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Background: Hello, I am trying to investigate the comovement of the logged retuurns of the Green Bond (GB) market and 4 other markets (Treasury Bonds, Corporate Bonds, MSCI World, and Carbon Emission Rights) over time. Previous studies within the field have used DCC-GARCH. I have struggled to understand the methodology of this model. Some sources explain an easy procedure in which you:

  1. Run GARCH on the market returns to get the parameters for volatility over time.
  2. Create a vector of the volatility over time.
  3. Use DCC on the vectors created in step 2.

From other sources it seems as DCC-GARCH is a multivariate GARCH model in which you get the DCC of the volatility over time in one procedure instead of the three intermediate steps as the solution above. The algebra/statistics of this procedure quickly becomes pretty hard and I have not understood it completely.

Questions:

  1. Can I use the three-step procedure to get accurate DCC of the volatility over time?
  2. I have struggled with the second step in this procedure (using R).
    In step 1, I created ARCH/GARCH models to see which best captures the volatility of the markets. Based on the Akaike Information Criterion, GARCH seem to be the better fit. However, how do I use these parameters to get a vector of the volatility over time (i.e. step 2)?
  3. As seen in the GARCH model of MSCI World, alpha + beta = 1.002>1 the assumption of unconditional variance being finite and positive is not fulfilled. How do I deal with this?

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Would be very greatful for any help!

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  • $\begingroup$ Would recommend posting on Quantitative Finance (QF) at StackExchange. $\endgroup$
    – user318288
    Apr 21, 2021 at 15:36
  • $\begingroup$ @nxglogic, good idea, though GARCH models (and their extensions) are more actively discussed on Cross Validated (see the count under the GARCH tag here versus there), so the OP's decision to post here first was a smart one. $\endgroup$
    – Richard Hardy
    Apr 21, 2021 at 17:10

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This is far from a full answer (as I cannot quite find the time for one now), but perhaps it will help you get going.

  1. Stepwise approach is commonly used, I think, so it should be OK.
  2. You do not have to manually extract results from univariate GARCH models to use them in the DCC model. Check out the rmgarch vignette explaining how the DCC-GARCH model can be implemented in R.
  3. I would look into the univariate GARCH specification and fitting routines. Perhaps it is possible to restrict the coefficients to lie in the stationary territory. Or perhaps the variance is really (mildly) explosive. In the latter case, it may not make sense to keep the asset in the multivariate model with DCC.
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  • $\begingroup$ Thank you for the quick answer yet again! 1. Great, then it acually feels as I do understand how the model is working. 2. Yes, I managed to use rmgarch and some other packages to get the DCC :). 3. The current mean model I am using is an arma(0,0). I used this model since previous literature, from my understanding, do so. However, my prelimary testing suggests that I have autoregressive time series so would it make more sence to make the mean model ARMA(1,0). If I test this, can I use Akaike to choose beteween the ARMA(1,0) - GARCH (1,1) model and the ARMA(1,0) - GARCH (1,1) model? $\endgroup$
    – Isac
    Apr 28, 2021 at 19:05
  • $\begingroup$ @Isac, you probably can (I think there is a typo in your last sentence as you are comparing identical models). In rmgarch, you can also use VAR + DCC-GARCH. That is, VAR is the model for the multivariate condtional mean instead of individual AR(1) models. $\endgroup$
    – Richard Hardy
    Apr 28, 2021 at 19:09
  • $\begingroup$ I ended up solving it with an ARMA-(1,1) -GARCH(1,1). I am summing up my master thesis and have created another thread in which I am asking some additonal quesitons, stats.stackexchange.com/questions/524436/…. Regardless if you have time to look at the new thread, I would appricate if you could send me information on how I should adress you in my aknowladgements to [email protected]. Please let me know if it is inappropiate to ask peopel to mail you through these forums. If so, I'll delete this message. $\endgroup$
    – Isac
    May 16, 2021 at 11:07

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