The diagnostics tests for multivariate GARCH

[jack]

Hi Tom.

I have a question regarding conducting diagnostics tests for multivariate GARCH models. I'm currently reading the GARCH WORKBOOK. As mentioned in the book, both univariate tests and multivariate diagnostics tests can be applied to multivariate GARCH models. When I use univariate diagnostics tests for multivariate GARCH models, I have a better understanding of the model because I can have a graphical look at the correlations of residuals and squared residuals (for each variable separately). I can also have a look at correlations coefficients of residuals and the squares. However, when I perform multivariate diagnostics tests, (@MVQstat and @MVARCHtest), I only get a single statistic and its p-value. I don't have a clear understanding because I can't see plots for correlations of standardized residuals and their squared terms. Please explain this because it's kind of a black box!
Specifically, my second question is, if I don't observe any evidence that residuals are serially correlated or show any ARCH effects when conducting univariate diagnostics tests, is it still necessary to perform multivariate tests?

My third question relates to the examples provided in the GARCH WORKBOOK. When I solve these examples step by step or run the relevant code, I still observe significant correlations in the errors and squared error terms based on multivariate diagnostics tests. What is the interpretation of these models now? Can I still interpret the estimation results and relationships between variables based on them, or are these results not reliable at all? What is the actual course of action? Because when I look at the examples, I see that they are based on articles published in reputable journals.

Thanks.

[TomDoan]

The multivariate ARCH test is a test of all pairwise interaction terms (the n(n+1)/2 second moment combinations) on their lags done as a multivariate regression; there's no really a simple way to display that, and even if there were, it's not clear that the information would be very helpful. The only MV-GARCH model which is designed to actually eliminate all of that is the VECH, so it's not surprising that more restricted models might fail a test. However, you also need to be aware of the effect on tests of the large sample sizes.

[jack]

Thank you for your kind reply.

Have you heard of the paper titled 'Volatility Spillovers between Energy and Agricultural Markets: A Critical Appraisal of Theory and Practice'? https://www.mdpi.com/1996-1073/11/6/1595 They strongly recommend diagonal BEKK model instead of full BEKK model. The authors refer to a paper that was produced using RATS6.3: https://academic.oup.com/erae/article-a ... m=fulltext: Price volatility in ethanol markets

They have done this paper too in this regard in Energy Economics: https://www.sciencedirect.com/science/a ... 8319301318

They also define new concepts of volatility spillovers as: Full and partial volatility and covolatility spillovers

[TomDoan]

Their "point" regarding BEKK is that no one has proved that it satisfies sufficient conditions for the covariance matrix of the estimates to be consistent, thus, it's not clear if a test based upon them is valid (as a general testing procedure). That's a completely different assertion than it not actually being a proper test for "spillover" as an economic construct. By constrast

https://estima.com/ratshelp/garchspillover.html

shows that BEKK (which admits a spillover test) and DVECH (which doesn't) can produce almost identical fits to a set of data---the BEKK has restrictions to allow it to produce a positive definite matrix by force, not to test any form of spillover.

[jack]

Do you mean that, contrary to what they claim, it is still possible to use the full BEKK model to measure the spillover effects?

They define the following three concepts of volatility spillovers:
1)Full volatility spillovers: d(Qiit)/d(εkt−1), k≠i

2)Full covolatility spillovers: d(Qijt)/d(εkt−1), i≠j, k≠i,j

3)Partial covolatility spillovers: d(Qijt)/(εit−1), i≠j

where Q is the conditional covariance matrix, and ε is the residual.

Furthermore, they argue that since, in their view, the full BEKK model is not a suitable method for investigating volatility spillovers, it is better to use the diagonal BEKK model naturally, and therefore, only the possibility of examining the concept of "partial covolatility spillovers" exists. Is this argument correct? Can I use their method for this purpose?

[TomDoan]

In my opinion, this is a paper which should be ignored. I don't think there is anything particularly helpful about it. "Partial co-volatility" is pretty much a meaningless concept. In the context of a DBEKK model it means simply that the "C" matrix isn't diagonal. But the C matrix will never be diagonal in practice---in a DBEKK there is no other way that the residuals can be correlated except with a non-diagonal C.

[jack]

I'm kind of confused!

You first mention that in the context of a DBEKK model, "partial co-volatility" means that the "C" matrix isn't diagonal. However, you then assert that the C matrix will never be diagonal in practice, so what's wrong with it?

[TomDoan]

Sorry. I got a bit confused by their notation. (I thought their Q was the correlation matrix, but it's the covariance matrix).

In a DBEKK model, there is no possibility of "full volatility spillover", basically by construction. There is no possibility of *NO* full or partial covolatility spillovers, also by construction, since the diagonal values in the A matrix have to be non-zero or you don't even have GARCH at all. So you would be testing for zero, something you know is non-zero.

To be perfectly honest, I don't see any real use for the covolatility spillovers. If there is full volatility spillover, there will almost certainly be covolatility spillover---it's hard to imagine how variances would change without covariances coming along.

[jack]

With the explanations you provided, I wonder how their recent article has been published in the Energy Economics journal: Volatility spillovers for spot, futures, and ETF prices in agriculture and energy https://www.sciencedirect.com/science/a ... 8319301318

or how several other articles like this one have been published (in the same journal) with references to it: Volatility spillovers for energy prices: A diagonal BEKK approach https://www.sciencedirect.com/science/a ... 8320303054

[TomDoan]

The fact that it's the same journal is hardly a surprise. The simple answer is that there are way more slots for papers in journals (even if you only count high and mid-range quality journals) than there are referees that are willing and able to review them. The first paper cited could best be described as "true, but uninteresting". (I'm not even sure about the technical merits since that all comes down to previous work of the same author). The paper, in fact, says that people have come up with sufficient conditions for full BEKK to give an asymptotic distribution for the coefficients, but the authors find them implausible (existence of 8th moments). I assume DBEKK only needs 4th moments. I'm not sure the difference is worth a paper. Besides, those are sufficient conditions, not necessary conditions. But, more important, the fact that a model which doesn't admit an actual test for spillover is preferable for testing spillover because it's easier to estimate is rather silly, isn't it? That's like the person searching for their keys on the wrong side of the street because the light is better there.

[jack]

Thank you so much for your valuable insights and guidance on the paper. I really appreciate your help.
It's quite concerning that some journals don't seem to be very thorough in their peer review process.

[jack]

Dear Tom,

I want to investigate the volatility spillovers between three markets (two foreign markets and one domestic market). However, the weekends in the foreign and domestic markets are different. Therefore, I have used the three-day average prices when all three markets are open, and then calculated the returns of the data. I don't know if this method is appropriate or not.

You can see the obtained results in the table below. I had a question. As you can see, the coefficients A(1,3) and B(1,3) are statistically significant, but their magnitudes are small, indicating that they may not be economically meaningful. Do you think there is really volatility spillovers from the first market to the third market? What would be an approximate range of the coefficients that would indicate a considerable overflow of fluctuations between the two markets?

MV-GARCH, BEKK - Estimation by BFGS
Convergence in    84 Iterations. Final criterion was  0.0000082 <=  0.0000100

With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations                       704
Log Likelihood                     -5146.6527

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
Mean Model(RTSE)
1.  RTSE{1}                       0.401723095  0.028921022     13.89035  0.00000000
2.  ROIL{1}                       0.000649371  0.014965506      0.04339  0.96538969
3.  RCOPPER{1}                    0.044745497  0.024356773      1.83709  0.06619713
4.  Constant                      0.241288304  0.062216982      3.87817  0.00010524
Mean Model(ROIL)
5.  RTSE{1}                       0.040524930  0.040305606      1.00544  0.31468429
6.  ROIL{1}                       0.124720343  0.033338770      3.74100  0.00018329
7.  RCOPPER{1}                    0.069437463  0.045867420      1.51387  0.13005797
8.  Constant                      0.160196462  0.135334846      1.18370  0.23653007
Mean Model(RCOPPER)
9.  RTSE{1}                      -0.005693650  0.028317129     -0.20107  0.84064593
10. ROIL{1}                       0.008732520  0.019915338      0.43848  0.66103680
11. RCOPPER{1}                    0.178838649  0.034297123      5.21439  0.00000018
12. Constant                      0.030332840  0.080580696      0.37643  0.70659865

13. C(1,1)                        0.218663564  0.077140251      2.83462  0.00458797
14. C(2,1)                       -0.071213538  0.375549266     -0.18963  0.84960298
15. C(2,2)                        1.198993790  0.782509301      1.53224  0.12546267
16. C(3,1)                       -0.029418668  0.097177444     -0.30273  0.76209454
17. C(3,2)                       -0.068547764  0.103748135     -0.66071  0.50879624
18. C(3,3)                        0.000016010  0.124306536  1.28794e-04  0.99989724
19. A(1,1)                        0.316808581  0.045144364      7.01768  0.00000000
20. A(1,2)                       -0.029293472  0.146972208     -0.19931  0.84201791
21. A(1,3)                        0.070008163  0.021599998      3.24112  0.00119062
22. A(2,1)                        0.008985584  0.022601156      0.39757  0.69094578
23. A(2,2)                        0.424664437  0.096952936      4.38011  0.00001186
24. A(2,3)                        0.021571443  0.018724183      1.15206  0.24929509
25. A(3,1)                       -0.025299949  0.020664954     -1.22429  0.22084189
26. A(3,2)                       -0.158637509  0.088901851     -1.78441  0.07435678
27. A(3,3)                        0.096099517  0.026567505      3.61718  0.00029783
28. B(1,1)                        0.949921394  0.014394717     65.99097  0.00000000
29. B(1,2)                        0.011331089  0.048595906      0.23317  0.81562970
30. B(1,3)                       -0.018813996  0.006924949     -2.71684  0.00659080
31. B(2,1)                       -0.003986134  0.010944411     -0.36422  0.71569643
32. B(2,2)                        0.859504618  0.113677488      7.56090  0.00000000
33. B(2,3)                       -0.001850452  0.008849590     -0.20910  0.83436993
34. B(3,1)                        0.006869882  0.006494929      1.05773  0.29017847
35. B(3,2)                        0.069188284  0.060699000      1.13986  0.25434516
36. B(3,3)                        0.993122321  0.003334592    297.82423  0.00000000
37. Shape                         8.128443814  1.116069943      7.28310  0.00000000


Multivariate Q Test
Test Run Over 2 to 705
Lags Tested         9
Degrees of Freedom 72
D of F Correction   9
Q Statistic        73.71287
Signif Level        0.42187


Multivariate ARCH Test
Statistic Degrees Signif
   760.03     360 0.00000

[TomDoan]

Dear Tom,

I want to investigate the volatility spillovers between three markets (two foreign markets and one domestic market). However, the weekends in the foreign and domestic markets are different. Therefore, I have used the three-day average prices when all three markets are open, and then calculated the returns of the data. I don't know if this method is appropriate or not.

I'm confused about what you are doing. If both trade M, T and W and one also trades Th and F, and the other Sa and Su, you are averaging the prices for M, T and W? And then what? How do you get returns from that? It makes more sense to do weekly Wednesday to Wednesday returns.

You can see the obtained results in the table below. I had a question. As you can see, the coefficients A(1,3) and B(1,3) are statistically significant, but their magnitudes are small, indicating that they may not be economically meaningful. Do you think there is really volatility spillovers from the first market to the third market? What would be an approximate range of the coefficients that would indicate a considerable overflow of fluctuations between the two markets?

Code: Select all

MV-GARCH, BEKK - Estimation by BFGS
Convergence in    84 Iterations. Final criterion was  0.0000082 <=  0.0000100

With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations                       704
Log Likelihood                     -5146.6527

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
Mean Model(RTSE)
1.  RTSE{1}                       0.401723095  0.028921022     13.89035  0.00000000
2.  ROIL{1}                       0.000649371  0.014965506      0.04339  0.96538969
3.  RCOPPER{1}                    0.044745497  0.024356773      1.83709  0.06619713
4.  Constant                      0.241288304  0.062216982      3.87817  0.00010524
Mean Model(ROIL)
5.  RTSE{1}                       0.040524930  0.040305606      1.00544  0.31468429
6.  ROIL{1}                       0.124720343  0.033338770      3.74100  0.00018329
7.  RCOPPER{1}                    0.069437463  0.045867420      1.51387  0.13005797
8.  Constant                      0.160196462  0.135334846      1.18370  0.23653007
Mean Model(RCOPPER)
9.  RTSE{1}                      -0.005693650  0.028317129     -0.20107  0.84064593
10. ROIL{1}                       0.008732520  0.019915338      0.43848  0.66103680
11. RCOPPER{1}                    0.178838649  0.034297123      5.21439  0.00000018
12. Constant                      0.030332840  0.080580696      0.37643  0.70659865

13. C(1,1)                        0.218663564  0.077140251      2.83462  0.00458797
14. C(2,1)                       -0.071213538  0.375549266     -0.18963  0.84960298
15. C(2,2)                        1.198993790  0.782509301      1.53224  0.12546267
16. C(3,1)                       -0.029418668  0.097177444     -0.30273  0.76209454
17. C(3,2)                       -0.068547764  0.103748135     -0.66071  0.50879624
18. C(3,3)                        0.000016010  0.124306536  1.28794e-04  0.99989724
19. A(1,1)                        0.316808581  0.045144364      7.01768  0.00000000
20. A(1,2)                       -0.029293472  0.146972208     -0.19931  0.84201791
21. A(1,3)                        0.070008163  0.021599998      3.24112  0.00119062
22. A(2,1)                        0.008985584  0.022601156      0.39757  0.69094578
23. A(2,2)                        0.424664437  0.096952936      4.38011  0.00001186
24. A(2,3)                        0.021571443  0.018724183      1.15206  0.24929509
25. A(3,1)                       -0.025299949  0.020664954     -1.22429  0.22084189
26. A(3,2)                       -0.158637509  0.088901851     -1.78441  0.07435678
27. A(3,3)                        0.096099517  0.026567505      3.61718  0.00029783
28. B(1,1)                        0.949921394  0.014394717     65.99097  0.00000000
29. B(1,2)                        0.011331089  0.048595906      0.23317  0.81562970
30. B(1,3)                       -0.018813996  0.006924949     -2.71684  0.00659080
31. B(2,1)                       -0.003986134  0.010944411     -0.36422  0.71569643
32. B(2,2)                        0.859504618  0.113677488      7.56090  0.00000000
33. B(2,3)                       -0.001850452  0.008849590     -0.20910  0.83436993
34. B(3,1)                        0.006869882  0.006494929      1.05773  0.29017847
35. B(3,2)                        0.069188284  0.060699000      1.13986  0.25434516
36. B(3,3)                        0.993122321  0.003334592    297.82423  0.00000000
37. Shape                         8.128443814  1.116069943      7.28310  0.00000000


Multivariate Q Test
Test Run Over 2 to 705
Lags Tested         9
Degrees of Freedom 72
D of F Correction   9
Q Statistic        73.71287
Signif Level        0.42187


Multivariate ARCH Test
Statistic Degrees Signif
   760.03     360 0.00000

You should never attempt to interpret the scales of the off-diagonals in a BEKK. Read the discussion on https://estima.com/ratshelp/garchmvrpf.html#GARCH_Output_BEKK

[jack]

Thank you for your excellent guide and help.

[jack]

Dear Tom,

I've encountered a fundamental question in estimating a simple GARCH model. I appreciate your guidance.

I have a dataset, and I intend to incorporate a dummy variable into the variance equation of the model. The estimation results vary depending on which of the normal or t-distributions I use and whether I employ robust standard errors to account for heteroscedasticity in the variance. When using the normal distribution, the estimated coefficient of the dummy variable is negative and significant. When using the normal distribution along with robust standard errors, the estimated coefficient becomes insignificant. However, when using the t-distribution, the coefficient is significant but suddenly becomes positive. If I use the t-distribution along with robust standard errors, the estimated coefficient becomes is positive but insignificant. The question is: which distribution estimation should I practically use, and should I use robust standard deviations? Also, if I use the normal distribution ,it seems that the long-term variance is negative because the dummy variable coefficient is negative and greater than the intercept.
Thank you for your guidance, as always.

Code: Select all

GARCH Model - Estimation by BFGS
Convergence in    20 Iterations. Final criterion was  0.0000078 <=  0.0000100

Dependent Variable X
Usable Observations                      3527
Log Likelihood                     -3872.0286

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                      0.018118210  0.010216396      1.77344  0.07615510
2.  X{1}                        0.368447164  0.017309202     21.28620  0.00000000

3.  C                             0.034327563  0.006968308      4.92624  0.00000084
4.  A                             0.147453426  0.015573093      9.46847  0.00000000
5.  B                             0.847459388  0.013281283     63.80855  0.00000000
6.  DUM                      -0.054459143  0.017253863     -3.15634  0.00159760


GARCH Model - Estimation by BFGS
Convergence in    20 Iterations. Final criterion was  0.0000077 <=  0.0000100

With Heteroscedasticity/Misspecification Adjusted Standard Errors
Dependent Variable X
Usable Observations                      3527
Log Likelihood                     -3872.0286

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                      0.018117739  0.015788361      1.14754  0.25115947
2.  X{1}                        0.368447517  0.019623949     18.77540  0.00000000

3.  C                             0.034327273  0.026130400      1.31369  0.18895019
4.  A                             0.147460375  0.030345439      4.85939  0.00000118
5.  B                             0.847452894  0.028125964     30.13063  0.00000000
6.  DUM                     -0.054457054  0.055056769     -0.98911  0.32261062


GARCH Model - Estimation by BFGS
Convergence in    32 Iterations. Final criterion was  0.0000007 <=  0.0000100

Dependent Variable X
Usable Observations                      3527
Log Likelihood                     -3594.0116

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                      0.027719642  0.006787359      4.08401  0.00004427
2.  X{1}                        0.340704222  0.014951948     22.78661  0.00000000

3.  C                            -0.009124384  0.005582018     -1.63460  0.10213224
4.  A                             0.206606228  0.028184502      7.33049  0.00000000
5.  B                             0.839764463  0.017157852     48.94345  0.00000000
6.  DUM                        0.032504334  0.015224199      2.13504  0.03275742
7.  Shape(t degrees)              3.758679405  0.279438573     13.45083  0.00000000


GARCH Model - Estimation by BFGS
Convergence in    32 Iterations. Final criterion was  0.0000007 <=  0.0000100

With Heteroscedasticity/Misspecification Adjusted Standard Errors
Dependent Variable X
Usable Observations                      3527
Log Likelihood                     -3594.0116

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                      0.027719642  0.006070636      4.56618  0.00000497
2.  X{1}                        0.340704222  0.014824037     22.98323  0.00000000

3.  C                            -0.009124384  0.007101532     -1.28485  0.19884565
4.  A                             0.206606228  0.038370996      5.38444  0.00000007
5.  B                             0.839764463  0.025175511     33.35640  0.00000000
6.  DUM                        0.032504334  0.019797520      1.64184  0.10062345
7.  Shape(t degrees)              3.758679405  0.308153951     12.19741  0.00000000

That's not that surprising. Given your estimate of the shape parameter, you must have rather noisy data. The t distribution reduces the influence of the biggest residuals relative to the use of the Normal. Note that the A+B is slightly greater than 1, which changes the behavior of the GARCH model. What kind of data do you have? The AR coefficient looks pretty big for the typical application of the GARCH.