplex,piters=10)
*
* GARCH-M model
*
dec symm[series] hhs(3,3)
clear(zeros) hhs
*
equation jpneq xjpn
# constant hhs(1,1) hhs(1,2) hhs(1,3)
equation fraeq xfra
# constant hhs(2,1) hhs(2,2) hhs(2,3)
equation suieq xsui
# constant hhs(3,1) hhs(3,2) hhs(3,3)
*
group garchm jpneq fraeq suieq
garch(model=garchm,p=1,q=1,pmethod=simplex,piters=10,iters=500,$
mvhseries=hhs)
*
* GARCH-M with using custom function of the variance (in this case, the
* square root of the variance of an equally weighted sum of the
* currencies).
*
set commonsd = 0.0
system(model=customm)
variables xjpn xfra xsui
det constant commonsd
end(system)
*
compute %nvar=%modelsize(customm)
compute weights=%fill(%nvar,1,1.0/%nvar)
garch(model=customm,p=1,q=1,mv=cc,variances=spillover,pmethod=simplex,piters=5,$
iters=500,hmatrices=hh,hadjust=(commonsd=sqrt(%qform(hh,weights))))
*
* VMA(1) mean model
*
dec vect[series] u(3)
clear(zeros) u
system(model=varma)
variables xjpn xfra xsui
det constant u(1){1} u(2){1} u(3){1}
end(system)
*
garch(model=varma,p=1,q=1,mv=bekk,asymmetric,rseries=u,$
pmethod=simplex,piters=10,iters=500)
*
* Estimate an asymmetric BEKK on the VAR1 with t errors, saving various statistics
* for diagnostics
*
nlpar(derive=fourth,exactline)
garch(model=var1,mv=bekk,asymmetric,p=1,q=1,distrib=t,$
pmethod=simplex,piters=10,iters=500,$
rseries=rs,mvhseries=hhs,stdresids=zu,derives=dd)
*
* Diagnostics on (univariate) standardized residuals
*
set z1 = rs(1)/sqrt(hhs(1,1))
set z2 = rs(2)/sqrt(hhs(2,2))
set z3 = rs(3)/sqrt(hhs(3,3))
@bdindtests(number=40) z1
@bdindtests(number=40) z2
@bdindtests(number=40) z3
*
* Multivariate Q statistic and ARCH test on jointly standardized
* residuals.
*
@mvqstat(lags=5)
# zu
@mvarchtest(lags=5)
# zu
*
* Fluctuations test
*
@flux
# dd
Output
This is generated by
garch(p=1,q=1,pmethod=simplex,piters=10) / xjpn xfra xsui
which is a standard (or DVECH) model (the default model for GARCH). Since the covariance matrix is symmetric, only the lower triangle needs to be modeled. In the output, C(i,j) is the variance constant, A(i,j) is the lagged squared residual ("ARCH") coefficient and B(i,j) is the lagged variance ("GARCH") coefficient. Because the components of the covariance matrix are modeled separately, it's possible for the H matrix to be non-positive definite for some parameters (even if all are positive). As a result, a DVECH can be hard to estimate, particularly if applied to data series which don't have relatively similar behavior.
MV-GARCH - Estimation by BFGS
Convergence in 122 Iterations. Final criterion was 0.0000063 <= 0.0000100
Usable Observations 6236
Log Likelihood -11835.6549
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.004660940 0.006798380 0.68560 0.49296810
2. Mean(XFRA) -0.003498719 0.006868955 -0.50935 0.61050525
3. Mean(XSUI) -0.002360634 0.008084067 -0.29201 0.77027846
4. C(1,1) 0.009016960 0.001272675 7.08504 0.00000000
5. C(2,1) 0.005697569 0.000749382 7.60303 0.00000000
6. C(2,2) 0.011503488 0.001287755 8.93298 0.00000000
7. C(3,1) 0.006015917 0.000806761 7.45688 0.00000000
8. C(3,2) 0.009933856 0.001236798 8.03192 0.00000000
9. C(3,3) 0.012775891 0.001565976 8.15842 0.00000000
10. A(1,1) 0.105800775 0.007736000 13.67642 0.00000000
11. A(2,1) 0.093889723 0.006062327 15.48741 0.00000000
12. A(2,2) 0.128144732 0.007036947 18.21027 0.00000000
13. A(3,1) 0.088759105 0.005470840 16.22404 0.00000000
14. A(3,2) 0.113620201 0.005943420 19.11697 0.00000000
15. A(3,3) 0.111522114 0.006146301 18.14459 0.00000000
16. B(1,1) 0.883968076 0.007890261 112.03280 0.00000000
17. B(2,1) 0.891020282 0.006097508 146.12859 0.00000000
18. B(2,2) 0.860886644 0.007137910 120.60766 0.00000000
19. B(3,1) 0.897670507 0.005628248 159.49376 0.00000000
20. B(3,2) 0.874866959 0.006324082 138.33897 0.00000000
21. B(3,3) 0.877454252 0.006493130 135.13579 0.00000000
This is generated by
garch(p=1,q=1,mv=bekk,pmethod=simplex,piters=10) / xjpn xfra xsui
which estimates a BEKK GARCH.
Because of the standard use of the transpose of \({\bf{A}}\) as the pre-multiplying matrix, the coefficients (unfortunately) have the opposite interpretation as they do for almost all other forms of GARCH models: A(i,j) is the effect of residual i on variable j, rather than j on i. However, note that it is very difficult to interpret the individual coefficients anyway.
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 86 Iterations. Final criterion was 0.0000086 <= 0.0000100
Usable Observations 6236
Log Likelihood -11821.7457
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.005284238 0.005779110 0.91437 0.36052308
2. Mean(XFRA) -0.002360430 0.004150748 -0.56868 0.56957616
3. Mean(XSUI) -0.002505826 0.004919824 -0.50933 0.61051925
4. C(1,1) 0.082827551 0.005082889 16.29537 0.00000000
5. C(2,1) 0.029966933 0.006774462 4.42352 0.00000971
6. C(2,2) 0.055802023 0.004748283 11.75204 0.00000000
7. C(3,1) 0.037995437 0.007654196 4.96400 0.00000069
8. C(3,2) -0.004017902 0.006806450 -0.59031 0.55498413
9. C(3,3) 0.058506480 0.006245748 9.36741 0.00000000
10. A(1,1) 0.359535262 0.012077169 29.76983 0.00000000
11. A(1,2) 0.102691494 0.009048917 11.34848 0.00000000
12. A(1,3) 0.111082248 0.011812194 9.40403 0.00000000
13. A(2,1) 0.038123247 0.014041254 2.71509 0.00662580
14. A(2,2) 0.403444341 0.016261862 24.80923 0.00000000
15. A(2,3) -0.066355330 0.013370391 -4.96286 0.00000069
16. A(3,1) -0.047522551 0.010449514 -4.54782 0.00000542
17. A(3,2) -0.125553482 0.012149506 -10.33404 0.00000000
18. A(3,3) 0.291344292 0.010526939 27.67607 0.00000000
19. B(1,1) 0.935272064 0.003791535 246.67373 0.00000000
20. B(1,2) -0.026717483 0.003105677 -8.60279 0.00000000
21. B(1,3) -0.028574502 0.004086792 -6.99192 0.00000000
22. B(2,1) -0.012475081 0.005562790 -2.24259 0.02492300
23. B(2,2) 0.909746081 0.006295125 144.51597 0.00000000
24. B(2,3) 0.029269416 0.005403727 5.41652 0.00000006
25. B(3,1) 0.016548608 0.004489013 3.68647 0.00022739
26. B(3,2) 0.048830173 0.004991829 9.78202 0.00000000
27. B(3,3) 0.946852761 0.004722801 200.48544 0.00000000
This is generated by
garch(p=1,q=1,mv=cc) / xjpn xfra xsui
which does a CC (Constant Correlation) model. The variances themselves are computed using separate equations for each variable. This uses the default for the latter, which is a simple univariate GARCH model for each variable.
In the output, C(i) is the variance constant, A(i) is the lagged squared residual ("ARCH") coefficient and B(i) is the lagged variance ("GARCH") coefficient. Only the off-diagonal of the constant correlation \({\bf{R}}\) matrix needs to be estimated, since it's symmetric with ones on the diagonal. R(i,j) is the correlation between the residuals for variables i and j.
MV-CC GARCH - Estimation by BFGS
Convergence in 52 Iterations. Final criterion was 0.0000029 <= 0.0000100
Usable Observations 6236
Log Likelihood -12817.3747
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) -0.000774432 0.006309040 -0.12275 0.90230536
2. Mean(XFRA) -0.004265646 0.006300943 -0.67699 0.49841526
3. Mean(XSUI) 0.003649864 0.007495332 0.48695 0.62629259
4. C(1) 0.016829173 0.001976098 8.51636 0.00000000
5. C(2) 0.028388500 0.002563720 11.07317 0.00000000
6. C(3) 0.032309814 0.003032845 10.65330 0.00000000
7. A(1) 0.164136506 0.010230563 16.04374 0.00000000
8. A(2) 0.133212898 0.008828392 15.08915 0.00000000
9. A(3) 0.112705972 0.006884732 16.37042 0.00000000
10. B(1) 0.812626460 0.011288583 71.98658 0.00000000
11. B(2) 0.804331051 0.011894292 67.62328 0.00000000
12. B(3) 0.831306635 0.010391077 80.00197 0.00000000
13. R(2,1) 0.564319761 0.008374994 67.38151 0.00000000
14. R(3,1) 0.579295418 0.008040597 72.04632 0.00000000
15. R(3,2) 0.828697521 0.003726935 222.35361 0.00000000
The next output is generated by
garch(p=1,q=1,mv=dcc) / xjpn xfra xsui
DCC ("Dynamic Conditional Correlations") was proposed by Engle(2002) to handle a bigger set of variables than the more fully parameterized models (such as DVECH and BEKK), without requiring the conditional correlations to be constant as in the CC. This adds two scalar parameters which govern a “GARCH(1,1)” model on the covariance matrix as a whole.
The output adds the DCC(A) and DCC(B) parameters which are the \(a\) and \(b\) in the \(Q\) recursion.
Note that, while CC and DCC are not nested, the fact that the log likelihood different is roughly +1000 in favor of DCC indicates the CC is clearly inadequate.
MV-DCC GARCH - Estimation by BFGS
Convergence in 48 Iterations. Final criterion was 0.0000054 <= 0.0000100
Usable Observations 6236
Log Likelihood -11814.4403
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.003986947 0.006357057 0.62717 0.53054863
2. Mean(XFRA) -0.003134785 0.005767888 -0.54349 0.58679300
3. Mean(XSUI) -0.003074034 0.006627697 -0.46382 0.64277935
4. C(1) 0.008500425 0.001063138 7.99560 0.00000000
5. C(2) 0.012488407 0.001255711 9.94528 0.00000000
6. C(3) 0.016570623 0.001779164 9.31371 0.00000000
7. A(1) 0.151681007 0.008998322 16.85659 0.00000000
8. A(2) 0.138405513 0.007920769 17.47375 0.00000000
9. A(3) 0.123714323 0.007255348 17.05147 0.00000000
10. B(1) 0.851988485 0.007806082 109.14419 0.00000000
11. B(2) 0.848502716 0.008012928 105.89172 0.00000000
12. B(3) 0.857976417 0.007724640 111.07009 0.00000000
13. DCC(A) 0.053241934 0.003272021 16.27188 0.00000000
14. DCC(B) 0.939057943 0.003968913 236.60331 0.00000000
The next output is generated by
garch(p=1,q=1,mv=cholesky) / xjpn xfra xsui
The Cholesky model is a rarely used alternative restricted covariance model which borrows some ideas from the structural VAR literature.
MV-Cholesky GARCH - Estimation by BFGS
Convergence in 45 Iterations. Final criterion was 0.0000021 <= 0.0000100
Usable Observations 6236
Log Likelihood -12173.5176
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.000188426 0.005612913 0.03357 0.97321995
2. Mean(XFRA) -0.006990799 0.005886029 -1.18769 0.23495414
3. Mean(XSUI) -0.004037401 0.007076942 -0.57050 0.56833812
4. C(1) 0.007428676 0.001051529 7.06464 0.00000000
5. C(2) 0.007284625 0.001161357 6.27251 0.00000000
6. C(3) 0.005181124 0.000836962 6.19039 0.00000000
7. A(1) 0.176287644 0.011581222 15.22185 0.00000000
8. A(2) 0.146646259 0.011453410 12.80372 0.00000000
9. A(3) 0.183556643 0.015164645 12.10425 0.00000000
10. B(1) 0.837268494 0.009216301 90.84648 0.00000000
11. B(2) 0.841732018 0.011376399 73.98932 0.00000000
12. B(3) 0.807020065 0.015552839 51.88892 0.00000000
13. F(2,1) 0.561328027 0.010939180 51.31354 0.00000000
14. F(3,1) 0.675957508 0.012167015 55.55656 0.00000000
15. F(3,2) 0.919538827 0.009235754 99.56294 0.00000000
The next output is generated by
nlpar(derive=fourth,exactline)
garch(p=1,q=1,mv=cc,variances=varma,iters=500,pmethod=simplex,piters=5) / $
xjpn xfra xsui
which does a CC model but with VARMA variances rather than the (default) variances used earlier. Note that VARIANCES=VARMA is a rather difficult model to fit which is why we use some rather high-end adjustments to the non-linear estimation process using NLPAR.
In the output, C(i) is the constant in the variance equation for variable i, A(i,j) is the coefficient for the "ARCH" term in the variance for i using the lagged squared residuals for variable j and B(i,j) is the coefficient in the "GARCH" term for computing the variance of i using the lagged variance of variable j.
MV-CC GARCH with VARMA Variances - Estimation by BFGS
Convergence in 102 Iterations. Final criterion was 0.0000000 <= 0.0000100
Usable Observations 6236
Log Likelihood -12679.6117
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) -0.000736365 0.005687626 -0.12947 0.89698745
2. Mean(XFRA) -0.004928502 0.006021537 -0.81848 0.41308370
3. Mean(XSUI) 0.004522104 0.007369896 0.61359 0.53948548
4. C(1) 0.011009851 0.001106113 9.95364 0.00000000
5. C(2) 0.011489265 0.001804615 6.36660 0.00000000
6. C(3) 0.021304610 0.001955298 10.89584 0.00000000
7. A(1,1) 0.129910659 0.009444814 13.75471 0.00000000
8. A(1,2) 0.032903256 0.004351554 7.56127 0.00000000
9. A(1,3) -0.002625706 0.000774117 -3.39187 0.00069416
10. A(2,1) 0.029611897 0.003823820 7.74406 0.00000000
11. A(2,2) 0.142254621 0.010294643 13.81832 0.00000000
12. A(2,3) -0.012085976 0.001802078 -6.70669 0.00000000
13. A(3,1) 0.002481219 0.005370488 0.46201 0.64407418
14. A(3,2) 0.032937641 0.007797229 4.22428 0.00002397
15. A(3,3) 0.089359548 0.007678690 11.63734 0.00000000
16. B(1,1) 0.860433868 0.008348932 103.05915 0.00000000
17. B(1,2) -0.032049330 0.004814092 -6.65740 0.00000000
18. B(1,3) 0.003083861 0.002378905 1.29634 0.19485967
19. B(2,1) -0.027854446 0.003415045 -8.15639 0.00000000
20. B(2,2) 0.787363196 0.017806848 44.21688 0.00000000
21. B(2,3) 0.054014152 0.010142159 5.32571 0.00000010
22. B(3,1) -0.007390051 0.006071894 -1.21709 0.22356937
23. B(3,2) -0.027511277 0.011884229 -2.31494 0.02061621
24. B(3,3) 0.877652457 0.012249078 71.65049 0.00000000
25. R(2,1) 0.559873267 0.008657402 64.66990 0.00000000
26. R(3,1) 0.573176839 0.008445591 67.86699 0.00000000
27. R(3,2) 0.830630501 0.003937859 210.93455 0.00000000
The next output is generated by
garch(p=1,q=1,mv=ewma,distrib=t) / xjpn xfra xsui
which does an EWMA (Exponentially Weighted Moving Average) model. This is a very tightly parameterized variance model which has just a single real parameter (Alpha in the output) governing the evolution of the covariance matrix.
Note that the log likelihood is substantially better than the previous models because this uses Student t errors (DISTRIB=T option)—all the previous examples assumed conditionally Normal residuals.
MV-GARCH, EWMA - Estimation by BFGS
Convergence in 12 Iterations. Final criterion was 0.0000058 <= 0.0000100
Usable Observations 6236
Log Likelihood -10516.1908
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) -0.005311325 0.003704962 -1.43357 0.15169474
2. Mean(XFRA) -0.004245558 0.004184190 -1.01467 0.31026475
3. Mean(XSUI) -0.005260662 0.005173644 -1.01682 0.30923931
4. Alpha 0.049804403 0.001974483 25.22402 0.00000000
5. Shape 5.253165403 0.139850219 37.56280 0.00000000
The next is from
garch(p=1,q=1,mv=cc,asymmetric,variances=exp) / xjpn xfra xsui
This is a CC model with the individual variances computed using E-GARCH with asymmetry, which is chosen using the combination of VARIANCES=EXP and ASYMMETRIC. In the output C(i) is the constant in the log variance equation for variable i, A(i) is the coefficient on the standardized lag absolute residual, B(i) is the coefficient on the lagged log variance, D(i) is the asymmetry coefficient, which will be zero if the variance responds identically to positive vs negative residuals, and will be negative if the variance increases more with a negative residual than with a similarly sized positive residual.
MV-CC GARCH with E-GARCH Variances - Estimation by BFGS
Convergence in 46 Iterations. Final criterion was 0.0000050 <= 0.0000100
Usable Observations 6236
Log Likelihood -12733.4127
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) -0.001061134 0.005995626 -0.17698 0.85952042
2. Mean(XFRA) -0.007507476 0.006656264 -1.12788 0.25937002
3. Mean(XSUI) 0.006646038 0.008248918 0.80569 0.42042391
4. C(1) -0.388302831 0.016520648 -23.50409 0.00000000
5. C(2) -0.280006611 0.014158350 -19.77678 0.00000000
6. C(3) -0.212549903 0.012904538 -16.47094 0.00000000
7. A(1) 0.390917065 0.016363203 23.89001 0.00000000
8. A(2) 0.265106474 0.013130072 20.19079 0.00000000
9. A(3) 0.216811643 0.012890599 16.81936 0.00000000
10. B(1) 0.894099733 0.007099411 125.93999 0.00000000
11. B(2) 0.912114139 0.006201980 147.06823 0.00000000
12. B(3) 0.926770474 0.006197222 149.54612 0.00000000
13. D(1) 0.019518794 0.009788554 1.99404 0.04614740
14. D(2) -0.011375872 0.006560111 -1.73410 0.08290078
15. D(3) 0.043824402 0.006302749 6.95322 0.00000000
16. R(2,1) 0.561037832 0.008673803 64.68187 0.00000000
17. R(3,1) 0.575786420 0.008422018 68.36680 0.00000000
18. R(3,2) 0.826049552 0.004068032 203.05878 0.00000000
The next is generated by
garch(p=1,q=1,mv=dcc,variances=koutmos,hmatrices=hh) / $
xjpn xfra xsui
This is a DCC model with the individual variances computed using the option VARIANCES=KOUTMOS. (This was introduced in Koutmos(1996)). This is an extension of the multivariate EGARCH but allows for "spillover" effects among the residuals.
In the output, A(i,j) is the loading of the index for variable j on the variance for i. D(j) is the asymmetry coefficient for variable j.
MV-DCC GARCH with Koutmos EGARCH Variances - Estimation by BFGS
Convergence in 230 Iterations. Final criterion was 0.0000098 <= 0.0000100
Usable Observations 6236
Log Likelihood -11667.5734
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.006024778 0.004630050 1.30123 0.19317842
2. Mean(XFRA) -0.000146329 0.005191954 -0.02818 0.97751558
3. Mean(XSUI) 0.003879846 0.006243991 0.62137 0.53435445
4. C(1) -0.333025563 0.014229055 -23.40462 0.00000000
5. C(2) -0.318645749 0.011582791 -27.51027 0.00000000
6. C(3) -0.256314323 0.011379009 -22.52519 0.00000000
7. A(1,1) 0.368281353 0.014178033 25.97549 0.00000000
8. A(1,2) 0.065355123 0.017493779 3.73591 0.00018704
9. A(1,3) -0.038663116 0.015996067 -2.41704 0.01564735
10. A(2,1) 0.089174917 0.008514294 10.47355 0.00000000
11. A(2,2) 0.312079544 0.016205921 19.25713 0.00000000
12. A(2,3) -0.021657608 0.015265041 -1.41877 0.15596558
13. A(3,1) 0.038346197 0.008701091 4.40706 0.00001048
14. A(3,2) 0.104655778 0.012043972 8.68947 0.00000000
15. A(3,3) 0.167712826 0.012842656 13.05905 0.00000000
16. B(1) 0.938873078 0.003994526 235.03992 0.00000000
17. B(2) 0.949210072 0.003458963 274.42045 0.00000000
18. B(3) 0.952207255 0.003918152 243.02459 0.00000000
19. D(1) 0.074075026 0.020036638 3.69698 0.00021818
20. D(2) -0.008025059 0.018932972 -0.42387 0.67166293
21. D(3) 0.164340488 0.028595478 5.74708 0.00000001
22. DCC(A) 0.050598072 0.003298070 15.34172 0.00000000
23. DCC(B) 0.943670624 0.003834867 246.07648 0.00000000
This estimates a BEKK with t errors, saving the residuals and the variances (in the VECT[SERIES] and SYMM[SERIES] forms). Those extra outputs will be used in the next calculation. The only addition to the output is the SHAPE parameter for degrees of freedom of the t.
garch(p=1,q=1,mv=bekk,pmethod=simplex,piters=10,distrib=t,$
rseries=rs,mvhseries=hhs) / xjpn xfra xsui
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 92 Iterations. Final criterion was 0.0000055 <= 0.0000100
Usable Observations 6236
Log Likelihood -10260.5399
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) -0.007107660 0.003858492 -1.84208 0.06546307
2. Mean(XFRA) -0.004612319 0.004092372 -1.12705 0.25972018
3. Mean(XSUI) -0.004487324 0.005066468 -0.88569 0.37578413
4. C(1,1) -0.017751180 0.004269828 -4.15735 0.00003220
5. C(2,1) 0.031717592 0.007622471 4.16106 0.00003168
6. C(2,2) 0.027368262 0.008516552 3.21354 0.00131111
7. C(3,1) 0.029154447 0.013872244 2.10164 0.03558493
8. C(3,2) -0.067679325 0.008945581 -7.56567 0.00000000
9. C(3,3) -0.000003986 0.083151803 -4.79320e-05 0.99996176
10. A(1,1) 0.291241803 0.013001384 22.40083 0.00000000
11. A(1,2) 0.001808702 0.012235897 0.14782 0.88248537
12. A(1,3) 0.008848857 0.016965762 0.52157 0.60196873
13. A(2,1) -0.004305175 0.009650004 -0.44613 0.65550192
14. A(2,2) 0.315102162 0.016932181 18.60966 0.00000000
15. A(2,3) -0.088257932 0.020523901 -4.30025 0.00001706
16. A(3,1) -0.007935956 0.006432005 -1.23382 0.21726880
17. A(3,2) -0.033315781 0.011216219 -2.97032 0.00297488
18. A(3,3) 0.347586918 0.018646535 18.64083 0.00000000
19. B(1,1) 0.963713100 0.002641411 364.84781 0.00000000
20. B(1,2) 0.000680630 0.002835135 0.24007 0.81027619
21. B(1,3) 0.002794142 0.004115180 0.67898 0.49714786
22. B(2,1) 0.000460633 0.003131357 0.14710 0.88305048
23. B(2,2) 0.941278157 0.005761179 163.38289 0.00000000
24. B(2,3) 0.045037314 0.007406098 6.08111 0.00000000
25. B(3,1) 0.002189996 0.002497809 0.87677 0.38061333
26. B(3,2) 0.021980487 0.004689952 4.68672 0.00000278
27. B(3,3) 0.927099189 0.006927691 133.82513 0.00000000
28. Shape(t degrees) 4.229200497 0.138321210 30.57521 0.00000000
This takes the residuals and covariance matrices just saved, and uses them compute the empirical probability of a residual (for Japan) being in the left .05 tail.
compute fixt=(%shape-2)/%shape
set trigger = %tcdf(rs(1)/sqrt(hhs(1,1)*fixt),%shape)<.05
sstats(mean) / trigger>>VaRp
disp "Probability of being below .05 level" #.#### VaRp
This is the output from that calculation.
Probability of being below .05 level 0.0420
This does a DCC variance model with a mean model with different explanatory variables for each equation (separate univariate AR(1)). The more complicated mean model is shown with a subheader for each equation followed by the coefficients which apply to it.
equation(constant) jpneq xjpn 1
equation(constant) fraeq xfra 1
equation(constant) suieq xsui 1
group ar1 jpneq fraeq suieq
garch(p=1,q=1,model=ar1,mv=dcc,pmethod=simplex,piter=10)
MV-DCC GARCH - Estimation by BFGS
Convergence in 50 Iterations. Final criterion was 0.0000000 <= 0.0000100
Usable Observations 6235
Log Likelihood -11810.9128
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. Constant 0.004012130 0.006527278 0.61467 0.53877188
2. XJPN{1} 0.025541364 0.012106316 2.10976 0.03487944
Mean Model(XFRA)
3. Constant -0.003121533 0.005934792 -0.52597 0.59890789
4. XFRA{1} 0.005552823 0.010310136 0.53858 0.59017737
Mean Model(XSUI)
5. Constant -0.003058080 0.006973783 -0.43851 0.66101596
6. XSUI{1} -0.001890715 0.010112046 -0.18698 0.85167907
7. C(1) 0.008296064 0.000941860 8.80817 0.00000000
8. C(2) 0.012328174 0.001261632 9.77161 0.00000000
9. C(3) 0.016479037 0.001792536 9.19314 0.00000000
10. A(1) 0.151627200 0.009158629 16.55567 0.00000000
11. A(2) 0.137359939 0.007952320 17.27294 0.00000000
12. A(3) 0.123129448 0.007012873 17.55763 0.00000000
13. B(1) 0.852578357 0.007747172 110.05026 0.00000000
14. B(2) 0.849687450 0.007917933 107.31177 0.00000000
15. B(3) 0.858564973 0.007545403 113.78650 0.00000000
16. DCC(A) 0.053146867 0.003102971 17.12774 0.00000000
17. DCC(B) 0.939177245 0.003732452 251.62473 0.00000000
This estimates a BEKK model with the mean model given by a VAR(1).
system(model=var1)
variables xjpn xfra xsui
lags 1
det constant
end(system)
*
garch(p=1,q=1,model=var1,mv=bekk,pmethod=simplex,piters=10)
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 96 Iterations. Final criterion was 0.0000080 <= 0.0000100
Usable Observations 6235
Log Likelihood -11809.4182
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. XJPN{1} 0.054047272 0.011942861 4.52549 0.00000603
2. XFRA{1} 0.023703172 0.012434190 1.90629 0.05661260
3. XSUI{1} -0.034353787 0.010046576 -3.41945 0.00062747
4. Constant 0.006198638 0.005824749 1.06419 0.28724271
Mean Model(XFRA)
5. XJPN{1} 0.028507734 0.010648194 2.67724 0.00742322
6. XFRA{1} 0.023745977 0.011737459 2.02309 0.04306350
7. XSUI{1} -0.009397730 0.009744988 -0.96437 0.33486271
8. Constant -0.001867249 0.004229528 -0.44148 0.65886617
Mean Model(XSUI)
9. XJPN{1} 0.040997212 0.013027364 3.14701 0.00164951
10. XFRA{1} 0.025073306 0.012911897 1.94188 0.05215209
11. XSUI{1} -0.018020929 0.009489028 -1.89913 0.05754694
12. Constant -0.001920828 0.005069808 -0.37888 0.70478004
13. C(1,1) 0.080435930 0.004793012 16.78192 0.00000000
14. C(2,1) 0.026820313 0.007480192 3.58551 0.00033642
15. C(2,2) 0.055106684 0.005020203 10.97698 0.00000000
16. C(3,1) 0.034208095 0.007693822 4.44618 0.00000874
17. C(3,2) -0.004050227 0.009476016 -0.42742 0.66907434
18. C(3,3) 0.058671874 0.005634198 10.41353 0.00000000
19. A(1,1) 0.356771446 0.011256418 31.69494 0.00000000
20. A(1,2) 0.099064871 0.009417503 10.51923 0.00000000
21. A(1,3) 0.107409312 0.012000034 8.95075 0.00000000
22. A(2,1) 0.033552159 0.014691203 2.28383 0.02238173
23. A(2,2) 0.398536021 0.016364355 24.35391 0.00000000
24. A(2,3) -0.070314051 0.018541044 -3.79235 0.00014923
25. A(3,1) -0.047183060 0.009893482 -4.76911 0.00000185
26. A(3,2) -0.122502522 0.012120930 -10.10669 0.00000000
27. A(3,3) 0.292916086 0.013683608 21.40635 0.00000000
28. B(1,1) 0.936382926 0.003618094 258.80557 0.00000000
29. B(1,2) -0.025339939 0.002987649 -8.48156 0.00000000
30. B(1,3) -0.027037423 0.003927755 -6.88368 0.00000000
31. B(2,1) -0.010670134 0.005503253 -1.93888 0.05251625
32. B(2,2) 0.911878410 0.006460352 141.14997 0.00000000
33. B(2,3) 0.030707222 0.006589697 4.65988 0.00000316
34. B(3,1) 0.016110359 0.003907448 4.12299 0.00003740
35. B(3,2) 0.047482297 0.005203515 9.12504 0.00000000
36. B(3,3) 0.946088060 0.004837787 195.56215 0.00000000
To do GARCH-M with a multivariate model, you have to plan ahead a bit. On your GARCH instruction, you need to use the option MVHSERIES=SYMM[SERIES] which will save the paths of the variances (example: MVHSERIES=HHS). Your regression equations for the means will include references to the elements of this array of series. Since those equations need to be created in advance, you need the to declare that as a SYMM[SERIES] first as well. The model here includes in each equation all the covariances which involve the residuals from an equation.
dec symm[series] hhs(3,3)
clear(zeros) hhs
*
equation jpneq xjpn
# constant hhs(1,1) hhs(1,2) hhs(1,3)
equation fraeq xfra
# constant hhs(2,1) hhs(2,2) hhs(2,3)
equation suieq xsui
# constant hhs(3,1) hhs(3,2) hhs(3,3)
*
group garchm jpneq fraeq suieq
garch(model=garchm,p=1,q=1,pmethod=simplex,piters=10,iters=500,$
mvhseries=hhs)
MV-GARCH - Estimation by BFGS
Convergence in 144 Iterations. Final criterion was 0.0000078 <= 0.0000100
Usable Observations 6236
Log Likelihood -11832.2625
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. Constant -0.007849113 0.009863216 -0.79580 0.42615031
2. HHS(1,1) 0.004435082 0.032556473 0.13623 0.89164156
3. HHS(2,1) -0.041757909 0.089146786 -0.46842 0.63948610
4. HHS(3,1) 0.091558025 0.076002136 1.20468 0.22832802
Mean Model(XFRA)
5. Constant -0.010751782 0.009107112 -1.18059 0.23776489
6. HHS(2,1) 0.069406060 0.044387649 1.56363 0.11790342
7. HHS(2,2) -0.026189907 0.038262864 -0.68447 0.49367638
8. HHS(3,2) 0.010818971 0.041245292 0.26231 0.79308398
Mean Model(XSUI)
9. Constant -0.004234646 0.011894647 -0.35601 0.72183103
10. HHS(3,1) 0.065188220 0.047639040 1.36838 0.17119375
11. HHS(3,2) 0.002382359 0.047947064 0.04969 0.96037159
12. HHS(3,3) -0.025794190 0.035166037 -0.73350 0.46325523
13. C(1,1) 0.008963132 0.000875514 10.23756 0.00000000
14. C(2,1) 0.005670072 0.000503384 11.26392 0.00000000
15. C(2,2) 0.011533285 0.000754783 15.28027 0.00000000
16. C(3,1) 0.005999971 0.000496130 12.09355 0.00000000
17. C(3,2) 0.009941202 0.000386110 25.74709 0.00000000
18. C(3,3) 0.012772167 0.000463267 27.56980 0.00000000
19. A(1,1) 0.105756073 0.006443928 16.41174 0.00000000
20. A(2,1) 0.093982155 0.005531661 16.98986 0.00000000
21. A(2,2) 0.128299751 0.006464150 19.84789 0.00000000
22. A(3,1) 0.088856383 0.005016732 17.71200 0.00000000
23. A(3,2) 0.113612844 0.005621794 20.20936 0.00000000
24. A(3,3) 0.111350277 0.005739693 19.40004 0.00000000
25. B(1,1) 0.884137252 0.006391777 138.32418 0.00000000
26. B(2,1) 0.891025082 0.005215409 170.84473 0.00000000
27. B(2,2) 0.860654492 0.005303599 162.27744 0.00000000
28. B(3,1) 0.897618057 0.004711600 190.51235 0.00000000
29. B(3,2) 0.874814093 0.004491115 194.78771 0.00000000
30. B(3,3) 0.877555588 0.004651904 188.64440 0.00000000
If you need some function of the variances or covariances, you need to use the HADJUST option to define it based upon the values that you save either with MVHSERIES or with HMATRICES. The following, for instance, adds the conditional standard deviation (generated into the series COMMONSD) of an equally weighted sum of the three currencies. This does a CC model using VARIANCES=SPILLOVER which includes cross terms (allowing for "spillover") on the lagged squared residuals.
set commonsd = 0.0
system(model=customm)
variables xjpn xfra xsui
det constant commonsd
end(system)
*
compute %nvar=%modelsize(customm)
compute weights=%fill(%nvar,1,1.0/%nvar)
garch(model=customm,p=1,q=1,mv=cc,variances=spillover,pmethod=simplex,piters=5,$
iters=500,hmatrices=hh,hadjust=(commonsd=sqrt(%qform(hh,weights))))
MV-CC GARCH with Spillover Variances - Estimation by BFGS
Convergence in 93 Iterations. Final criterion was 0.0000088 <= 0.0000100
Usable Observations 6236
Log Likelihood -12740.4462
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. Constant -0.029842586 0.019071561 -1.56477 0.11763711
2. COMMONSD 0.053667188 0.035746474 1.50133 0.13327071
Mean Model(XFRA)
3. Constant -0.047162588 0.021717377 -2.17165 0.02988191
4. COMMONSD 0.082817429 0.040949910 2.02241 0.04313421
Mean Model(XSUI)
5. Constant -0.028134340 0.025811939 -1.08997 0.27572462
6. COMMONSD 0.060123324 0.047615815 1.26268 0.20670582
7. C(1) 0.013447146 0.001870851 7.18772 0.00000000
8. C(2) 0.023003846 0.002332264 9.86331 0.00000000
9. C(3) 0.032222689 0.002844326 11.32876 0.00000000
10. A(1,1) 0.165972437 0.010884665 15.24828 0.00000000
11. A(1,2) 0.013450984 0.002765010 4.86471 0.00000115
12. A(1,3) -0.002086714 0.000322548 -6.46946 0.00000000
13. A(2,1) 0.023299804 0.003531985 6.59680 0.00000000
14. A(2,2) 0.153443254 0.010282048 14.92341 0.00000000
15. A(2,3) 0.009559265 0.002955966 3.23389 0.00122117
16. A(3,1) -0.000452464 0.003489263 -0.12967 0.89682495
17. A(3,2) 0.026380541 0.005891969 4.47737 0.00000756
18. A(3,3) 0.121021238 0.008060161 15.01474 0.00000000
19. B(1) 0.811204171 0.010975985 73.90719 0.00000000
20. B(2) 0.772015008 0.011621376 66.43061 0.00000000
21. B(3) 0.808656548 0.010370612 77.97578 0.00000000
22. R(2,1) 0.561778133 0.008001589 70.20833 0.00000000
23. R(3,1) 0.576805047 0.007896678 73.04401 0.00000000
24. R(3,2) 0.830563269 0.003711043 223.80858 0.00000000
This is an asymmetric BEKK model with a VMA (Vector Moving Average) mean model. The residuals (the U(1), U(2) and U(3) series) are generated recursively as the function is evaluated and saved used the RSERIES option. They are then used in computing the mean model for the next period.
dec vect[series] u(3)
clear(zeros) u
system(model=varma)
variables xjpn xfra xsui
det constant u(1){1} u(2){1} u(3){1}
end(system)
*
garch(model=varma,p=1,q=1,mv=bekk,asymmetric,rseries=u,$
pmethod=simplex,piters=10,iters=500)
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 98 Iterations. Final criterion was 0.0000091 <= 0.0000100
Usable Observations 6236
Log Likelihood -11750.8048
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. Constant 0.000517049 0.005483577 0.09429 0.92487840
2. U(1){1} 0.056542853 0.010905316 5.18489 0.00000022
3. U(2){1} 0.023799767 0.011324202 2.10167 0.03558195
4. U(3){1} -0.034939036 0.008870599 -3.93875 0.00008191
Mean Model(XFRA)
5. Constant -0.004620079 0.003787255 -1.21990 0.22250207
6. U(1){1} 0.026529499 0.009291297 2.85531 0.00429954
7. U(2){1} 0.024502516 0.011042365 2.21896 0.02648976
8. U(3){1} -0.004608662 0.007019772 -0.65653 0.51148581
Mean Model(XSUI)
9. Constant -0.003730446 0.004553395 -0.81927 0.41263412
10. U(1){1} 0.039552904 0.011341454 3.48746 0.00048763
11. U(2){1} 0.024216883 0.010049424 2.40978 0.01596222
12. U(3){1} -0.011612455 0.009982280 -1.16331 0.24470501
13. C(1,1) 0.071440677 0.004666243 15.31011 0.00000000
14. C(2,1) 0.019904067 0.006920749 2.87600 0.00402751
15. C(2,2) 0.049758322 0.005031042 9.89026 0.00000000
16. C(3,1) 0.029643632 0.007841235 3.78048 0.00015653
17. C(3,2) -0.006359708 0.010108110 -0.62917 0.52923851
18. C(3,3) 0.057138519 0.005507015 10.37559 0.00000000
19. A(1,1) 0.337768489 0.009543796 35.39142 0.00000000
20. A(1,2) 0.101314155 0.008208917 12.34196 0.00000000
21. A(1,3) 0.095841256 0.011174256 8.57697 0.00000000
22. A(2,1) 0.001144970 0.014701361 0.07788 0.93792201
23. A(2,2) 0.376879863 0.017268295 21.82496 0.00000000
24. A(2,3) -0.092997279 0.017098999 -5.43876 0.00000005
25. A(3,1) -0.038116822 0.009680120 -3.93764 0.00008229
26. A(3,2) -0.126887422 0.012663180 -10.02019 0.00000000
27. A(3,3) 0.303054636 0.013129243 23.08242 0.00000000
28. B(1,1) 0.934735090 0.003065463 304.92461 0.00000000
29. B(1,2) -0.027758350 0.002866379 -9.68412 0.00000000
30. B(1,3) -0.027667110 0.003421433 -8.08641 0.00000000
31. B(2,1) -0.006977941 0.005582143 -1.25005 0.21128239
32. B(2,2) 0.907298431 0.006209658 146.11086 0.00000000
33. B(2,3) 0.035183606 0.006772729 5.19489 0.00000020
34. B(3,1) 0.014406438 0.004140365 3.47951 0.00050233
35. B(3,2) 0.052042648 0.005061386 10.28229 0.00000000
36. B(3,3) 0.942431126 0.005114451 184.26827 0.00000000
37. D(1,1) 0.184171852 0.021219408 8.67941 0.00000000
38. D(1,2) 0.019941521 0.019292952 1.03362 0.30131529
39. D(1,3) 0.108275699 0.025745289 4.20565 0.00002603
40. D(2,1) 0.092772704 0.016025850 5.78894 0.00000001
41. D(2,2) 0.108997998 0.021529893 5.06264 0.00000041
42. D(2,3) 0.105291651 0.024522423 4.29369 0.00001757
43. D(3,1) -0.027926024 0.015632011 -1.78646 0.07402416
44. D(3,2) 0.072495801 0.012881016 5.62811 0.00000002
45. D(3,3) -0.025898035 0.021433408 -1.20830 0.22693102
This estimates an asymmetric BEKK, with the VAR(1) mean model and t distributed errors, saving various statistics for diagnostics. This again requires some extra care with NLPAR settings to properly estimate.
nlpar(derive=fourth,exactline)
garch(model=var1,mv=bekk,asymmetric,p=1,q=1,distrib=t,$
pmethod=simplex,piters=10,iters=500,$
rseries=rs,mvhseries=hhs,stdresids=zu,derives=dd)
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 128 Iterations. Final criterion was 0.0000096 <= 0.0000100
Usable Observations 6235
Log Likelihood -10204.7792
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. XJPN{1} -0.013922256 0.011718449 -1.18806 0.23480856
2. XFRA{1} 0.015596795 0.009916931 1.57274 0.11577808
3. XSUI{1} -0.008653617 0.007005863 -1.23520 0.21675742
4. Constant -0.008317954 0.003622105 -2.29644 0.02165061
Mean Model(XFRA)
5. XJPN{1} -0.010041997 0.010743524 -0.93470 0.34994172
6. XFRA{1} -0.004013781 0.008893109 -0.45134 0.65174735
7. XSUI{1} 0.020068634 0.008117448 2.47228 0.01342529
8. Constant -0.003343760 0.004088702 -0.81780 0.41346871
Mean Model(XSUI)
9. XJPN{1} -0.000913145 0.013654436 -0.06688 0.94668097
10. XFRA{1} 0.033303662 0.010263716 3.24480 0.00117535
11. XSUI{1} -0.032648518 0.010025903 -3.25642 0.00112828
12. Constant -0.003634622 0.005161392 -0.70419 0.48131189
13. C(1,1) 0.015388925 0.004323636 3.55926 0.00037191
14. C(2,1) -0.029941041 0.007329806 -4.08483 0.00004411
15. C(2,2) 0.021794519 0.008962191 2.43183 0.01502278
16. C(3,1) -0.015010306 0.016045852 -0.93546 0.34954960
17. C(3,2) -0.063157209 0.008171624 -7.72884 0.00000000
18. C(3,3) 0.000003465 0.057349090 6.04191e-05 0.99995179
19. A(1,1) 0.280572594 0.014279402 19.64876 0.00000000
20. A(1,2) 0.010020816 0.013238732 0.75693 0.44909081
21. A(1,3) 0.011505341 0.018402599 0.62520 0.53183854
22. A(2,1) -0.003963025 0.009942508 -0.39859 0.69019231
23. A(2,2) 0.312372338 0.018349368 17.02360 0.00000000
24. A(2,3) -0.075759059 0.021118346 -3.58736 0.00033405
25. A(3,1) -0.006188012 0.006596246 -0.93811 0.34818729
26. A(3,2) -0.023218635 0.012992926 -1.78702 0.07393404
27. A(3,3) 0.345681109 0.019083399 18.11423 0.00000000
28. B(1,1) 0.959879223 0.002999470 320.01626 0.00000000
29. B(1,2) -0.000179505 0.003237216 -0.05545 0.95577955
30. B(1,3) 0.001566655 0.004553897 0.34403 0.73082734
31. B(2,1) 0.000665128 0.003233844 0.20568 0.83704312
32. B(2,2) 0.939704790 0.006320441 148.67709 0.00000000
33. B(2,3) 0.041077450 0.008056670 5.09856 0.00000034
34. B(3,1) 0.001479810 0.002558464 0.57840 0.56299552
35. B(3,2) 0.020074720 0.005238068 3.83247 0.00012686
36. B(3,3) 0.928790712 0.007365728 126.09626 0.00000000
37. D(1,1) 0.185748046 0.024666145 7.53049 0.00000000
38. D(1,2) -0.030184227 0.027193491 -1.10998 0.26700771
39. D(1,3) 0.055188975 0.042029155 1.31311 0.18914531
40. D(2,1) 0.008546134 0.014572386 0.58646 0.55756587
41. D(2,2) 0.037431586 0.035568899 1.05237 0.29263055
42. D(2,3) 0.117933534 0.035422148 3.32937 0.00087042
43. D(3,1) -0.002750566 0.009722616 -0.28290 0.77725054
44. D(3,2) 0.071494173 0.020963580 3.41040 0.00064868
45. D(3,3) -0.088057871 0.034007411 -2.58937 0.00961509
46. Shape(t degrees) 4.185348587 0.135853155 30.80789 0.00000000
These do "univariate" diagnostics on the residuals. The Z variables are the residuals standardized by their variances, so they should (if the model is correct) be mean zero, variance one and be serially uncorrelated. The one diagnostic which fails badly across all three variables is the Ljung-Box Q statistic which tests for serial correlation in the mean. This might indicate that the VAR(1) mean model isn't adequate, though it can also mean more serious problems exist.
set z1 = rs(1)/sqrt(hhs(1,1))
set z2 = rs(2)/sqrt(hhs(2,2))
set z3 = rs(3)/sqrt(hhs(3,3))
@bdindtests(number=40) z1
@bdindtests(number=40) z2
@bdindtests(number=40) z3
Independence Tests for Series Z1
Test Statistic P-Value
Ljung-Box Q(40) 121.34208 0.0000
McLeod-Li(40) 56.02381 0.0476
Turning Points -1.32167 0.1863
Difference Sign -0.76761 0.4427
Rank Test -1.05722 0.2904
Independence Tests for Series Z2
Test Statistic P-Value
Ljung-Box Q(40) 100.66701 0.0000
McLeod-Li(40) 9.85788 1.0000
Turning Points 0.00000 1.0000
Difference Sign -0.24125 0.8094
Rank Test -0.42783 0.6688
Independence Tests for Series Z3
Test Statistic P-Value
Ljung-Box Q(40) 96.737938 0.0000
McLeod-Li(40) 45.299177 0.2604
Turning Points -1.231559 0.2181
Difference Sign -0.021932 0.9825
Rank Test -2.172303 0.0298
These are tests on the jointly standardized residuals which are (if the model is correct) mutually serially uncorrelated, mean zero with an identity covariance matrix. (The univariate standardized residuals don't have any predictable "cross-variable" properties). @MVQSTAT checks for serial correlation in the mean, while @MVARCHTEST checks for residual cross-variable ARCH. These also reject very strongly. The failure of the multivariate Q is not a surprise given the univariate Q results. The failure of the multivariate ARCH test is more of a surprise given that the univariate McLeod-Li tests weren't too much of a problem.
@mvqstat(lags=5)
# zu
@mvarchtest(lags=5)
# zu
Multivariate Q Test
Test Run Over 3 to 6237
Lags Tested 5
Degrees of Freedom 45
Q Statistic 143.1558
Signif Level 0.0000
Multivariate ARCH Test
Statistic Degrees Signif
520.49 180 0.00000
This does a Nyblom fluctuations test which is a fairly general test for structural breaks in the time sequence. This has a joint test on the entire coefficient vector, plus tests on the individual parameters. You can check the full GARCH output above to see which each coefficient is. The individual coefficient for which stability is really rejected is the SHAPE (#46). #28 (B(1,1), which is the own variance persistence on Japan) and #35 (B(3,2) which is the variance term for France from Switzerland) are also way over the 0.00 p-value limit.
In practice, this will almost always reject stability in a GARCH model if you have this large a model with this much data. (All GARCH models are approximations in some form.) The real question with any diagnostic rejection is whether they point you to a better model or other adjustment. In this case, a look at the Japanese returns data and Japanese residuals (the Z1 series) will show that roughly the first 1500 observations just don't look at all like the last 75% of the data set—overall the data are much "quieter" with a few very, very large changes. Most likely, this is due to central bank intervention. It is very unlikely that you can make a minor adjustment to the model to make a GARCH model fit both the period of relatively tight control and a later period with a generally freer market. Instead, a better approach would be to choose a subrange during which the market structure is closer to being uniform.
@flux
# dd
Test Statistic P-Value MaxBreak
Joint 33.7516152 0.00 1881
1 0.1349037 0.42 2526
2 1.8625182 0.00 723
3 1.0381692 0.00 856
4 0.2837077 0.15 731
5 0.0804031 0.67 3779
6 0.1188751 0.48 2460
7 0.0811202 0.67 5439
8 0.6483756 0.02 3403
9 0.1165895 0.49 4024
10 0.0412048 0.92 4489
11 0.0604203 0.80 3891
12 1.1238170 0.00 2700
13 1.2118427 0.00 867
14 1.1811254 0.00 1950
15 0.4665242 0.05 3408
16 1.1236611 0.00 3409
17 0.8525660 0.01 2666
18 0.7245221 0.01 4230
19 0.8873992 0.00 867
20 0.6174838 0.02 2511
21 0.2002908 0.26 4114
22 0.3249779 0.11 813
23 0.9102751 0.00 2479
24 0.4951248 0.04 3324
25 0.2702358 0.16 1304
26 1.9144147 0.00 2605
27 0.2420621 0.19 4339
28 2.8383894 0.00 867
29 0.5976814 0.02 2511
30 0.2141506 0.23 3918
31 0.8854576 0.00 754
32 1.9153297 0.00 2608
33 0.4733080 0.05 4339
34 0.8525894 0.01 513
35 2.6453120 0.00 2608
36 0.3470989 0.10 4339
37 0.6553186 0.02 1293
38 0.1300776 0.44 1795
39 0.2167433 0.23 4107
40 0.5152266 0.04 1293
41 0.1353293 0.42 1937
42 0.1415885 0.40 3114
43 0.2394900 0.20 4107
44 0.1191656 0.48 1937
45 0.3434281 0.10 2364
46 7.7646334 0.00 1685
Graphs
garch(p=1,q=1,mv=dcc,variances=koutmos,hmatrices=hh) / $
xjpn xfra xsui
*
set jpnfra = %cvtocorr(hh(t))(1,2)
set jpnsui = %cvtocorr(hh(t))(1,3)
set frasui = %cvtocorr(hh(t))(2,3)
*
spgraph(vfields=3,footer="Conditional Correlations")
graph(header="Japan with France",min=-1.0,max=1.0)
# jpnfra
graph(header="Japan with Switzerland",min=-1.0,max=1.0)
# jpnsui
graph(header="France with Switzerland",min=-1.0,max=1.0)
# frasui
spgraph(done)
This generates and graphs the conditional correlations for the series from the DCC-Koutmos estimates. This can be done with any type of multivariate model (though a CC will be flat, so it won't be very interesting) as long as you save the series of covariance matrices using the HMATRICES option. In the SET instructions, %CVTOCORR(HH(T)) returns the covariance matrix converted to correlations, and the (i,j) subscripts pull that particular element out of the covariance matrix, so %CVTOCORR(HH(T))(1,2) is the conditional correlation of Japan (variable 1) with France (variable 2) at time T.
Copyright © 2024 Thomas A. Doan