This paper investigates the asymptotic theory for a vector autoregressive moving average–generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The paper establishes conditions for strict stationarity, ergodicity, and higher-order moments of the model. It proves the consistency of the quasi-maximum-likelihood estimator (QMLE) under only the second-order moment condition, which is a new result even for univariate ARCH and GARCH models. The asymptotic normality of the QMLE for the vector ARCH model is obtained under only the second-order moment of the unconditional errors and the finite fourth-order moment of the conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also obtained for the vector ARMA-ARCH and ARMA-GARCH models, along with a consistent estimator of the asymptotic covariance. The paper also discusses the properties of the vector ARMA-GARCH model, including its causal representation and sufficient conditions for strict stationarity and ergodicity. The paper provides a uniform convergence result, which is used to prove the consistency of the QMLE under the second-order moment condition. The asymptotic normality of the QMLE is established using the second derivative of the likelihood function, which simplifies the proof and reduces the requirement for higher-order moments. The paper concludes with a discussion of the implications of the results for the asymptotic theory of the vector ARMA-GARCH model.This paper investigates the asymptotic theory for a vector autoregressive moving average–generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The paper establishes conditions for strict stationarity, ergodicity, and higher-order moments of the model. It proves the consistency of the quasi-maximum-likelihood estimator (QMLE) under only the second-order moment condition, which is a new result even for univariate ARCH and GARCH models. The asymptotic normality of the QMLE for the vector ARCH model is obtained under only the second-order moment of the unconditional errors and the finite fourth-order moment of the conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also obtained for the vector ARMA-ARCH and ARMA-GARCH models, along with a consistent estimator of the asymptotic covariance. The paper also discusses the properties of the vector ARMA-GARCH model, including its causal representation and sufficient conditions for strict stationarity and ergodicity. The paper provides a uniform convergence result, which is used to prove the consistency of the QMLE under the second-order moment condition. The asymptotic normality of the QMLE is established using the second derivative of the likelihood function, which simplifies the proof and reduces the requirement for higher-order moments. The paper concludes with a discussion of the implications of the results for the asymptotic theory of the vector ARMA-GARCH model.