This paper develops the asymptotic theory for parametric and nonparametric regression models when the errors have a fractional local to unity root (FLUR) model structure. FLUR models are stationary time series with semi-long range dependence property in the sense that their covariance function resembles that of a long memory model for moderate lags but eventually diminishes exponentially fast according to the presence of a decay factor governed by a noncentrality parameter. When this parameter is sample size dependent, the asymptotic normality for these regression models admit a wide range of stochastic processes with behavior that includes long, semi-long, and short memory processes.