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I've used DCC-ARMA(1,0) -GARCH(1,1) to model green bond co-movement with some other marekts. In the output, I get the parameters "dccalpha" and "dccbeta". However, I do not know how to interpret these. Don't know if the output is needed to answer my quesiton but included it in the bottom in case someone is interested.

From previous literature, I have understood that alpha1 and beta1, aka jointalpha and jointbeta, tell me the degree of volatility spillovers in-between the time series. Is this true or does they rather show the correlation of the time series? The reason why I distinguish these two is that “to me” correlation indicates the strength of which two variables tend to co-move which does not have to imply a causal relationship whereas spillovers means that volatility in prices of one market has a causal effect on price movements in another market.

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*** Below is an update form the orignially posted question. It was made in order to answer Richard Hardy's response with an equation containing subscripts and denotations. These details makes the formula easier to understand.

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  • $\begingroup$ What do you think about my answer? If it is helpful and clear, you may accept it by clicking on the tick mark to the left. Otherwise, you may ask for further clarification. This is how Cross Validated works. $\endgroup$
    – Richard Hardy
    Commented Jun 9, 2021 at 17:42

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As I have said in "DCC GARCH - specifying ARCH and GARCH parameter matrices in Stata", I do not see how DCC could generate spillovers in the sense that $\sigma_{i,t}^2$ be a function of $\sigma_{j,t}^2$ for assets $\{i,j\}$ where $i\neq j$. For any asset in the set, its volatility is generated by a univariate GARCH model. Thus the asset's volatility depends only on its own past, not the past of other asset's volatilities. In that sense there are no spillovers or causal effects. There are only contemporaneous correlations between the standardized innovations.

dccalpha and dccbeta tell you how the correlations are evolving over time in an autoregressive manner. dccalpha provides the contribution of the realized correlation matrix from last period while dccbeta provides the contribution of a "long-run" (intentionally in quotation marks as this is likely not the standard name for it) correlation matrix that is due to all previous periods.

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  • $\begingroup$ Okey, gotcha with the "no spillover effect part", thanks! However, what does it mean that they are evolving in an autoregressive manner? Seems wierd to me that there is two seperate parameters if both of them reflective how autoregressive the correlation is. What is then the difference betweem the dccalpha and the dccbeta :) $\endgroup$
    – Isac
    Commented May 16, 2021 at 15:58
  • $\begingroup$ @Isac, updated. $\endgroup$
    – Richard Hardy
    Commented May 16, 2021 at 16:48
  • $\begingroup$ It makes sense “on the surface” but have a hard time grasping the dccalpha matrix. Since I am not to familiar with matrices I like the algebraic approach of explaining how to retrieve DCC. The formula in the updated question shows this approach. That is, the standardized residuals are calculated in Formula 1. Then the DCC is calculated in formula 2. Since the information set It-1 is used, this would be the beta? How would the alpha look like? Also, I couldn't understand this from Engle (2002). It is from that paper or do you have any other paper that would be appropiate to refer to? :) $\endgroup$
    – Isac
    Commented May 20, 2021 at 17:35
  • $\begingroup$ @Isac, think about GARCH(1,1): there is a weighted combination of two components: the "realized" variance (the last error squared) plus the last conditional variance (the one underlying the last error). In DCC, we have an analogy: a weighted combination of the "realized" empirical correlation matrix from the last period (the error vector multiplied by itself to make a matrix) plus the conditional correlation matrix from last period (the one supposedly underlying the error vector from the last period). Does that make sense? $\endgroup$
    – Richard Hardy
    Commented May 20, 2021 at 17:50
  • $\begingroup$ maybe. In GARCH(1,1) , the larger the coefficient of the ARCH parmeter, the more similar we expect the volatility of t and t-1 to be. The greater the GARCH coefficient, the greater impact yestardays "distance" from u(t-2) to u(t-1) is expected to have on todays voaltility. So the greater dccalpha, the more similar we expect todays correlation to be that of yestardays correlation? A high dccbeta means that we expect the that the flucuation of DCC (t-2) to DCC (t-1) will have a high impact on DCC(t)? Also, when you say "empirical correlation", does that mean Pearson rather than DCC? $\endgroup$
    – Isac
    Commented May 20, 2021 at 19:16

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