Phase-shifting common cycles and common trends

In this paper, we outline a framework for modelling and analysing economic fluctuations and dynamics. It is assumed that there may exist common trends and common cycles in the time series to be analysed It is further generalised that common cycles may have non-coincident, or phase-shifting attributes These attributes are examined via the Markov transition matrix in a VAR system, revealing the way in which the phase-shifting works with the reduced rank Markov transition matrix. The links with the structural common trend model are also presented.