Estimation for partially nonstationary multivariate autoregressive models with conditional heteroscedasticity

Wai Keung LI, Shiqing LING, H. WONG

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23 Citations (Scopus)


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. Copyright © 2001 Biometrika Trust.
Original languageEnglish
Pages (from-to)1135-1152
Issue number4
Publication statusPublished - Dec 2001


Conditional Heteroscedasticity
Multivariate Models
Autoregressive Model
Least-Squares Analysis
Maximum Likelihood Estimator
Maximum likelihood
least squares
Autoregressive Conditional Heteroscedasticity
Asymptotic distribution
Least Squares Estimator
time series analysis
Multivariate Time Series
Brownian movement
Covariance matrix
Brownian motion
Time series
Conditional heteroscedasticity


Li, W. K., Ling, S., & Wong, H. (2001). Estimation for partially nonstationary multivariate autoregressive models with conditional heteroscedasticity. Biometrika, 88(4), 1135-1152. doi: 10.1093/biomet/88.4.1135


  • Brownian motion
  • Cointegration
  • Full-rank and reduced-rank maximum likelihood estimator
  • Least squares estimator
  • Multivariate ARCH process
  • Partially nonstationary
  • Unit root