Time series convergence within I(2) models: the case of daily long term bond yields in the euro area

The purpose of the paper is to suggest a modelling strategy that can be used to study the process of pairwise convergence within multiple time series analysis. The scenario of interest is the one in which the observed time series are characterized by non-stationary behaviour driven by common stochastic trends, but with different linear deterministic trends, over a first sub-period, and non-stationary behaviour, common stochastic trends and no deterministic linear trends over a second sub-period. A typical example of such scenario is represented by the ten-years zero-coupon bond yields of the four largest euro area countries that show a clear convergence behaviour, though following different time paths, as if a common driving force had led them to a point from which they have moved together, still sharing the same driving force. The point corresponds to the time of the introduction of euro. Therefore, the idea of convergence we have in mind is the one according to which series, following different dynamic behaviours and time paths, get to converge through narrowing their differences. 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 assuming an I(2) common stochastic trend as the driving force. The results obtained with this second specification are encouraging and satisfactory. According to theory, the bond market unification process would imply a single latent factor, that is a single common stochastic trend as a driving force underlying the co-movements of yields of the same maturities across different countries' market. Analysing weekly observations of long term bond yields relative to France, Germany, Italy and Spain within the I(1) model we detect some clear signals that the convergence process is more complex than expected and that, in order to investigate the empirical regularities behind the swings in the series, we have to use an I(2) model. In fact, analysing the observations within an I(2) model, we find evidence of the existence of the expected common stochastic trend. Such results indicate the importance of modelling the smoothing behaviour shown by the series along their process of convergence, using an approach which takes into accounts the curvature of the co-movements in the series, as well as the level and the slope. As a conclusion, it's reasonable to state that the chosen I(2) approach has allowed a better modelling of the convergence process, giving evidence of expected characteristics that the I(1) model wouldn't show.