This paper investigates the asymptotic theory for a vector autoregressive moving average-generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The authors establish conditions for strict stationarity, ergodicity, and higher-order moments of the model. They prove that the quasi-maximum-likelihood estimator (QMLE) is consistent under only the second-order moment condition, which is a new result even for univariate ARCH and GARCH models. Additionally, they obtain the asymptotic normality of the QMLE for the vector ARCH model under the second-order moment of unconditional errors and the finite fourth-order moment of conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also established for the vector ARMA-ARCH and ARMA-GARCH models, along with a consistent estimator of the asymptotic covariance. The paper provides a comprehensive asymptotic theory for the QMLE in these models, extending previous work on univariate and multivariate time series models.This paper investigates the asymptotic theory for a vector autoregressive moving average-generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The authors establish conditions for strict stationarity, ergodicity, and higher-order moments of the model. They prove that the quasi-maximum-likelihood estimator (QMLE) is consistent under only the second-order moment condition, which is a new result even for univariate ARCH and GARCH models. Additionally, they obtain the asymptotic normality of the QMLE for the vector ARCH model under the second-order moment of unconditional errors and the finite fourth-order moment of conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also established for the vector ARMA-ARCH and ARMA-GARCH models, along with a consistent estimator of the asymptotic covariance. The paper provides a comprehensive asymptotic theory for the QMLE in these models, extending previous work on univariate and multivariate time series models.