Fractional Cointegration

Abstract

Economic and financial time series data frequently exhibit autocorrelation or trending behavior. Frequently, two or more such time series appear to be "cointegrated" or to "move together", in such a way that there is a linear combination of the series (the "cointegrating error") that has weaker autocorrelation. Much of the cointergration literature has assumed that the raw time series have a "unit root", their first differences behaving like a short memory, stationary time series, while the cointegrating errors also behave like a short memory, stationary time series. These are restrictive notions of time series behavior. It is possible that cointegration can exist between time series that are stationary, or exhibit different forms of nonstationarity, and that cointegrating errors can have long memory or even be non-stationary. In such circumstances standard methods developed under the assumption of unit root raw series and short memory cointegrating errors can lose validity, and the underlying cointegrating structure not be detected. We consider the possibility of cointegration between more general, fractional, series, which allow for the possibility of raw series that are stationary, or more or less nonstationary, than unit root ones, and cointegrating errors that can have long memory. Precise methods of estimating such cointegrating relations are presented, allowing degrees of nonstationarity or memory to be unknown.