Asymptotic properties of least squares estimators of cointegrating vectors. (English) Zbl 0651.62105

Stock, James H.

Econometrica 55, 1035-1056 (1987).

A vectorial time series is said to be cointegrated if there are one or more linear combinations of the components which have a lower order of integration than do any of the individual scalar random variables. A multiplying vector that reduces the order of integration of the system is said to be a cointegrating vector. Here the special case is considered when the order of integration can be reduced from 1 to zero, i.e. the linear combinations of the process are stationary.
Least squares estimators of the parameters of a cointegrating vector are shown to converge in probability to their true values at the rate \(T^{1-\delta}\) for any positive \(\delta\), sharply contrasting with the conventional rate \(T^{1/2}\). These estimators can be written asymptotically in terms of relatively simple nonnormal random matrices which do not depend on the parameters of the system. These asymptotic representations form the basis for simple and fast Monte Carlo calculations of the limiting distributions of these estimators. Asymptotic distributions thus computed are tabulated for several cointegrated processes. Relations to error correction models and possible econometric consequences are also mentioned.

Reviewer: J.Tóth

MSC:

62P20	Applications of statistics to economics
62E20	Asymptotic distribution theory in statistics
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
91B84	Economic time series analysis
65C05	Monte Carlo methods

Keywords:

tables; convergence in probability; vectorial time series; linear combinations; order of integration; cointegrating vector; Least squares estimators; nonnormal random matrices; asymptotic representations; limiting distributions; cointegrated processes; error correction models