On Asymptotic Inference in Linear Cointegrated Time Series Systems

Abstract

This paper considers vector-valued nonstationary time series models, in particular, autoregressive models, whose nonstationarity is driven by a few nonstationary (induced by “unit roots”) trends, in such a way that some of the linear combinations of the components of the vector model will be stationary. Models of this form are called cointegrated models. These stationary linear combinations are called cointegrating relationships. Asymptotic inference problems associated with the parameters of the cointegrating relationships when the remaining parameters are treated as unknown nuisance parameters are considered. Similarly, inference problems associated with the unit roots are considered. All possible unit roots, including complex ones, together with their possible multiplicities, are allowed. The framework under which the asymptotic inference problems are dealt with is the one described in LeCam (1986, Asymptotic Methods in Statistical Decision Theory ) and LeCam and Yang (1990, Asymptotics in Statistics: Some Basic Concepts ), though it will be seen that the usual normal or mixed normal situations do not apply in the present context.