Least Squares Bias in Time Series with Moderate Deviations from a Unit Root

Abstract

This articles derives the approximate bias of the least squares estimator of the autoregressive coefficient in discrete autoregressive time series where the autoregressive coefficient is given by αT=1+c/kT, with kT being a deterministic sequence increasing to infinity at a rate slower than T, such that kT=o(T) as T→∞. The cases in which c<0, c=0 and c>0 are considered correspond to (moderately) stationary, non-stationary and (moderately) explosive series respectively. The result is used to derive the limiting distribution of the indirect inference method for such processes with moderate deviations from a unit root and for explosive series with a fixed coefficient, which does not depend on the sample size. Second, the result demonstrates why the jackknife estimator cannot be constructed for explosive time series for values of the autoregressive parameter close to unity in view of the discontinuity of the bias function, which the article derives. Finally, the expression is used to construct a bias-corrected estimator, and simulations are carried out to assess the three estimators' bias reduction capabilities.