A Survey of Recent Theoretical Results for Time Series Models with GARCH Errors,

This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA-GARCH are summarized. Various new ARCH-type models, including double threshold ARCH and GARCH, ARFIMA-GARCH, CHARMA and vector ARMA-GARCH, are also reviewed.

Asymptotic Theory for a Vector ARMA-GARCH Model,

This paper investigates the asymptotic theory for a vector ARMA-GARCH model. The conditions for the strict stationarity, ergodicity, and the higherorder moments of the model are established. Consistency of the quasi- maximum likelihood estimator (QMLE) is proved under only the second-order moment condition. This consistency result is new, even for the univariate 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, as well as a consistent estimator of the asymptotic covariance.

A univariate time varying analysis of periodic ARMA processes

The standard approach for studying the periodic ARMA model with coefficients that vary over the seasons is to express it in a vector form. In this paper we introduce an alternative method which views the periodic formulation as a time varying univariate process and obviates the need for vector analysis. The specification, interpretation, and solution of a periodic ARMA process enable us to formulate a forecasting method which avoids recursion and allows us to obtain analytic expressions of the optimal predictors. Our results on periodic models are general, analogous to those for stationary specifications, and place the former on the same computational basis as the latter.Comment: 26 pages, no figures. arXiv admin note: text overlap with arXiv:1403.335

A Note on an Iterative Least Squares Estimation Method for ARMA and VARMA Models

In this note we suggest a new iterative least squares method for estimating scalar and vector ARMA models. A Monte Carlo study shows that the method has better small sample properties than existing least squares methods and compares favourably with maximum likelihood estimation as well.ARMA models

Temporal aggregation of univariate and multivariate time series models: A survey

We present a unified and up-to-date overview of temporal aggregation techniques for univariate and multivariate time series models explaining in detail how these techniques are employed. Some empirical applications illustrate the main issues.Temporal aggregation, ARIMA, Seasonality, GARCH, Vector ARMA, Spurious causality, Multivariate GARCH

A survey of recent theoretical results for time series models with GARCH errors

This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA-GARCH are summarized. Various new ARCH-type models, including double threshold ARCH and GARCH, ARFIMA-GARCH, CHARMA and vector ARMA-GARCH, are also reviewed

Asymptotic theory for a vector ARMA-GARCH model

This paper investigates the asymptotic theory for a vector ARMA-GARCH model. The conditions for the strict stationarity, ergodicity, and the higherorder moments of the model are established. Consistency of the quasi-maximum likelihood estimator (QMLE) is proved under only the second-order moment condition. This consistency result is new, even for the univariate 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, as well as a consistent estimator of the asymptotic covariance

Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes

The class of multivariate L\'{e}vy-driven autoregressive moving average (MCARMA) processes, the continuous-time analogs of the classical vector ARMA processes, is shown to be equivalent to the class of continuous-time state space models. The linear innovations of the weak ARMA process arising from sampling an MCARMA process at an equidistant grid are proved to be exponentially completely regular ($\beta$-mixing) under a mild continuity assumption on the driving L\'{e}vy process. It is verified that this continuity assumption is satisfied in most practically relevant situations, including the case where the driving L\'{e}vy process has a non-singular Gaussian component, is compound Poisson with an absolutely continuous jump size distribution or has an infinite L\'{e}vy measure admitting a density around zero.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ329 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm