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We provide new limit theory for functionals of a general class of processeslying at the boundary between stationarity and nonstationarity -- what we termweakly nonstationary processes (WNPs). This includes, as leading examples,fractional processes with d=1/2, and arrays of autoregressive processes withroots drifting slowly towards unity. We first apply the theory to studyinference in parametric and nonparametric regression models involving WNPs ascovariates. We then use these results to develop a new specification test forparametric regression models. By construction, our specification test statistichas a chi-squared limiting distribution regardless of the form and extent ofpersistence of the regressor, implying that a practitioner can validly performthe test using a fixed critical value, while remaining agnostic about themechanism generating the regressor. Simulation exercises confirm that the testcontrols size across a wide range of data generating processes, and outperformsa comparable test due to Wang and Phillips (2012, Ann. Stat.) against manyalternatives.

Estimation and Inference in the Presence of Fractional d=1/2 and Weakly Nonstationary Processes