Regressions with Fractional d=1/2 and Weakly Nonstationary Processes
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Despite major theoretical advances, important classes of fractional models (e.g. ARFIMA models) have not yet been fully characterised in terms of asymptotic theory. In particular, no limit theory is available for general additive functionals of fractionally integrated processes of order d=1/2. Such processes cannot be handled by existing asymptotic results for either stationary or nonstationary processes, i.e. by existing LLNs or FCLTs. Their asymptotics must therefore be derived by novel arguments, such as are developed in this paper. In the course of doing so, we develop new limit theory for a broader class of linear processes lying on the boundary between stationarity and nonstationarity -- what we term weakly nonstationary processes. This includes, as leading examples, both fractional processes with d=1/2, and arrays of autoregressive processes with roots drifting slowly towards unity. We apply our new results to the asymptotics of both parametric and kernel regression estimators.
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