GARCHMV.RPF is an illustrative example which includes several variants on multivariate GARCH 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. It's discussed in considerable detail in Section 9.4 of the User's Guide.
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.
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=choleski) / xjpn xfra xsui
*
* CC with VARMA variances
*
garch(p=1,q=1,mv=cc,variances=varma,pmethod=simplex,piters=10) / $
xjpn xfra xsui
*
* 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) / $
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,$
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,$
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)
*
* Diagnostics on (univariate) standardized residuals
*
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)
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
garch(p=1,q=1,pmethod=simplex,piters=10) / xjpn xfra xsui
For a standard (or DVECH) model (the default model for GARCH), there's a separate equation for each of component of the covariance matrix:
\({\bf{H}}_{ij,t} = {c_{ij}} + {a_{ij}}{\kern 1pt} {\kern 1pt} {u_{i,t - 1}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {u_{j,t - 1}} + {b_{ij}}{\kern 1pt} {{\bf{H}}_{ij,t - 1}}\)
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 169 Iterations. Final criterion was 0.0000051 <= 0.0000100
Usable Observations 6236
Log Likelihood -11835.6510
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.004646969 0.007064672 0.65778 0.51068231
2. Mean(XFRA) -0.003473262 0.007241505 -0.47963 0.63148870
3. Mean(XSUI) -0.002331525 0.008693786 -0.26818 0.78855853
4. C(1,1) 0.009026181 0.001051883 8.58097 0.00000000
5. C(2,1) 0.005697145 0.000701699 8.11907 0.00000000
6. C(2,2) 0.011473368 0.001290261 8.89229 0.00000000
7. C(3,1) 0.006003505 0.000744173 8.06735 0.00000000
8. C(3,2) 0.009885913 0.001210674 8.16562 0.00000000
9. C(3,3) 0.012694509 0.001481653 8.56780 0.00000000
10. A(1,1) 0.105889668 0.007467950 14.17921 0.00000000
11. A(2,1) 0.093909954 0.005676249 16.54437 0.00000000
12. A(2,2) 0.127781756 0.005754532 22.20541 0.00000000
13. A(3,1) 0.088713290 0.005240428 16.92863 0.00000000
14. A(3,2) 0.113209884 0.004860551 23.29158 0.00000000
15. A(3,3) 0.111076067 0.004654186 23.86584 0.00000000
16. B(1,1) 0.883884251 0.007285389 121.32286 0.00000000
17. B(2,1) 0.891015563 0.005782334 154.09272 0.00000000
18. B(2,2) 0.861224733 0.006389700 134.78329 0.00000000
19. B(3,1) 0.897759886 0.005427159 165.41986 0.00000000
20. B(3,2) 0.875295362 0.005333268 164.11990 0.00000000
21. B(3,3) 0.877950585 0.004919708 178.45584 0.00000000
garch(p=1,q=1,mv=bekk,pmethod=simplex,piters=10) / xjpn xfra xsui
For a BEKK, the variance constant is represented by a lower triangular matrix whose outer product is used. (Note that this is often represented as an upper triangular matrix—the resulting product matrix will be the same in either case). The "ARCH" and "GARCH" terms are formed by a sandwich product with an \(n \times n\) matrix of coefficients around a symmetric matrix. In keeping with the literature, the pre-multiplying matrix is the transposed one:
\({{\bf{H}}_t}{\rm{ = }}{\bf{CC'}}{\rm{ + }}{\bf{A'}}{{\bf{u}}_{t - 1}}{{\bf{u'}}_{t - 1}}{\bf{A}}{\rm{ + }}{\bf{B'}}{{\bf{H}}_{t - 1}}{\bf{B}}\)
It's important to note that the signs of each of the coefficient matrices aren't identified statistically—multiply \({\bf{A}}\) (or \({\bf{B}}\) or \({\bf{C}}\)) by -1 and you get exactly the same recursion. The guess values used by RATS tend to force them towards positive values on the diagonal, but if a GARCH model doesn't fit the data particularly well, it's possible for them to "flip" signs at some point in the estimation. Also, it is not unreasonable (and in fact not unexpected) for some off-diagonal elements of \({\bf{A}}\) (in particular) and \({\bf{B}}\) to be negative even where the diagonal elements are positive. This is most easily seen with \({\bf{A}}\): define \({{\bf{v}}_{t - 1}} = {\bf{A'}}{{\bf{u}}_{t - 1}}\), which is an \(n\) vector. The contribution of the "ARCH" term to the covariance matrix is then \({{\bf{v}}_{t - 1}}{{{\bf{v'}}}_{t - 1}}\) which means that the squares of the elements of \({\bf{v}}\) will be the contributions to the variances themselves. To have "spillover" effects, so that shocks in one component affect the variance of another, \({\bf{v}}\) will have to be a linear combination of the different components of \({\bf{u}}\). Negative coefficients in the off-diagonals of \({\bf{A}}\) mean that the variance is affected more when the shocks move in opposite directions than when they move in the same direction, which probably isn’t unreasonable in many situations. It's also possible (unlikely in practice, but still possible), for the diagonal elements in \({\bf{A}}\) (or less likely \({\bf{B}}\)) to have opposite signs. For instance, if the correlation between two components is near zero, the sign of any column of \({\bf{A}}\) (or \({\bf{B}}\)) has little effect on the likelihood. In most applications, the correlations among the variables tends to be high and positive, but increasingly GARCH models are being applied to series for which that is not the case.
Another common question is how it’s possible for the off-diagonals in the \({\bf{A}}\) and \({\bf{B}}\) matrices to be larger than the diagonals, since one would expect that the “own” effect would be dominant. However, the values of the coefficients are sensitive to the scales of the variables, since nothing in the recursion is standardized to a common variance. If you multiply component i by .01 relative to j, its residuals also go down by a factor of .01, so the coefficient A(i,j) which applies residual i to the variance of j has to go up by a factor of 100. Rescaling a variable keeps the diagonals of \({\bf{A}}\) and \({\bf{B}}\) the same, but forces a change in scale of the off-diagonals. Even without asymmetrical scalings, the tendency will be for (relatively) higher variance series to have lower off-diagonal coefficients than lower variance series.
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 99 Iterations. Final criterion was 0.0000095 <= 0.0000100
Usable Observations 6236
Log Likelihood -11821.7455
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.005274621 0.005993779 0.88002 0.37885072
2. Mean(XFRA) -0.002371319 0.005337535 -0.44427 0.65684579
3. Mean(XSUI) -0.002518778 0.006411297 -0.39287 0.69441873
4. C(1,1) 0.082828746 0.005178298 15.99536 0.00000000
5. C(2,1) 0.029936102 0.007492155 3.99566 0.00006451
6. C(2,2) 0.055798917 0.005548758 10.05611 0.00000000
7. C(3,1) 0.037973459 0.008612781 4.40897 0.00001039
8. C(3,2) -0.003978644 0.010336388 -0.38492 0.70029944
9. C(3,3) 0.058513092 0.006529872 8.96083 0.00000000
10. A(1,1) 0.359533638 0.011779071 30.52309 0.00000000
11. A(1,2) 0.102646412 0.009938548 10.32811 0.00000000
12. A(1,3) 0.111026735 0.012909390 8.60046 0.00000000
13. A(2,1) 0.038173251 0.015381725 2.48173 0.01307472
14. A(2,2) 0.403485330 0.016928098 23.83524 0.00000000
15. A(2,3) -0.066141107 0.018546755 -3.56618 0.00036222
16. A(3,1) -0.047563779 0.010789285 -4.40843 0.00001041
17. A(3,2) -0.125567115 0.012877006 -9.75127 0.00000000
18. A(3,3) 0.291199552 0.014670994 19.84866 0.00000000
19. B(1,1) 0.935266231 0.003837748 243.70182 0.00000000
20. B(1,2) -0.026704236 0.003267176 -8.17349 0.00000000
21. B(1,3) -0.028562880 0.004291962 -6.65497 0.00000000
22. B(2,1) -0.012475984 0.006006868 -2.07695 0.03780587
23. B(2,2) 0.909756881 0.006645208 136.90420 0.00000000
24. B(2,3) 0.029206613 0.006931742 4.21346 0.00002515
25. B(3,1) 0.016556233 0.004503752 3.67610 0.00023683
26. B(3,2) 0.048816557 0.005701999 8.56131 0.00000000
27. B(3,3) 0.946900974 0.005611772 168.73475 0.00000000
garch(p=1,q=1,mv=cc) / xjpn xfra xsui
For a CC (Constant Correlation) model, the variances are computed using separate equations for each variable. With the CC model, these are used to form the overall covariance matrix by using:
\({{\bf{H}}_{ij,t}} = {{\bf{R}}_{ij}}\sqrt {{{\bf{H}}_{ii,t}}{\kern 1pt} {\kern 1pt} {{\bf{H}}_{jj,t}}} \)
where the \({\bf{R}}\)'s are the constant correlations.
The default variance model (governed by the VARIANCES option, so this would be VARIANCES=SIMPLE) takes the form
\({H_{ii,t}} = {c_i} + {a_i}u_{i,t - 1}^2 + {b_i}{H_{ii,t - 1}}\)
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 47 Iterations. Final criterion was 0.0000099 <= 0.0000100
Usable Observations 6236
Log Likelihood -12817.3747
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) -0.000775585 0.007081544 -0.10952 0.91278842
2. Mean(XFRA) -0.004266997 0.007222367 -0.59080 0.55465240
3. Mean(XSUI) 0.003648561 0.008713667 0.41872 0.67542294
4. C(1) 0.016827996 0.002079419 8.09264 0.00000000
5. C(2) 0.028385668 0.002678169 10.59891 0.00000000
6. C(3) 0.032306057 0.003053511 10.57997 0.00000000
7. A(1) 0.164130916 0.011353043 14.45700 0.00000000
8. A(2) 0.133203602 0.008892333 14.97960 0.00000000
9. A(3) 0.112696962 0.007653219 14.72543 0.00000000
10. B(1) 0.812634601 0.012102769 67.14452 0.00000000
11. B(2) 0.804346693 0.012209911 65.87654 0.00000000
12. B(3) 0.831322084 0.010479082 79.33158 0.00000000
13. R(2,1) 0.564320088 0.008522088 66.21852 0.00000000
14. R(3,1) 0.579295504 0.008318390 69.64034 0.00000000
15. R(3,2) 0.828697594 0.003977519 208.34536 0.00000000
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:
\({{\bf{Q}}_t} = (1 - a - b){\kern 1pt} {{\bf{\bar Q}}} + a{u_{t - 1}}{u'_{t - 1}} + b{\kern 1pt} {{\bf{Q}}_{t - 1}}\)
where \({{\bf{\bar Q}}}\) is the unconditional covariance matrix. However, \({\bf{Q}}\) isn’t the sequence of covariance matrices. Instead, it is used solely to provide the correlation matrix. The actual \({\bf{H}}\) matrix is generated using univariate GARCH models for the variances (controlled by the VARIANCES option), combined with the correlations produced by the \({\bf{Q}}\):
\({{\bf{H}}_{ij}}_{,t} = {{\bf{Q}}_{ij,t}}\frac{{\sqrt {{{\bf{H}}_{ii,t}}{{\bf{H}}_{jj,t}}} }}{{\sqrt {{{\bf{Q}}_{ii,t}}{{\bf{Q}}_{jj,t}}} }}\)
Engle's proposal was to estimate this in two stages, with the univariate GARCH models done first followed by estimating the a and b (and thus the dynamic correlations) in a second stage. This gives consistent, but not efficient estimates, but has the advantage that it can be applied to very large numbers of variables. The disadvantage (other than statistically inefficiency) is that only a limit number of univariate variance models can be used in the two step approach: of the five offered by RATS, only VARIANCES=SIMPLE and VARIANCES=EXPONENTIAL would work since the others have interaction terms. As a result, the GARCH instruction with MV=DCC does a full maximum likelihood estimation of all the coefficients.
The output adds the DCC(A) and DCC(B) parameters which are the a and b in the Q recursion. Note that while apparently a generalization of the CC model, the two don't formally nest because CC estimates freely the correlation matrix and (if \(n > 2\)) actually has more free parameters that the DCC model. While you can't do a standard likelihood ratio test to compare CC with DCC, in this case, DCC has a higher log likelihood by roughly 1000, so CC is clearly quite inadequate. The @TSECCTEST procedure offers a Lagrange Multiplier test for CC against a time-varying alternative, though that alternative is not DCC.
MV-DCC GARCH - Estimation by BFGS
Convergence in 42 Iterations. Final criterion was 0.0000033 <= 0.0000100
Usable Observations 6236
Log Likelihood -11814.4403
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.003987506 0.006060129 0.65799 0.51054435
2. Mean(XFRA) -0.003133447 0.006152936 -0.50926 0.61056964
3. Mean(XSUI) -0.003070966 0.007508317 -0.40901 0.68253341
4. C(1) 0.008499092 0.001136736 7.47675 0.00000000
5. C(2) 0.012485541 0.001314571 9.49780 0.00000000
6. C(3) 0.016566320 0.001742031 9.50977 0.00000000
7. A(1) 0.151662080 0.009226265 16.43808 0.00000000
8. A(2) 0.138382721 0.008036770 17.21870 0.00000000
9. A(3) 0.123692585 0.006958449 17.77588 0.00000000
10. B(1) 0.852005668 0.008113585 105.00977 0.00000000
11. B(2) 0.848527776 0.008240241 102.97366 0.00000000
12. B(3) 0.858001790 0.007341456 116.87079 0.00000000
13. DCC(A) 0.053230310 0.003340516 15.93476 0.00000000
14. DCC(B) 0.939072327 0.003985276 235.63544 0.00000000
garch(p=1,q=1,mv=cholesky) / xjpn xfra xsui
The Cholesky model has some similarities to the CC and DCC models, but one important difference—while the other models apply a univariate model to observed data, the Cholesky model uses ideas from structural VAR modeling to map the observable residuals (\({\bf{u}}\)) to uncorrelated residuals (\({\bf{v}}\)) using \({{\bf{u}}_t} = {\bf{F}}{{\bf{v}}_t}\), where \(\bf{F}\) is lower triangular. The difference with the VAR literature is that the components of v are assumed to follow (univariate) GARCH processes rather than having a fixed (identity) covariance matrix. As in the VAR literature, it’s necessary to come out with some normalization between the \(\bf{F}\) and the variances of \({\bf{v}}\). For the Cholesky model in the VAR, the obvious choice is to fix the variances at 1. However, here the variances aren’t fixed, so it’s simpler to make the diagonals of \(\bf{F}\) equal to 1 and leave the component GARCH processes free, so the free parameters in \(\bf{F}\) will be the elements below the diagonal.
The Cholesky model is sensitive to the order of listing—by construction, the first \({\bf{v}}\) is identical to the first \({\bf{u}}\). Any of the choices for the VARIANCE option are permitted with MV=CHOLESKY; this example uses VARIANCES=SIMPLE.
MV-Cholesky GARCH - Estimation by BFGS
Convergence in 114 Iterations. Final criterion was 0.0000025 <= 0.0000100
Usable Observations 6236
Log Likelihood -12173.5176
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.000188257 0.006440919 0.02923 0.97668251
2. Mean(XFRA) -0.006990951 0.006479695 -1.07890 0.28063176
3. Mean(XSUI) -0.004037282 0.007499684 -0.53833 0.59035134
4. C(1) 0.007428328 0.001043410 7.11928 0.00000000
5. C(2) 0.007284154 0.000984301 7.40033 0.00000000
6. C(3) 0.005180700 0.000874994 5.92084 0.00000000
7. A(1) 0.176281319 0.009392566 18.76817 0.00000000
8. A(2) 0.146640647 0.011034329 13.28949 0.00000000
9. A(3) 0.183550187 0.014973932 12.25798 0.00000000
10. B(1) 0.837274108 0.007741453 108.15464 0.00000000
11. B(2) 0.841738507 0.010952723 76.85199 0.00000000
12. B(3) 0.807028041 0.015512532 52.02426 0.00000000
13. F(2,1) 0.561329167 0.010339981 54.28725 0.00000000
14. F(3,1) 0.675958192 0.010563103 63.99239 0.00000000
15. F(3,2) 0.919538839 0.007987286 115.12532 0.00000000
nlpar(derive=fourth,exactline)
garch(p=1,q=1,mv=cc,variances=varma,iters=500,pmethod=simplex,piters=5) / $
xjpn xfra xsui
VARIANCES=VARMA is a particular model for calculating the variances (only) as part of a CC, DCC or similar model. It includes not just own lagged squared residuals and own variances, but all the "other" lagged squared residuals and variances in each variance equation:
\({{\bf{H}}_{ii,t}} = {c_{ii}} + \sum\limits_j {{a_{ij}}u_{j,t - 1}^2 + } \sum\limits_j {{b_{ij}}{{\bf{H}}_{jj,t - 1}}} \)
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.
Note that VARIANCES=VARMA is rather difficult model to fit which is why we use some rather high-end adjustments to the non-linear estimation process using NLPAR. It has numerical problems because the coefficients can be either positive or negative as you can see below. (If you try to impose non-negativity, the model fits only barely better than the model without the cross variable terms). If high residual or high volatility entries don't align very well among variables, a variance for some variable can approach zero at some observation. For instance, here A(2,3) is fairly large negative, so a very big residual in the Swiss data (variable 3) might push the variance of France (variable 2) negative in the next entry. Any zero (or negative) value for H at any data point results in an uncomputable likelihood resulting in quite a few "dead-end" parameter paths, where you approach a local maximum near a computability boundary. The more dissimilar the data series are, the harder it will be to get estimates with VARMA variances.
MV-CC GARCH with VARMA Variances - Estimation by BFGS
Convergence in 100 Iterations. Final criterion was 0.0000044 <= 0.0000100
Usable Observations 6236
Log Likelihood -12679.6117
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) -0.000736526 0.005800641 -0.12697 0.89896154
2. Mean(XFRA) -0.004932286 0.006020712 -0.81922 0.41266110
3. Mean(XSUI) 0.004520785 0.007389388 0.61179 0.54067386
4. C(1) 0.011010474 0.001103742 9.97559 0.00000000
5. C(2) 0.011489506 0.001829405 6.28046 0.00000000
6. C(3) 0.021305583 0.001988511 10.71434 0.00000000
7. A(1,1) 0.129922935 0.009331420 13.92317 0.00000000
8. A(1,2) 0.032904582 0.004297358 7.65693 0.00000000
9. A(1,3) -0.002626674 0.000726780 -3.61413 0.00030136
10. A(2,1) 0.029612149 0.003842227 7.70703 0.00000000
11. A(2,2) 0.142268555 0.010452185 13.61137 0.00000000
12. A(2,3) -0.012089233 0.001905454 -6.34454 0.00000000
13. A(3,1) 0.002480943 0.005348943 0.46382 0.64277718
14. A(3,2) 0.032942902 0.007925060 4.15680 0.00003227
15. A(3,3) 0.089358992 0.007689536 11.62086 0.00000000
16. B(1,1) 0.860420594 0.008337012 103.20491 0.00000000
17. B(1,2) -0.032051355 0.004687804 -6.83718 0.00000000
18. B(1,3) 0.003086356 0.002261622 1.36466 0.17235850
19. B(2,1) -0.027854598 0.003416978 -8.15182 0.00000000
20. B(2,2) 0.787331480 0.017259356 45.61766 0.00000000
21. B(2,3) 0.054032656 0.009604093 5.62600 0.00000002
22. B(3,1) -0.007389152 0.006045852 -1.22219 0.22163756
23. B(3,2) -0.027518045 0.012180167 -2.25925 0.02386782
24. B(3,3) 0.877652592 0.012406316 70.74240 0.00000000
25. R(2,1) 0.559872016 0.008597643 65.11924 0.00000000
26. R(3,1) 0.573176258 0.008452819 67.80889 0.00000000
27. R(3,2) 0.830632130 0.003877817 214.20094 0.00000000
garch(p=1,q=1,mv=ewma,distrib=t) / xjpn xfra xsui
MV=EWMA (Exponentially Weighted Moving Average) is a very tightly parameterized variance model. There is just a single real parameter (α) governing the evolution of the variance:
\({{\bf{H}}_t} = (1 - \alpha ){{\bf{H}}_{t - 1}} + \alpha {{\bf{u}}_{t - 1}}{{{\bf{u'}}}_{t - 1}}\)
This is a special case of the DVECH model with all the coefficients equal across all components and an "I-GARCH" restriction. 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 conditional 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
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. The individual variable variances are computing using
\(\log \,\,{H_{ii,t}} = {c_{ii}} + {a_i}\frac{{\left| {{u_{i,t - 1}}} \right|}}{{\sqrt {{H_{ii,t - 1}}} }} + {d_i}\frac{{{u_{i,t - 1}}}}{{\sqrt {{H_{ii,t - 1}}} }} + {b_i}\log \,{\kern 1pt} {H_{ii,t - 1}}\)
This, by construction, produces a positive value for \(H\) regardless of the signs of the coefficients. 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. Note that the C coefficients will not look at all like the corresponding coefficients in a standard GARCH recursion, since this is an equation for \(log H\) rather than \(H\) itself. The A (and D) coefficients are on standardized residuals and don't affect the overall persistence of the variance—only the B coefficient does that. Note that this is often written with the expected value subtracted off from the \(\left| u \right|/H\) term. That's omitted above since it just washes into the constant and it depends rather strongly on the distribution of u. (If you use t or GED errors, for instance, it will change with the shape parameter, and again, its presence or absence affects only the values of C.)
MV-CC GARCH with E-GARCH Variances - Estimation by BFGS
Convergence in 85 Iterations. Final criterion was 0.0000044 <= 0.0000100
Usable Observations 6236
Log Likelihood -12733.4127
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) -0.001060332 0.004956105 -0.21394 0.83059028
2. Mean(XFRA) -0.007506138 0.004511126 -1.66392 0.09612906
3. Mean(XSUI) 0.006647523 0.006109784 1.08801 0.27658944
4. C(1) -0.388295949 0.014171400 -27.39997 0.00000000
5. C(2) -0.279987895 0.013542497 -20.67476 0.00000000
6. C(3) -0.212536916 0.012447451 -17.07473 0.00000000
7. A(1) 0.390913505 0.013837912 28.24946 0.00000000
8. A(2) 0.265094072 0.012079158 21.94640 0.00000000
9. A(3) 0.216801697 0.012399646 17.48451 0.00000000
10. B(1) 0.894103282 0.006442689 138.77797 0.00000000
11. B(2) 0.912123027 0.005926409 153.90821 0.00000000
12. B(3) 0.926777214 0.005831397 158.92885 0.00000000
13. D(1) 0.019517848 0.007972415 2.44817 0.01435829
14. D(2) -0.011373668 0.006065338 -1.87519 0.06076643
15. D(3) 0.043824490 0.005322117 8.23441 0.00000000
16. R(2,1) 0.561039801 0.008207476 68.35717 0.00000000
17. R(3,1) 0.575788380 0.007999527 71.97780 0.00000000
18. R(3,2) 0.826050110 0.003391556 243.56082 0.00000000
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 asymmetric VARIANCES=EGARCH but allows for "spillover" effects among the residuals. The variance recursion takes the form:
\(\log \,\,{{\bf{H}}_{ii,t}} = {c_{ii}} + \sum\limits_j {{a_{ij}}{\kern 1pt} \left( {\frac{{\left| {{u_{j,t - 1}}} \right|}}{{\sqrt {{H_{jj,t - 1}}} }} + {d_j}\frac{{{u_{j,t - 1}}}}{{\sqrt {{H_{jj,t - 1}}} }}} \right) + \,} {b_i}\log \,{\kern 1pt} {{\bf{H}}_{ii,t - 1}}\)
The EGARCH model would be this without the off-diagonal A terms. Note that the asymmetry comes in in a tightly restricted way—each variable has one asymmetrical "index" at a given point in time which applies to all terms which use it. 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 42 Iterations. Final criterion was 0.0000061 <= 0.0000100
Usable Observations 6236
Log Likelihood -11667.5715
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. Mean(XJPN) 0.005989776 0.005036023 1.18939 0.23428774
2. Mean(XFRA) -0.000170947 0.004443345 -0.03847 0.96931087
3. Mean(XSUI) 0.003852159 0.005970849 0.64516 0.51882282
4. C(1) -0.333034803 0.010777764 -30.90017 0.00000000
5. C(2) -0.318301003 0.011343927 -28.05916 0.00000000
6. C(3) -0.256016986 0.011908892 -21.49797 0.00000000
7. A(1,1) 0.368494853 0.007279952 50.61776 0.00000000
8. A(1,2) 0.088850652 0.006908113 12.86178 0.00000000
9. A(1,3) 0.038046669 0.009960447 3.81978 0.00013357
10. A(2,1) 0.065570711 0.015020497 4.36542 0.00001269
11. A(2,2) 0.312389935 0.008568187 36.45928 0.00000000
12. A(2,3) 0.104876318 0.010641738 9.85519 0.00000000
13. A(3,1) -0.039139969 0.015169378 -2.58020 0.00987443
14. A(3,2) -0.022110875 0.013142124 -1.68244 0.09248296
15. A(3,3) 0.167400052 0.013712578 12.20777 0.00000000
16. B(1) 0.938816563 0.003642546 257.73637 0.00000000
17. B(2) 0.949220617 0.003384572 280.45516 0.00000000
18. B(3) 0.952244390 0.003716941 256.19032 0.00000000
19. D(1) 0.074120039 0.022222215 3.33540 0.00085176
20. D(2) -0.007741055 0.020207464 -0.38308 0.70166119
21. D(3) 0.164427268 0.025485235 6.45186 0.00000000
22. DCC(A) 0.050573164 0.003246238 15.57900 0.00000000
23. DCC(B) 0.943697656 0.003785154 249.31555 0.00000000
garch(p=1,q=1,mv=bekk,pmethod=simplex,piters=10,distrib=t,$
rseries=rs,mvhseries=hhs) / xjpn xfra xsui
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.
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 174 Iterations. Final criterion was 0.0000017 <= 0.0000100
Usable Observations 6236
Log Likelihood -10260.5399
Variable Coeff Std Error T-Stat Signif
*************************************************************************************
1. Mean(XJPN) -0.007107571 0.003520367 -2.01899 0.04348872
2. Mean(XFRA) -0.004612078 0.004012708 -1.14937 0.25040435
3. Mean(XSUI) -0.004487020 0.004795857 -0.93560 0.34947740
4. C(1,1) 0.017750872 0.004250848 4.17584 0.00002969
5. C(2,1) -0.031718066 0.007177749 -4.41894 0.00000992
6. C(2,2) 0.027367999 0.007390060 3.70335 0.00021277
7. C(3,1) -0.029154724 0.012310787 -2.36823 0.01787363
8. C(3,2) -0.067679807 0.008805753 -7.68586 0.00000000
9. C(3,3) -0.000002633 0.092546709 -2.84488e-005 0.99997730
10. A(1,1) 0.291241915 0.013392552 21.74656 0.00000000
11. A(1,2) 0.001808593 0.012405060 0.14579 0.88408337
12. A(1,3) 0.008848868 0.017283922 0.51197 0.60867128
13. A(2,1) -0.004304572 0.009865373 -0.43633 0.66259625
14. A(2,2) 0.315102049 0.016579256 19.00580 0.00000000
15. A(2,3) -0.088257323 0.021391275 -4.12586 0.00003694
16. A(3,1) -0.007936246 0.006790998 -1.16864 0.24254785
17. A(3,2) -0.033315032 0.012407640 -2.68504 0.00725208
18. A(3,3) 0.347587032 0.018458040 18.83120 0.00000000
19. B(1,1) 0.963713123 0.002617505 368.18005 0.00000000
20. B(1,2) 0.000680668 0.002804945 0.24267 0.80826316
21. B(1,3) 0.002794144 0.003981563 0.70177 0.48282229
22. B(2,1) 0.000460476 0.003022010 0.15237 0.87889201
23. B(2,2) 0.941278388 0.005686360 165.53266 0.00000000
24. B(2,3) 0.045037363 0.008192856 5.49715 0.00000004
25. B(3,1) 0.002190083 0.002451443 0.89339 0.37165084
26. B(3,2) 0.021980135 0.004998175 4.39763 0.00001094
27. B(3,3) 0.927099000 0.007229271 128.24238 0.00000000
28. Shape 4.229194265 0.130653096 32.36964 0.00000000
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 uses 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. The output is shown below.
Probability of being below .05 level 0.0420
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)
This is a DCC variance model with 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.
MV-DCC GARCH - Estimation by BFGS
Convergence in 46 Iterations. Final criterion was 0.0000000 <= 0.0000100
Usable Observations 6235
Log Likelihood -11810.8881
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. Constant 0.004014562 0.005991287 0.67007 0.50281525
2. XJPN{1} 0.025553751 0.011332247 2.25496 0.02413591
Mean Model(XFRA)
3. Constant -0.003110255 0.005798073 -0.53643 0.59166204
4. XFRA{1} 0.005578945 0.008777423 0.63560 0.52503606
Mean Model(XSUI)
5. Constant -0.003047090 0.006896665 -0.44182 0.65861887
6. XSUI{1} -0.001867027 0.008487257 -0.21998 0.82588673
7. C(1) 0.008298121 0.001065518 7.78787 0.00000000
8. C(2) 0.012349739 0.001345125 9.18111 0.00000000
9. C(3) 0.016510023 0.001705979 9.67774 0.00000000
10. A(1) 0.151651156 0.008649248 17.53345 0.00000000
11. A(2) 0.137409295 0.007843544 17.51878 0.00000000
12. A(3) 0.123174851 0.006552406 18.79842 0.00000000
13. B(1) 0.852560638 0.007249044 117.61008 0.00000000
14. B(2) 0.849633627 0.008057941 105.44053 0.00000000
15. B(3) 0.858506129 0.007250363 118.40871 0.00000000
16. DCC(A) 0.053195549 0.003431160 15.50366 0.00000000
17. DCC(B) 0.939101210 0.004130209 227.37380 0.00000000
variables xjpn xfra xsui
lags 1
det constant
end(system)
*
garch(p=1,q=1,model=var1,mv=bekk,pmethod=simplex,piters=10)
This estimates a BEKK model with the mean given by a VAR(1).
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 92 Iterations. Final criterion was 0.0000096 <= 0.0000100
Usable Observations 6235
Log Likelihood -11809.4143
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. XJPN{1} 0.054037654 0.011182386 4.83239 0.00000135
2. XFRA{1} 0.023682621 0.013255290 1.78665 0.07399337
3. XSUI{1} -0.034327923 0.010181658 -3.37155 0.00074748
4. Constant 0.006201575 0.005449407 1.13803 0.25510902
Mean Model(XFRA)
5. XJPN{1} 0.028493123 0.007798861 3.65350 0.00025869
6. XFRA{1} 0.023713659 0.010571715 2.24312 0.02488888
7. XSUI{1} -0.009351366 0.008488737 -1.10162 0.27062673
8. Constant -0.001862777 0.003988431 -0.46705 0.64046759
Mean Model(XSUI)
9. XJPN{1} 0.040977801 0.009376496 4.37027 0.00001241
10. XFRA{1} 0.025030406 0.012992675 1.92650 0.05404179
11. XSUI{1} -0.017960025 0.009721875 -1.84738 0.06469167
12. Constant -0.001914884 0.004623543 -0.41416 0.67875739
13. C(1,1) 0.080420869 0.004610689 17.44227 0.00000000
14. C(2,1) 0.026790278 0.005889919 4.54850 0.00000540
15. C(2,2) 0.055096257 0.004810186 11.45408 0.00000000
16. C(3,1) 0.034166293 0.007852979 4.35074 0.00001357
17. C(3,2) -0.004060758 0.008276084 -0.49066 0.62366564
18. C(3,3) -0.058667313 0.005427901 -10.80847 0.00000000
19. A(1,1) 0.356745196 0.011327049 31.49498 0.00000000
20. A(1,2) 0.099016521 0.009757899 10.14732 0.00000000
21. A(1,3) 0.107351092 0.012190861 8.80587 0.00000000
22. A(2,1) 0.033518944 0.015356110 2.18278 0.02905233
23. A(2,2) 0.398503867 0.015871681 25.10785 0.00000000
24. A(2,3) -0.070336064 0.018314828 -3.84039 0.00012284
25. A(3,1) -0.047181157 0.009773714 -4.82735 0.00000138
26. A(3,2) -0.122497792 0.011869847 -10.32008 0.00000000
27. A(3,3) 0.292921264 0.013931915 21.02520 0.00000000
28. B(1,1) 0.936389658 0.003620075 258.66581 0.00000000
29. B(1,2) -0.025326977 0.003018928 -8.38939 0.00000000
30. B(1,3) -0.027019383 0.003971902 -6.80263 0.00000000
31. B(2,1) -0.010660383 0.005715656 -1.86512 0.06216461
32. B(2,2) 0.911888451 0.006231551 146.33410 0.00000000
33. B(2,3) 0.030709661 0.006744664 4.55318 0.00000528
34. B(3,1) 0.016113537 0.004129170 3.90237 0.00009526
35. B(3,2) 0.047484087 0.004999802 9.49719 0.00000000
36. B(3,3) 0.946092066 0.005254531 180.05260 0.00000000
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)
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.
MV-GARCH - Estimation by BFGS
Convergence in 307 Iterations. Final criterion was 0.0000065 <= 0.0000100
Usable Observations 6236
Log Likelihood -11832.2529
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. Constant -0.007826198 0.009458987 -0.82738 0.40802045
2. HHS(1,1) 0.004330142 0.035786484 0.12100 0.90369152
3. HHS(2,1) -0.041418733 0.096875687 -0.42755 0.66898229
4. HHS(3,1) 0.091415703 0.079952015 1.14338 0.25287998
Mean Model(XFRA)
5. Constant -0.010748239 0.009321273 -1.15309 0.24887470
6. HHS(2,1) 0.069507553 0.048175312 1.44280 0.14907557
7. HHS(2,2) -0.026451605 0.035767541 -0.73954 0.45957783
8. HHS(3,2) 0.011003924 0.037422503 0.29405 0.76872302
Mean Model(XSUI)
9. Constant -0.004246433 0.011234855 -0.37797 0.70545320
10. HHS(3,1) 0.065304691 0.049066030 1.33096 0.18320373
11. HHS(3,2) 0.002358563 0.052118159 0.04525 0.96390473
12. HHS(3,3) -0.025805514 0.037254256 -0.69269 0.48850642
13. C(1,1) 0.008904091 0.000955334 9.32040 0.00000000
14. C(2,1) 0.005617118 0.000510199 11.00965 0.00000000
15. C(2,2) 0.011424107 0.000975179 11.71488 0.00000000
16. C(3,1) 0.005944968 0.000599493 9.91666 0.00000000
17. C(3,2) 0.009836654 0.000927038 10.61084 0.00000000
18. C(3,3) 0.012644626 0.001257515 10.05525 0.00000000
19. A(1,1) 0.105317240 0.005856045 17.98436 0.00000000
20. A(2,1) 0.093487282 0.003615995 25.85382 0.00000000
21. A(2,2) 0.127516767 0.005670313 22.48849 0.00000000
22. A(3,1) 0.088424588 0.003935099 22.47074 0.00000000
23. A(3,2) 0.112939266 0.004930254 22.90740 0.00000000
24. A(3,3) 0.110733029 0.005216830 21.22611 0.00000000
25. B(1,1) 0.884651950 0.006372744 138.81805 0.00000000
26. B(2,1) 0.891673687 0.003993003 223.30906 0.00000000
27. B(2,2) 0.861543181 0.005714278 150.77027 0.00000000
28. B(3,1) 0.898187207 0.004150039 216.42864 0.00000000
29. B(3,2) 0.875620493 0.005011153 174.73433 0.00000000
30. B(3,3) 0.878296876 0.005263227 166.87422 0.00000000
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))))
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.
VARIANCES=SPILLOVER is another choice for the VARIANCE option. This includes the cross terms (allowing for "spillover") on the lagged squared residuals, but just an "own" term on the lagged variance:
\({H_{ii,t}} = {c_{ii}} + \sum\limits_j {{a_{ij}}{\kern 1pt} u_{j,t - 1}^2 + \,} {b_i}{\kern 1pt} {H_{ii,t - 1}}\)
Thus it's a generalization of VARIANCES=SIMPLE and a restriction on VARIANCES=VARMA. In the output, A(i,j) is the contribution of the residuals for variable j to the variance for i. B(i) is the cofficient on the lagged variance in computing the variance of i.
MV-CC GARCH with Spillover Variances - Estimation by BFGS
Convergence in 119 Iterations. Final criterion was 0.0000086 <= 0.0000100
Usable Observations 6236
Log Likelihood -12756.3977
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. Constant -0.022568963 0.025081358 -0.89983 0.36821063
2. COMMONSD 0.041373595 0.043324744 0.95496 0.33959562
Mean Model(XFRA)
3. Constant -0.053071880 0.027336040 -1.94146 0.05220228
4. COMMONSD 0.098511447 0.048195888 2.04398 0.04095549
Mean Model(XSUI)
5. Constant -0.037672632 0.032147471 -1.17187 0.24124955
6. COMMONSD 0.082253879 0.057070245 1.44127 0.14950720
7. C(1) 0.017700054 0.002151240 8.22784 0.00000000
8. C(2) 0.027393116 0.002443877 11.20888 0.00000000
9. C(3) 0.031550488 0.002699347 11.68819 0.00000000
10. A(1,1) 0.167750546 0.012382366 13.54754 0.00000000
11. A(1,2) 0.017118685 0.010184291 1.68089 0.09278404
12. A(1,3) -0.041865504 0.008373446 -4.99979 0.00000057
13. A(2,1) -0.053353465 0.006970518 -7.65416 0.00000000
14. A(2,2) 0.162574231 0.011612094 14.00042 0.00000000
15. A(2,3) -0.007820467 0.008080816 -0.96778 0.33315330
16. A(3,1) -0.030341878 0.005180577 -5.85685 0.00000000
17. A(3,2) -0.007273839 0.007633572 -0.95287 0.34065348
18. A(3,3) 0.125687264 0.008003167 15.70469 0.00000000
19. B(1) 0.822608394 0.012289923 66.93357 0.00000000
20. B(2) 0.814725928 0.010651896 76.48647 0.00000000
21. B(3) 0.839040858 0.009307537 90.14639 0.00000000
22. R(2,1) 0.571210137 0.007732364 73.87264 0.00000000
23. R(3,1) 0.584929593 0.007706896 75.89691 0.00000000
24. R(3,2) 0.831118224 0.003520824 236.05785 0.00000000
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)
This is a 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.
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 108 Iterations. Final criterion was 0.0000033 <= 0.0000100
Usable Observations 6236
Log Likelihood -11750.8005
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(XJPN)
1. Constant 0.000523303 0.005813198 0.09002 0.92827146
2. U(1){1} 0.056554362 0.009808178 5.76604 0.00000001
3. U(2){1} 0.023803688 0.011465595 2.07610 0.03788498
4. U(3){1} -0.034958557 0.009508359 -3.67661 0.00023635
Mean Model(XFRA)
5. Constant -0.004613971 0.004694026 -0.98295 0.32563432
6. U(1){1} 0.026529120 0.007421725 3.57452 0.00035087
7. U(2){1} 0.024498547 0.010730269 2.28313 0.02242300
8. U(3){1} -0.004617997 0.007544712 -0.61208 0.54048223
Mean Model(XSUI)
9. Constant -0.003721936 0.005520153 -0.67425 0.50015553
10. U(1){1} 0.039553340 0.008607083 4.59544 0.00000432
11. U(2){1} 0.024237283 0.011630158 2.08400 0.03715991
12. U(3){1} -0.011646145 0.010604136 -1.09826 0.27208907
13. C(1,1) 0.071422771 0.004516114 15.81509 0.00000000
14. C(2,1) 0.019883625 0.006323327 3.14449 0.00166378
15. C(2,2) 0.049749451 0.004647989 10.70343 0.00000000
16. C(3,1) 0.029611499 0.005290450 5.59716 0.00000002
17. C(3,2) -0.006374633 0.008271129 -0.77071 0.44087948
18. C(3,3) -0.057137171 0.005141634 -11.11265 0.00000000
19. A(1,1) 0.337710951 0.008658665 39.00266 0.00000000
20. A(1,2) 0.101237507 0.004177367 24.23476 0.00000000
21. A(1,3) 0.095746615 0.005565063 17.20495 0.00000000
22. A(2,1) 0.001127453 0.012262610 0.09194 0.92674386
23. A(2,2) 0.376841707 0.015655157 24.07141 0.00000000
24. A(2,3) -0.093041436 0.015675601 -5.93543 0.00000000
25. A(3,1) -0.038118765 0.009517395 -4.00517 0.00006197
26. A(3,2) -0.126869832 0.012222708 -10.37985 0.00000000
27. A(3,3) 0.303089438 0.012548885 24.15270 0.00000000
28. B(1,1) 0.934752661 0.002876731 324.93568 0.00000000
29. B(1,2) -0.027739103 0.002345306 -11.82750 0.00000000
30. B(1,3) -0.027638434 0.002440450 -11.32514 0.00000000
31. B(2,1) -0.006973578 0.005422110 -1.28614 0.19839511
32. B(2,2) 0.907310830 0.006752667 134.36333 0.00000000
33. B(2,3) 0.035199366 0.006556973 5.36823 0.00000008
34. B(3,1) 0.014409585 0.003967165 3.63221 0.00028100
35. B(3,2) 0.052041664 0.005570214 9.34285 0.00000000
36. B(3,3) 0.942421445 0.005391111 174.81026 0.00000000
37. D(1,1) 0.184149303 0.023016895 8.00061 0.00000000
38. D(1,2) 0.019930251 0.020548357 0.96992 0.33208666
39. D(1,3) 0.108268772 0.026514164 4.08343 0.00004438
40. D(2,1) 0.092760590 0.017406768 5.32900 0.00000010
41. D(2,2) 0.108974929 0.020496916 5.31665 0.00000011
42. D(2,3) 0.105254202 0.022420676 4.69452 0.00000267
43. D(3,1) -0.027936003 0.017903316 -1.56038 0.11866975
44. D(3,2) 0.072493050 0.015363244 4.71860 0.00000237
45. D(3,3) -0.025909857 0.015084676 -1.71763 0.08586455
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)
This is an asymmetric BEKK with a VAR(1) mean model. It saves quite a bit of extra information for use in the diagnostics which follow.
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 345 Iterations. Final criterion was 0.0000037 <= 0.0000100
Usable Observations 6235
Log Likelihood -10204.7966
Variable Coeff Std Error T-Stat Signif
*************************************************************************************
Mean Model(XJPN)
1. XJPN{1} -0.013921151 0.011194265 -1.24360 0.21364801
2. XFRA{1} 0.015594486 0.008867192 1.75867 0.07863317
3. XSUI{1} -0.008652645 0.006518585 -1.32738 0.18438277
4. Constant -0.008318846 0.003284624 -2.53266 0.01131998
Mean Model(XFRA)
5. XJPN{1} -0.010038844 0.009868755 -1.01724 0.30904162
6. XFRA{1} -0.004017891 0.008766931 -0.45830 0.64673638
7. XSUI{1} 0.020068910 0.009535854 2.10457 0.03532842
8. Constant -0.003345475 0.003861700 -0.86632 0.38631370
Mean Model(XSUI)
9. XJPN{1} -0.000910601 0.012528476 -0.07268 0.94205878
10. XFRA{1} 0.033291046 0.014106353 2.36000 0.01827475
11. XSUI{1} -0.032641038 0.014289210 -2.28431 0.02235310
12. Constant -0.003636505 0.004782305 -0.76041 0.44701052
13. C(1,1) -0.015392658 0.004343307 -3.54400 0.00039411
14. C(2,1) 0.029951772 0.007645844 3.91739 0.00008951
15. C(2,2) -0.021807417 0.008736865 -2.49602 0.01255945
16. C(3,1) 0.015015021 0.015619442 0.96130 0.33639970
17. C(3,2) 0.063187456 0.008216776 7.69005 0.00000000
18. C(3,3) -0.000018067 0.087576711 -2.06294e-004 0.99983540
19. A(1,1) 0.280604635 0.014683508 19.11019 0.00000000
20. A(1,2) 0.010016035 0.013843954 0.72350 0.46937566
21. A(1,3) 0.011491007 0.019573140 0.58708 0.55714971
22. A(2,1) -0.003966151 0.010040058 -0.39503 0.69281880
23. A(2,2) 0.312460200 0.018676057 16.73052 0.00000000
24. A(2,3) -0.075822038 0.022894095 -3.31186 0.00092678
25. A(3,1) -0.006185871 0.006748768 -0.91659 0.35935611
26. A(3,2) -0.023251431 0.013402764 -1.73482 0.08277204
27. A(3,3) 0.345784467 0.020605075 16.78152 0.00000000
28. B(1,1) 0.959869243 0.002985897 321.46766 0.00000000
29. B(1,2) -0.000180508 0.003202131 -0.05637 0.95504610
30. B(1,3) 0.001570903 0.004669265 0.33643 0.73654301
31. B(2,1) 0.000666509 0.003214894 0.20732 0.83576056
32. B(2,2) 0.939664925 0.006161669 152.50170 0.00000000
33. B(2,3) 0.041120339 0.008141994 5.05040 0.00000044
34. B(3,1) 0.001479435 0.002533738 0.58389 0.55929140
35. B(3,2) 0.020100283 0.005007705 4.01387 0.00005973
36. B(3,3) 0.928741374 0.007323588 126.81508 0.00000000
37. D(1,1) -0.185790914 0.025936078 -7.16342 0.00000000
38. D(1,2) 0.030160685 0.031051947 0.97130 0.33140006
39. D(1,3) -0.055270115 0.047589871 -1.16138 0.24548577
40. D(2,1) -0.008550264 0.015076597 -0.56712 0.57063159
41. D(2,2) -0.037446781 0.037124812 -1.00867 0.31313167
42. D(2,3) -0.117934969 0.035581661 -3.31449 0.00091811
43. D(3,1) 0.002752208 0.010059619 0.27359 0.78439993
44. D(3,2) -0.071491809 0.023186020 -3.08340 0.00204649
45. D(3,3) 0.088110333 0.035118148 2.50897 0.01210842
46. Shape 4.185325813 0.130614594 32.04332 0.00000000
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
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.
Independence Tests for Series Z1
Test Statistic P-Value
Ljung-Box Q(40) 121.33672 0.0000
McLeod-Li(40) 55.97498 0.0480
Turning Points -1.32167 0.1863
Difference Sign -0.76761 0.4427
Rank Test -1.05736 0.2903
Independence Tests for Series Z2
Test Statistic P-Value
Ljung-Box Q(40) 100.67880 0.0000
McLeod-Li(40) 9.87095 1.0000
Turning Points 0.00000 1.0000
Difference Sign -0.24125 0.8094
Rank Test -0.42778 0.6688
Independence Tests for Series Z3
Test Statistic P-Value
Ljung-Box Q(40) 96.734741 0.0000
McLeod-Li(40) 45.290929 0.2607
Turning Points -1.231559 0.2181
Difference Sign -0.021932 0.9825
Rank Test -2.172267 0.0298
@mvqstat(lags=5)
# zu
@mvarchtest(lags=5)
# zu
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.
Multivariate Q(5)= 143.15596
Significance Level as Chi-Squared(45)= 3.68438e-012
Test for Multivariate ARCH
Statistic Degrees Signif
520.29 180 0.00000
@flux
# dd
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 of 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.
Test Statistic P-Value
Joint 33.7520472 0.00
1 0.1348914 0.42
2 1.8624746 0.00
3 1.0380136 0.00
4 0.2837474 0.15
5 0.0804013 0.67
6 0.1188408 0.48
7 0.0811365 0.67
8 0.6485388 0.02
9 0.1165840 0.49
10 0.0412141 0.92
11 0.0604180 0.80
12 1.1239040 0.00
13 1.2120879 0.00
14 1.1816754 0.00
15 0.4669988 0.05
16 1.1253165 0.00
17 0.8529359 0.01
18 0.7246583 0.01
19 0.8872770 0.00
20 0.6176391 0.02
21 0.2002857 0.26
22 0.3249375 0.11
23 0.9103172 0.00
24 0.4956675 0.04
25 0.2701695 0.16
26 1.9152490 0.00
27 0.2420097 0.19
28 2.8383045 0.00
29 0.5980212 0.02
30 0.2142316 0.23
31 0.8854486 0.00
32 1.9161060 0.00
33 0.4738197 0.05
34 0.8524883 0.01
35 2.6470491 0.00
36 0.3472104 0.10
37 0.6553268 0.02
38 0.1302127 0.44
39 0.2168067 0.23
40 0.5151937 0.04
41 0.1351453 0.42
42 0.1417446 0.40
43 0.2395982 0.20
44 0.1190260 0.48
45 0.3437533 0.10
46 7.7629884 0.00
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.
