This paper investigates the asymptotic theory for a vector autoregressive moving average-generalized autoregressive conditional heteroskedasticity~ARMA-GARCH! model+ The conditions for the strict stationarity, the ergodicity, and the higher order moments of the model are established+ Consistency of the quasimaximum-likelihood estimator~QMLE! is proved under only the second-order moment condition+ This consistency result is new, even for the univariate autoregressive conditional heteroskedasticity~ARCH! and GARCH models+ Moreover, 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 and also a consistent estimator of the asymptotic covariance+
How to undertake statistical inference for infinite variance autoregressive models has been a long-standing open problem. To solve this problem, we propose a self-weighted least absolute deviation estimator and show that this estimator is asymptotically normal if the density of errors and its derivative are uniformly bounded. Furthermore, a Wald test statistic is developed for the linear restriction on the parameters, and it is shown to have non-trivial local power. Simulation experiments are carried out to assess the performance of the theory and method in finite samples and a real data example is given. The results are entirely different from other published results and should provide new insights for future research on heavy-tailed time series. Copyright 2005 Royal Statistical Society.
Although econometricians have been using Bollerslev's~1986, Journal of Econometrics 31, 307-327! GARCH~r, s! model for over a decade, the higher order moment structure of the model remains unresolved+ The sufficient condition for the existence of the higher order moments of the GARCH~r, s! model was given by Ling~1999a, Journal of Applied Probability 36, 688-705!+ This paper shows that Ling's condition is also necessary+ As an extension, the necessary and sufficient moment conditions are established for Ding, Granger, and Engle's~1993, Journal of Empirical Finance, 1, 83-106! asymmetric power GARCH~r, s! model+
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