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New asymptotic approximations are established for the Wald and t statistics in the presence of unknown but strong autocorrelation. The asymptotic theory extends the usual fixed-smoothing asymptotics under weak dependence to allow for near-unit-root and weak-unit-root processes. As the locality parameter that characterizes the neighborhood of the autoregressive root increases from zero to infinity, the new fixed-smoothing asymptotic distribution changes smoothly from the unit-root fixed-smoothing asymptotics to the usual fixed-smoothing asymptotics under weak dependence. Simulations show that the new approximation is more accurate than the usual fixed-smoothing approximation.

This chapter derives asymptotic properties of the least squares (LS) estimator of the autoregressive (AR) parameter in local to unity processes with errors being fractional Gaussian noise (FGN) with the Hurst parameter H∈(0,1). It is shown that the estimator is consistent for all values of H∈(0,1). Moreover, the rate of convergence is n−1 when H∈[0.5,1). The rate of convergence is n−2H when H∈(0,0.5). Furthermore, the limiting distribution of the centered LS estimator depends on H. When H=0.5, the limiting distribution is the same as that obtained in Phillips (1987a) for the local to unity model with errors for which the standard functional central limit theorem is applicable. When H > 0.5 or when H < 0.5, the limiting distributions are new to the literature. The asymptotic properties of the LS estimator with fitted intercept are also derived. Simulation studies are performed to check the reliability of the asymptotic approximation for different values of sample size.

In this study, the authors investigate methods of sequential analysis to test prospectively for the existence of a unit root against stationary or explosive states in a p-th order autoregressive (AR) process monitored over time. Our sequential sampling schemes use stopping times based on the observed Fisher information of a local-to-unity parameter. In contrast to the Dickey–Fuller (DF) test statistic, the sequential test statistic has asymptotic normality. The authors derive the joint limit of the test statistic and the stopping time, which can be characterized using a 3/2-dimensional Bessel process driven by a time-changed Brownian motion. The authors obtain their limiting joint Laplace transform and density function under the null and local alternatives. In addition, simulations are conducted to show that the theoretical results are valid.

This article investigates the robustness of impulse response estimators to near unit roots and near cointegration in vector autoregressive (VAR) models. We compare estimators based on VAR specifications determined by pretests for unit roots and cointegration as well as unrestricted VAR specifications in levels. Our main finding is that the impulse response estimators obtained from the levels specification tend to be most robust when the magnitude of the roots is not known. The pretest specification works well only when the restrictions imposed by the model are satisfied. Its performance deteriorates even for small deviations from the exact unit root for one or more model variables. We illustrate the practical relevance of our results through simulation examples and an empirical application.

This paper proposes a new class of estimators for the autoregressive coefficient of a dynamic panel data model with random individual effects and nonstationary initial condition. The new estimators we introduce are weighted averages of the well-known first difference (FD) GMM/IV estimator and the pooled ordinary least squares (POLS) estimator. The proposed procedure seeks to exploit the differing strengths of the FD GMM/IV estimator relative to the pooled OLS estimator. In particular, the latter is inconsistent in the stationary case but is consistent and asymptotically normal with a faster rate of convergence than the former when the underlying panel autoregressive process has a unit root. By averaging the two estimators in an appropriate way, we are able to construct a class of estimators which are consistent and asymptotically standard normal, when suitably standardized, in both the stationary and the unit root case. The results of our simulation study also show that our proposed estimator has favorable finite sample properties when compared to a number of existing estimators.