© 2001 by Biometrika Trust
Estimation for partially nonstationary multivariate autoregressive models with conditional heteroscedasticity
1 Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong hrntlwk{at}hku.hk 2 Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong maling{at}ust.hk 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong mathwong{at}polyu.edu.hk
This paper investigates a partially nonstationary multivariate autoregressive model, which allows its innovations to be generated by a multivariate ARCH, autoregressive conditional heteroscedastic, process.Three estimators, including the least squares estimator, a full-rank maximum likelihood estimator and a reduced-rank maximum likelihood estimator, are considered and their asymptotic distributions are derived. When the multivariate ARCH process reduces to the innovation with a constant covariance matrix, these asymptotic distributions are the same as those given by Ahn & Reinsel (1990). However, in the presence of multivariate ARCH innovations, the asymptotic distributions of the full-rank maximum likelihood estimator and the reduced-rank maximum likelihood estimator involve two correlated multivariate Brownian motions, which are dierent from those given by Ahn & Reinsel (1990). Simulation results show that the full-rank and reduced-rank maximum likelihood estimator are more ecient than the least squares estimator. An empirical example shows that the two features of multivariate conditional heteroscedasticity and partial nonstationarity may be present simultaneously in a multivariate time series.
Key Words: Brownian motion; Cointegration; Full-rank and reduced-rank maximum likelihood estimators; Least squares estimator; Multivariate ARCH process; Partially nonstationary; Unit root
Received October 1999. Revised May 2001