Strong consistency and asymptotic normality of the quasi-maximum likelihood estimator for multivariate generalized autoregressive conditional heteroskedasticity (GARCH) is proven in this paper. The asymptotic normality is proven under the condition that the initial state of the process either obeys the stationary law, or is fixed. GARCH has become popular recently in modeling financial time series with volatility (time-varying variance), and its properties are essential in acquiring an accurate estimate of the underlying parameters describing the process.

My background is in pattern recognition and this paper is grounded in econometrics; thus, I am not in a position to provide a complete technical evaluation of the work. However, the two fields share many fundamental concepts in mathematics and statistics, and often produce fruitful interactions to solve practical problems; thus, I read this paper with considerable interest.

The proofs are lengthy, involve an extensive number of matrix manipulations, and are not easy to follow strictly. However, the authors have put them in appendices, and provided only their skeletons in the main body. It is easy to grasp the big picture of the problems discussed. The contributions of the authors’ work, and its relation to past developments in the field, are also clearly described. A reader with basic knowledge of statistics and linear algebra can enjoy the paper without an in-depth knowledge of econometrics. The paper is purely theoretical, and a practitioner who is interested in applying GARCH may not find it very useful.