The purpose of the paper is to suggest a modelling strategy that can be used to study the process of pairwise convergence within time series analysis. Moving from the works of Bernard (1992) and Bernard and Durlauf (1995), we specify an I(1) cointegrated model characterized by broken linear trends, and we identify the driving force leading to convergence as a common stochastic trend, but the results are unsatisfactory. Then we deal the same question of time series convergence within I(2) cointegration analysis, allowing for broken linear trends and an I(2) common stochastic trend as the driving force. The results obtained with this second specification are encouraging and satisfactory. The suggested modelling strategy is applied to the convergence of long-term bond markets in the Economic and Monetary Union (EMU), that we observe during the years covering the second stage, that is the period from 1993 to the end of 1998, before the introduc-tion of euro. During the third stage, started in 1999 and continuing, the markets show a tendency to move together and to behave similarly.