Convergence to Stochastic Integrals with Fractionally Integrated Integrator Processes: Theory, and Applications to Cointegrating Regression

The weak limit is derived of the sample covariance of a pair of fractionally integrated processes, one I(dY ) for −2 < dY < 12 , and the other I(1 + dX) for −2 < dX < 12 . The stochastic component (mean deviation) of the limit has the representation of a RiemannLiouville fractional integral of a functional of regular Brownian motion, B, having the form φ(t) = R t 0 XdB. In the case dX < 0, the mean deviation is of small order relative to the mean, if the latter is non-zero. The results have applications in the analysis of cointegrating regressions, in which the cointegrating residual has long memory. A fractional variant of the fully modified least squares estimator (FFMLS) is derived, which both converges more rapidly than least squares in certain cases, and is asymptotically mixed Gaussian, permitting standard inference procedures.