Weak Unit Roots

This paper develops the large sample theory for econometric models with time series having roots in proximity of unity. In particular, a special attention is given to the time series with roots outside the n-1-neighborhood of unity, called the weak unit roots. They are the processes with roots approaching to unity as sample size increases, but not too fastly. It is shown that the weak unit root processes yield the standard law of large numbers and central limit theorem-like results, and as a consequence, the usual large sample theory of inference based on normal asymptotics is applicable for models with weak unit root processes. This suggests that we may rely on the conventional statistical theory also for models with roots close to unity, as long as the roots are not too close to unity. In practice, it seems that we may safely use the standard normal theory, unless the roots are very close to one in a metric proportional to the magnitude of sample size. We consider a wide class of models including autoregressions and nonlinear, as well as linear, cointegrated models with weak unit roots.