13 - Separation in Cointegrated Systems and Persistent-Transitory Decompositions

Summary

INTRODUCTION

It is a frequent empirical finding in macroeconomics that several cointegration relations may exist amongst economic variables but in the particular way that the single relations appear to have no variables in common. It is also sometimes found in such systems that the error correction terms or other stationary variables from one set of variables may have important explanatory power for variables in another set. For example Konishi et al. (1993) considered three types of variables of US data: real, financial and interest rate variables. They found that cointegration existed between variables in each subset but not across the variables such that the different sectors did not share a common stochastic trend. On the other hand, it was also found that the error correction terms of the interest rate relation and the sector of financial aggregates had predictive power with respect to the real variables of the system. As argued by Konishi et al. (1993) the situation sketched above may extend the usual “partial equilibrium” cointegration set-up to a more “general equilibrium” setting although in a limited sense.

The notion of separation initially developed by Konishi and Granger (1992) and Konishi (1993) provides a useful way of describing formally the above possibility: Consider two groups of I(1)-variables, X_1t and X_2t of dimension p₁ and p₂, respectively, X₁ and X₂ are assumed to have no variables in common and in each sub-system there is cointegration with the cointegration ranks being r₁ < p₁ and r₁ < p₂.