The concept of cointegration has principally been developed under the assumption that the raw data vector Zt is I(1) and the cointegrating residual et is I(0); we call this framework the CI(l) case. The purpose of this thesis is to consider more general fractional circumstances, where Zt is stationary with long memory and et is stationary with less memory, or where Zt is nonstationary while et is less nonstationary or stationary, possibly with long memory. First we establish weak convergence to what we term "type II fractional Brownian motion" for a wide class of nonstationary fractionally integrated processes, then we go on to investigate the behaviour of the discretely averaged periodogram for processes that are not second order stationary. These results are exploited for the analysis of a procedure originally proposed by Robinson (1994a), which we call Frequency Domain Least Squares (FDLS). FDLS yield estimates of the cointe-grating vector that are consistent for stationary and nonstationary Zt, asymptotically equivalent to OLS in some circumstances, and superior in many others, including the standard CI(1) case; a semiparametric methodology for fractional cointegration analysis is applied to data sets on eleven US macroeconomic variables. Finally, we investigate an alternative definition of fractional cointegration, for which we introduce a continuously averaged version of FDLS, obtaining consistent estimates in both the stationary and the nonstationary case. Asymptotic distributions and Monte Carlo evidence on finite sample performance are also provided.
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