A fractional cointegration analysis of the long-run relationship between black and official foreign exchange rates: the case of the Brazilian cruzeiro

This paper applies a relatively new but generalized concept of fractional cointegration to shed some light on the validity of a long-run relationship between monthly black and official US dollar rates of the Brazilian cruzeiro. An investigation of the stochastic properties of these rates reveals that, while the relationship is not cointegrated in their logs, they appear to be fractionally cointegrated if we allow for mean reverting processes that are CI (1, d) with 0< d < 1. The paper demonstrates that relaxing the condition that the residual from the cointegration equation must be a I (0) process, captures a much wider class of mean-reversion behaviour. Furthermore, an analysis of the short-run dynamics propelling the long-run relationship tends to imply that although the official rate influences the changes in the black rate, this is purely over the short-run. In the longer term, the black rate is found to be the initial receptor of any exogenous shock to the equilibrium and it is the official exchange rate that bears the brunt of short-run adjustment to re-establish the long-run equilibrium relationship. The approach illustrated in this paper is shown to hold enormous potential for tests of mean reversion involving hypotheses popular to economic and financial research in general, where the dynamics of time-series data are under constant scrutiny.