Conventional methods to test for long-term PPP based on the theory of cointegration are typically undertaken in the framework of vector error correction models (VECM). The standard approach in the use of VECMs is to employ a model of full-order, which assumes nonzero entries in all the coefficient matrices. But, the use of full-order VECM models may lead to incorrect inferences if zero entries are required in the coefficient matrices. Specifically, if we wish to test for indirect causality, instantaneous causality, or Granger non-causality, and employ "overparameterised" full-order VECM models that ignore entries assigned a priori to be zero, then the power of statistical inference is weakened and the resultant specifications can produce different conclusions concerning the cointegrating relationships among the variables. In this paper, an approach is presented that incorporates zero entries in the VECM analysis. This approach is a more straightforward and effective means of testing for causality and cointegrating relations. The paper extends prior work on PPP through an investigation of causality between the U.S. Dollar and the Japanese Yen. The results demonstrate the inconsistencies that can arise in the area and show that bi-directional feedback exists between prices, interest rates and the exchange rate such that adjustment mechanisms are complete within the context of PPP.