Neither univariate not multivariate EGARCH has an underlying stochastic process that leads to its derivation, no regularity conditions, no likelihood equation, no Jacobian or Hessian matrices, and hence no asymptotic properties of the QMLE.
In short, EGARCH has no statistical properties.
It might be useful to read the following short papers on EGARCH:
“A one line derivation of EGARCH”, Econometrics, 2(2), 2014, 92-97.
“The correct regularity condition and interpretation of asymmetry in EGARCH”, Economics Letters, 161, 2017, 52-55.
“On the invertibility of EGARCH(p,q)”, Econometric Reviews, 37(8), 2018, 824-849.
As for DCC, the following paper shows it has fatal flaws and should not be used:
“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.