Moment Approximation for Unit Root Models with Nonnormal Errors

Phillips (1977a, 1977b) made seminal contributions to time series finite-sample theory, and then, he was among the first to develop the distributions of estimators and forecasts in stationary time series models, see Phillips (1978, 1979), among others. From the mid-eighties Phillips (1987a, 1987b), through his fundamental papers, opened the path of asymptotic (large-sample) theory for the unit root type non-stationary models. This has certainly created a large literature of important papers, including many of Phillips’’own papers. However, not much is known about the analytical finite-sample properties of estimators under the unit root, although see Kiviet and Phillips (2005) for the case when the errors are normally distributed. An objective of this paper is to analyze the …finite-sample behavior of the estimator in the first-order autoregressive model with unit root and nonnormal errors. In particular, we derive analytical approximations for the first two moments in terms of model parameters and the distribution parameters. Through Monte Carlo simulations, we find that our approximate formula perform quite well across different distribution specifications in small samples. However, when the noise to signal ratio is huge, and bias distortion can be quite substantial, and our approximations do not fare well.