The prime objective of this thesis is to provide an innovative methodology on how to study convergence, in the context of the European Monetary Union (EMU). In particular two aspects are examined, convergence in incomes and monetary pass-through. With concern to incomes, empirical investigations have primarily used cointegration methodology. The view expressed here is that cointegration might not be a satisfactory method, in the sense that one is not able to see if two countries are in the process of convergence. By using cointegration, only the following binary question may be answered: Is there convergence between two countries or not? If the residuals obtained from the cointegrating regression are stationary, then convergence is achieved. However, it not possible to determine if the two countries are on the transition path. In the second part monetary pass-through is investigated. More particularly, this part examines how regime changes in EMU have affected the way in which changes in the policy determined rates are passed through to bank lending rates and if this process has become more homogeneous in European countries. Most of the empirical work up to now splits the sample into pre-EMU and post-EMU periods and compares the estimates. Others, claiming that the change need not to occur at this exact date, use statistical procedures to find a break date, then compare the pre-brake and post-break estimates. However, interest rates dynamics may be more complex than a simple jump from one regime to the next. Over the last two decades significant structural changes have taken place in the European economy. This implies a gradual transition from one regime to the next maybe expected. The starting premise of this thesis is that the 'the only constant aspect of convergence is change' and thus is claimed that constant coefficient methods might not be the optimal way to tackle these problems. Instead time-varying coefficient methods are suggested. Convergence and in general the transition path to Europe is viewed as a continuous change in the parameters of a model. Such methods might add valuable information concerning the transition of the individual European countries to a single integrated economy.