The Wald-type Test of a Normalization of Cointegrating Vectors

Vector autoregressive (VAR) models have often been used in the econometric literature as useful models to describe stationary/non-stationary time series. Additionally cointegrating vectors are of primary interest for researchers who investigate the long-run stable relationship between economic variables. In VAR models it is well known that cointegrating vectors are identifiable only up to their column space. Although Johansen’s (1988, 1991) maximum likelihood (ML) method solves the identification problem by imposing the just-identifying restrictions on cointegrating vectors, it is difficult to interpret the identified estimator in an economic sense because Johansen’s identifying restrictions are imposed from a statistical point of view. See Johansen and Juselius (1994), Boswijk (1996), Luukkonen et al. (1999) and Pesaran and Shin (2002) among others. To obtain an interpretable estimator we may impose some restrictions on cointegrating vectors from an economic point of view, but such restrictions are not necessarily identifiable. We then need to check whether the restrictions imposed on cointegrating vectors are valid or not. General identifying conditions are given by Boswijk (1995), Johansen (1995a), Pesaran and Shin (2002), and Boswijk and Doornik (2003). One of the useful identifying or normalizing conditions is expressed as c′β = Ir where β consists of n× r cointegrating vectors and c is an n× r known matrix of full column rank, as investigated by Stock (1987), Johansen (1991, 1995b), and Paruolo (1997) among others. Tests for the validity of this normalization are proposed by Boswijk (1996), Luukkonen et al. (1999), Saikkonen (1999), and Paruolo (2005). The null hypothesis of the failure of the normalization is considered in Boswijk (1996) and Paruolo (2005), while the other papers propose tests for the null of the validity of the normalization. In this paper we propose a test for the null hypothesis of the invalid normalization of cointegrating vectors. Since our test is based on the unrestricted ML