Shocks and Volatilties

[Zankawa]

Hi Tom,
In the BEKK model, my understanding of the A and G matrices is that the elements of matrix A measure the shock spillovers whilst the elements of matrix G measure the volatility spillovers. If this is correct, I want to know what is the difference between shocks and volatilities when examining the linkages between variables. If the elements of A (Aii and Aij) are significant for example, but the elements of G are not or the vice versa, how will the results be interpreted?
Thank you

[TomDoan]

First of all, BEKK parameters don't really "measure" anything---the relationship between the parameters and their effects on the variance is complicated and non-linear. But, both the A and G(?) (do you mean D?) are different parts of the "spillover".

[Zankawa]

Many thanks for the reply. Can you give me a brief description of what parts of the spillover are the A and the G?
Thank you

[TomDoan]

The A has effects of both signs, the D has only the negative signs. BEKK allows for some form of "spillover" (which is a very vaguely defined term to start), but again, the parameters don't have any individual interpretations.

[Zankawa]

Dear Tom,
In your replies to my questions about the shocks and volatilities of the BEKK model, you mentioned that the parameters of the BEKK model don't have any individual interpretations because the relationship between the parameters are complicated and non-linear. However, I am writing a dissertation and the BEKK is among the models I am using in the dissertation, and so I am required to explain exactly the terms 'shocks' and 'volatilities' in the BEKK model. The issue here is not the interpretation of the parameters per se, but the actual definitions of the terms. Therefore, I wanted to ask again if you could help me with the actual definitions of the terms 'shocks' and 'volatilities' in the BEKK model and the differences between the two.
Thank you

[TomDoan]

Shocks are the errors (difference between actual and predicted) and volatilities are the variances (or full covariance matrix). All GARCH models (not just BEKK) predict the covariance matrix given past shocks---they just use a different mapping from one to the other.

[Zankawa]

Thank you for the reply. Do you know if there is any document where the shocks and volatilities are discussed that I can make reference to? Based on the explanations you have given about shocks and volatilities, do the results of a model make any sense if the shocks are significant but the volatilities are not significant?
Thank you

[TomDoan]

You seem to be confusing the objects (the volatilities and the shocks) with the coefficients in the GARCH model on them. The "ARCH" coefficients (the ones on the lagged shocks) are generally quite a bit smaller than the "GARCH" coefficients (the ones on the lagged variances/covariances), because, in practice, the volatility seems to be quite a bit more persistent than would be produced by a low order ARCH model. At minimum you can go back to Bollerslev's original paper which lays out the case for GARCH vs ARCH. There's certainly nothing all that unexpected about the "ARCH" coefficients being "statistically insignificant", particularly if the data set isn't large or at least isn't large relative to the number of parameters you're trying to estimate. They tend to be numerically small, and their immediate effects are often not really well-estimated, which means that they can have a standard error fairly high relative to their value.