ISSUE:

How valuable is the insignificant coefficient in MGARCH?

BASIC & APPLIED RESEARCH

Strictly speaking, in basic research significance test would dictate the removal or deletion of insignificant variables in the model. Although this strict application of methodological regime is doctrinal correct; in non-social science, such as medical science or pharmaceutical research, strict application of significance test would be appropriate---it many places, it may be dictated by law. Or instance, the test for new pharmaceutical products would require strict application of significance test in many cases if the danger from the new drug outweighs the benefits. Yet, at the same time less than strict statistical significant, i.e. 0.9975 in medical science would still be treated as acceptable even if the result is less than 99.75%. For instance, in a cancer treatment of 100 patients where without treatment the probability of death within a year is 100% and with the proposed treatment the survival rate passing one year is 20%---although far below the required statistical significance level, a 20% survival rate when compared to 100% death rate is still worth while despite the treatment efficiency is only 0.80 not meeting the standard of 0.9975. In this case, the coefficient that is statistically insignificant may still be reported.

   In economics, some research products will be used for policy formulation. In macro-issue policy modeling, less than conventional 0.95 significance test may still be worth while to report if the implementation of such variable in policy tool would benefit the population. For instance, social welfare, affirmative action, and many posterity measures would fail strict statistical significance test, yet we still implement them because their benefits are significant to the people.

   The rationale for reporting these insignificant coefficients may be due to the potential benefits of their inclusion (not in statistical significance terms but in practical terms). This is applied research consideration. Under basic research consideration, strict interpretation of significance test would be followed. Variable with insignificant coefficient would be removed. Some times the intended contribution or application of the research also influence what the researchers report.

From your question I imagine you look at MGARCH models for the entire covariance, such as BEKK, as opposed to models for a decomposition of the covariance, such as CCC or DCC. The latter combine univariate models for the marginal volatilities (thus univariate GARCH) with a model for the correlation.

In BEKK-type models, coefficients might be many, and are normally reported in matrices where the most relevant role is given to coefficients on the main diagonal. Those must be statistically significant otherwise we might have some doubt on the need of an MGARCH structure. The off-diagonal coefficient might be non-significant, suggesting they might be zero, and this open the door to the specification of restricted models. Then one might decide if to move from a fully parameterized model to a restricted specification.

What is not common is the imposition of zero restrictions on single coefficients in BEKK-type models.

I have some general reservations concerning GARCH incl.MGARCH:

I think autocorrelation has its roots in the phenomena of financial markets 

that indexes and stocks react on similar external or exogen informations

in a similar way. Autocorrelation doesn't cover the events and doesn't predict them.

As far as mono oder multi factoriell models  is concerned, the latter  ones allow e better calibration.  Bette keep potential factors in the model since their influence may become important.

I found the stalagmites effect: Comparing a format of cours data of - say - a quarter of a year with all other periods one may find high  frequencies (60 repetitions and more) and times with unique formats  of no "autocorrelation" respectively - "white noise". Considering prediction I didn't find any formats to forecast the next similar format of interest (like the begin of jums).

pls see:

https://ideas.repec.org/p/wpa/wuwpif/0510024.html

Thanks

If you are serious about multivariate GARCH models, individually insignificant estimated parameters are not of particular concern as the emphasis is on the covariances and correlations to calculate optimal hedge ratios.

If you do a simple search of the following two papers in RePEc (and elsewhere), the main issues associated with multivariate conditional volatility models are explained in detail:

“What they did not tell you about algebraic (non-)existence, mathematical (ir-)regularity and (non-)asymptotic properties of the dynamic conditional correlation (DCC) model, Journal of Risk and Financial Management, 12(2:61), 2019, 1-9.

“What they did not tell you about algebraic (non-)existence, mathematical (ir-)regularity and (non-)asymptotic properties of the full BEKK dynamic conditional covariance model, Journal of Risk and Financial Management, 12, 2019, 1-7.