Summary

Fractionally integrated and fractionally cointegrated time series are classes of models that generalize standard notions of integrated and cointegrated time series. The fractional models are characterized by a small number of memory parameters that control the degree of fractional integration and/or cointegration. In classical work, the memory parameters are assumed known and equal to 0, 1, or 2. In the fractional integration and fractional cointegration context, however, these parameters are real-valued and are typically assumed unknown and estimated. Thus, fractionally integrated and fractionally cointegrated time series can display very general types of stationary and nonstationary behavior, including long memory, and this more general framework entails important additional challenges compared to the traditional setting. Modeling, estimation, and testing in the context of fractional integration and fractional cointegration have been developed in time and frequency domains. Related to both alternative approaches, theory has been derived under parametric or semiparametric assumptions, and as expected, the obtained results illustrate the well-known trade-off between efficiency and robustness against misspecification. These different developments form a large and mature literature with applications in a wide variety of disciplines.