DCC-GARCH: selection of error distribution and extraction of volatility decay I am in a hesitation of detecting which indicators from maximum likelihood (ML) estimates of the Gaussian DCC model tell the volatility parameters' decaying. Another question is, how to know which model is best between Gaussian DCC and $t$-DCC. I would appreciate if you give an answer with an example.
 A: Volatility decay
In a DCC-GARCH model, conditional variance of each individual series is modelled by a separate univariate GARCH model (vanilla GARCH or some more exotic flavour of GARCH). Therefore, volatility decay is characterized by the GARCH parameters. Consider a vanilla GARCH($q$,$p$) model for simplicity, the conditional variance equation of which is
$$ \sigma_t^2 = \omega + \sum_{i=1}^p \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^q \beta_j \sigma_{t-j}^2. $$
Decay is characterized by the so-called persistence and half-life. 


*

*Persistence is the sum of the GARCH coefficient values, $\lambda = \sum_{i=1}^p \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^q \beta_j \sigma_{t-j}^2$. 

*Half-life is $K = \frac{\log(0.5)}{\log{\lambda}}$. 


For example, if $\lambda=0.9$, then $K\approx 6.6$, which means that
half of the initial gap between the current conditional variance and its long-term mean is closed in less than seven periods, absent new shocks. 
Gaussian DCC or $t$-DCC?
I would compare the values of the information criteria for the competing models to decide which one is a better approximation of the underlying process. If you are interested in forecasting, AIC (or AICc) is a relevant measure as it is efficient (in a certain sense), which is useful for forecasting. If you are interested in explanatory modelling, you may choose BIC as it is a consistent model selector (in a certain sense). See e.g. Claeskens "Statistical Model Choice" (2016) or Hyndman "Facts and fallacies of the AIC" (2013, blog post).
