We provide new limit theory for functionals of a general class of
processes lying at the boundary between stationarity and nonstationarity – what we term weakly nonstationary processes (WNPs). This
includes, as leading examples, fractional processes with d = 1/2, and
arrays of autoregressive processes with roots drifting slowly towards
unity. We first apply the theory to study inference in parametric and
nonparametric regression models involving WNPs as covariates. We
then use these results to develop a new specification test for parametric regression models. By construction, our specification test statistic
has a χ
2
limiting distribution regardless of the form and extent of
persistence of the regressor, implying that a practitioner can validly
perform the test using a fixed critical value, while remaining agnostic
about the mechanism generating the regressor. Simulation exercises
confirm that the test controls size across a wide range of data generating processes, and outperforms a comparable test due to Wang
and Phillips (2012, Ann. Stat.) against many alternatives.
