Mgarch coefficients

[kula]

Hi Tom,
The RATS software coupled with your nice guidance helped me lot in learning the basics of MGARCH. I express my sincere thanks for this.
The attached file shaped by your comments throughout the forum is the one with, in my opinion at least, the proper diognastic tests. I also learned from the discussions in the forum that the individual coefficients are nearly impossible to interpret. But, is not it possible to report, for example, based on the relative magnitudes of the coefficients, that the volatility spillover from asset 1 to asset 2 (B(1,2), with the coefficient of 0.572074115) is stronger than that of from 2 to 1 ( B(2,1), with the coefficient of 0.006732749)?

xkury and xu100.RPF

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[TomDoan]

No. The off-diagonals depend upon scale. See the second-to-last paragraph before the output in https://estima.com/ratshelp/garchmvrpf. ... utput_BEKK.

[kula]

Hi Tom,
Thanks for your reply,
I now understand that higher variance series tend to have lower off-diagonal coefficients than lower variance series. I think, based on the output, I could just say that the residuals and volatility of each series have a statistically significant impact on the variance of the other series. Is it impossible to comment on the relative spillover effects of each variable? Also, as for providing further comments for and implications of the output, what is the next step? For example, could we produce and comment on conditional correlation graph? Is there anything that could be done?

[TomDoan]

Hi Tom,
Thanks for your reply,
I now understand that higher variance series tend to have lower off-diagonal coefficients than lower variance series. I think, based on the output, I could just say that the residuals and volatility of each series have a statistically significant impact on the variance of the other series. Is it impossible to comment on the relative spillover effects of each variable?

Not really. The BEKK GARCH coefficients (particularly on the H term) are really hard to interpret.

Also, as for providing further comments for and implications of the output, what is the next step? For example, could we produce and comment on
conditional correlation graph?

You can. It's described in the User's Guide. It just won't tell you anything special about the model---even a DBEKK model (where the A and B matrices are diagonal, and hence there are no "spillover" effects) will have time-varying correlations because of the fixed (and non-diagonal) C matrix.

Is there anything that could be done?

You can do Volatility Impulse Response Functions.

[kula]

Hi Tom,

I firstly extend my sincere thanks for your help.

As stated in GARCHMV.RPF, negative coefficients in the off-diagonals of A mean that the variance is affected more when the shocks move in opposite directions than when they move in the same direction.

As for B coefficients, the off-diagonal ones might have negative coefficients, too. I understand that this can not be interpreted as the inverse relationship. Is there any possible explanation for the negative sign of off-diagonal B coefficients?

Similarly, how could we explain the statistically significant negative sign of off-diagonal D coefficients?

[TomDoan]

Hi Tom,

I firstly extend my sincere thanks for your help.

As stated in GARCHMV.RPF, negative coefficients in the off-diagonals of A mean that the variance is affected more when the shocks move in opposite directions than when they move in the same direction.

As for B coefficients, the off-diagonal ones might have negative coefficients, too. I understand that this can not be interpreted as the inverse relationship. Is there any possible explanation for the negative sign of off-diagonal B coefficients?

Not really.

Similarly, how could we explain the statistically significant negative sign of off-diagonal D coefficients?

That will be similar to A, but because it only comes into play when both residuals have the chosen sign, it means that the added variance is actually smaller than it would be if only the own residuals had the chosen sign.

[humyra]

Dear Tom,

Could you please elaborate further on this statement "negative coefficients in the off-diagonals of A mean that the variance is affected more when the shocks move in opposite directions than when they move in the same direction". What would be an example of shocks moving in the opposite direction?

Also, does it mean that an asymmetric BEKK would be a better model to run?

Thanks.

[TomDoan]

Dear Tom,
Could you please elaborate further on this statement "negative coefficients in the off-diagonals of A mean that the variance is affected more when the shocks move in opposite directions than when they move in the same direction". What would be an example of shocks moving in the opposite direction?

Literally, one is positive while the other is negative. It happens all the time even for a pair of series where the shocks are generally positively correlated. As to what type of situation would have the BEKK A off-diagonal negative, suppose you have two exchange rates of currencies in a particular region. It's quite possible that the two rates moving in opposite directions would give a stronger signal of coming volatility in the currency market than the two moving together.

Also, does it mean that an asymmetric BEKK would be a better model to run?

Not necessarily.

[humyra]

Thank you Tom, I don't think my thesis would proceed without your constant help

[humyra]

Also, am I right in assuming that in RATS, the asymmetric term in the Asym BEKK-GARCH models capture the positive shocks instead of the negative ones? So, a significant D12 would then indicate that positive news has a greater impact on volatility from series i to series j than negative news does?

[TomDoan]

Also, am I right in assuming that in RATS, the asymmetric term in the Asym BEKK-GARCH models capture the positive shocks instead of the negative ones? So, a significant D12 would then indicate that positive news has a greater impact on volatility from series i to series j than negative news does?

No. The default handling of asymmetry in BEKK models is to include only the negative shocks. By construction, the covariance matrix is non-decreasing in the asymmetry term so if variance is assumed to go up with negative shocks, you need to include only the negative shocks in that. There have been models where positive or even mixed signs have been used which is why RATS has the SIGNS option. (For instance, if one of the series is inflation, a positive shock is "bad" news).

[humyra]

If I'm testing spillovers from the return series of a bank's stock to a financial market, then a significant D12 means that negative shocks from the bank to the financial market will increase volatility more.
Also, what does a negative, significant D12 indicate?

[TomDoan]

The signs of BEKK coefficient matrices aren't statistically identified---multiply any one of the matrices by -1 and you get exactly the same model. The A and B matrices conventionally aim for positive diagonal elements; those matrices will pretty much always will have non-zero diagonals or you end up with really odd behavior. However, there's no obvious sign convention for the D matrix and again, the sign of the matrix doesn't actually matter.