Shiqing Ling

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.

Stationarity and the existence of moments of a family of GARCH processes

This paper investigates some structural properties of a family of GARCH processes. A simple sufficient condition for the existence of the alpha delta-order stationary solution of the processes is derived, where alpha belongs to (0,1] and delta > 0. The solution is strictly stationary and ergodic, and the causal expansion of the family of GARCH processes is also established. Furthermore, the necessary and sufficient condition for the existence of the moments is obtained. The technique used in this paper for the moment conditions is different to that used in He and Terasvirta (1999a), and avoids the assumption that the process started at some finite value infinitely many periods ago. Moreover, the conditions for the strict stationarity of the model and the existence of its moments are simple to check and should prove useful in practice.

Necessary and sufficient moment conditions for the GARCH(r,s) and asymmetric power GARCH(r,s) models

Although econometricians have been using Bollerslevs (1986) 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 higherorder moments of the GARCH (r, s) model was given by Ling (1999a). This paper shows that Lings condition is also necessary. As an extension, the necessary and sufficient moment conditions are established for Ding, Granger and Engles (1993) asymmetric power GARCH (r, s) model.

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.

On Fractionally Integrated Autoregressive Moving-Average Time Series Models with Conditional Heteroscedasticity

Abstract This article considers fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity, which combines the popular generalized autoregressive conditional heteroscedastic (GARCH) and the fractional (ARMA) models. The fractional differencing parameter d can be greater than 1/2, thus incorporating the important unit root case. Some sufficient conditions for stationarity, ergodicity, and existence of higher-order moments are derived. An algorithm for approximate maximum likelihood (ML) estimation is presented. The asymptotic properties of ML estimators, which include consistency and asymptotic normality, are discussed. The large-sample distributions of the residual autocorrelations and the square-residual autocorrelations are obtained, and two portmanteau test statistics are established for checking model adequacy. In particular, non-stationary FARIMA(p, d, q)-GARCH(r, s) models are also considered. Some simulation results are reported. As an illustration,...

Diagnostic checking of nonlinear multivariate time series with multivariate arch errors

Multivariate time series with multivariate ARCH errors have been found useful in many applications. In order to check the adequacy of these models, we define the sum of squared (standardized) residual autocorrelations and derive their asymptotic distribution. The results are used to derive several new multivariate portmanteau tests. Simulation results show that the asymptotic standard errors are quite satisfactory compared with empirical standard errors and that the tests have reasonable empirical size and power. The distribution of the standardized residual autocorrelations is also derived.

Estimation and Testing for Unit Root Processes with GARCH (1, 1) Errors: Theory and Monte Carlo Evidence

Abstract Least squares (LS) and maximum likelihood (ML) estimation are considered for unit root processes with GARCH (1, 1) errors. The asymptotic distributions of LS and ML estimators are derived under the condition α + β < 1. The former has the usual unit root distribution and the latter is a functional of a bivariate Brownian motion, as in Ling and Li [Ling, S., Li, W. K. (1998). Limiting distributions of maximum likelihood estimators for unstable autoregressive moving‐average time series with GARCH errors. Ann. Statist.26:84–125]. Several unit root tests based on LS estimators, ML estimators, and mixing LS and ML estimators, are constructed. Simulation results show that tests based on mixing LS and ML estimators perform better than Dickey–Fuller tests which are based on LS estimators, and that tests based on the ML estimators perform better than the mixed estimators.

Fitting an error distribution in some heteroscedastic time series models

This paper addresses the problem of fltting a known distribution to the innovation distribution in a class of stationary and ergodic time series models. The asymptotic null distribution of the usual Kolmogorov-Smirnov test based on the residuals generally depends on the underlying model parameters and the error distribution. To overcome the dependence on the underlying model parameters, we propose that tests be based on a vector of certain weighted residual empirical processes. Under the null hypothesis and under minimal moment conditions, this vector of processes is shown to converge weakly to a vector of independent copies of a Gaussian process whose covariance function depends only on the fltted distribution and not on the model. Under certain local alternatives, the proposed test is shown to have nontrivial asymptotic power. The Monte Carlo critical values of this test are tabulated when fltting standard normal and double exponential distributions. The results obtained are shown to be applicable to GARCH and ARMA-GARCH models, the often used models in econometrics and flnance. A simulation study shows that the test has satisfactory size and power for flnite samples at these models. The paper also contains an asymptotic uniform expansion result for a general weighted residual empirical process useful in heteroscedastic models under minimal moment conditions, a result of independent interest.

Testing for a linear MA model against threshold MA models

This paper investigates the (conditional) quasi-likelihood ratio test for the threshold in MA models. Under the hypothesis of no threshold, it is shown that the test statistic converges weakly to a function of the centred Gaussian process. Under local alternatives, it is shown that this test has nontrivial asymptotic power. The results are based on a new weak convergence of a linear marked empirical process, which is independently of interest. This paper also gives an invertible expansion of the threshold MA models.

Asymptotic Inference for Unit Root Processes with GARCH(1,1) Errors

This paper investigates the so-called one-step local quasi‐maximum likelihood estimator for the unit root process with GARCH~1,1! errors+ When the scaled conditional errors ~the ratio of the disturbance to the conditional standard deviation! follow a symmetric distribution, the asymptotic distribution of the estimated unit root is derived only under the second-order moment condition+ It is shown that this distribution is a functional of a bivariate Brownian motion as in Ling and Li ~1998, Annals of Statistics 26, 84‐125! and can be used to construct the unit root test+