Long memory and fractional cointegration with deterministic trends.

We discuss the estimation of the order of integration of a fractional process that may be contaminated by a time-varying deterministic component, or subject to a break in the dynamics of the zero-mean stochastic component, and the estimation of the cointegrating parameter in a bivariate system generated by fractionally integrated processes and by additive polynomial trends. In Chapter 1 we review the theoretical literature on fractional integration and cointegration, and we analyse a situation in which a fractional model reconciles two apparently conflicting economic theories. In Chapter 2 we consider local Whittle estimation of the order of integration when the process is contaminated by a deterministic trend or by a break in the mean. We propose a simple condition to assess whether the asymptotic properties of the estimate are unaffected by the time-varying mean, and a test, with asymptotically normal test statistic under the null, to detect if that condition is met. In Chapter 3 we discuss local Whittle estimation when the zero-mean stochastic component is subject to a break: we show that the estimate is robust to instability in the short term dynamics, while in presence of a break in the long term dynamics only the highest order of integration is consistently estimated. We propose a test to detect that break: the limit distribution of the test statistic under the null is not standard, but it is well known in the literature. We also propose a procedure to estimate the location of a break when it is present. In Chapter 4 we consider a cointegrating relation in which a nonstationary, bivariate process is augmented by a deterministic trend. We derive the limit properties of the Ordinary Least Squares and Generalised Least Squares estimates: these depend on the comparison between the deterministic and the stochastic components.