Semiparametric Frequency Domain Analysis of Fractional Cointegration - (Revised version forthcoming in P M Robinson: Time Series with Long Memory (Oxford University Press).

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), but is also of interest in more general, including fractional, circumstances, where zt is stationary with long memory and et is stationay with less memory, or where zt is nonstationary while et is either less nonstationay or stationary, possibly with long memory. Inference rules based on estimates of the cointegrating vector that have been developed in the I(1) / I(0) case appear to lose validity in the above circumstances, while the estimates themselves, including ordinary least squares, are typically inconsistent when zt is stationary. Partitioning zt into a scalar yt and vector xt , we consider a narrow-band frequency domain least squares estimate of yt on xt . This estimate is consistent under stationary zt , whereas least squares is inconsistent due to correlation between xt and et . This correlation does not prevent consistency of least squares when zt is nonstationary, but it produces a larger second order bias relative to the frequency domain estimate in the I(1) / I(0) case, and a slower rate of convergence in many circumstances in which zt exhibits less-than-I(1) nonstationarity. When et is itself nonstationary, the two estimates have a common limit distribution. Our conclusions in the I(1) / I(0) case are supported by Monte Carlo simulations. A semiparametric methodology for fractional cointegration analysis is applied to data analysed by Engle and Granger (1987) and Campbeell and Shiller (1987).