We develop semi-/non-parametric methods the yield tractable limit distribution theory in parametric and non-parametric regressions, where the covariate is either a nonstationary fractional process and/or a nearly integrated array. The fractional parameter under consideration is either d=1-1/alpha (weakly nonstationary) or 1-1/alpha < d < 2-1/alpha, with alpha in (0,2]. The alpha parameter corresponds to the stability index of innovations that drive the covariates and are in the domain of attraction of an alpha-stable law.