This article analyzes the identification and normalization of cointegrating vectors. Normalizing a cointegrating relation with respect to one of the relevant variables is with loss of generality; and restrictions that are supposed to identify a vector may fail to do so for particular parameter values. I propose to tackle both problems by testing whether particular rank conditions are violated. It is shown that Johansen and Juselius's class of likelihood ratio statistics for structural hypotheses in a cointegrated Gaussian vector autoregression may be used for this purpose. The tests are applied to a model of the demand for money in the United Kingdom.