Asymptotics of tests for a unit root in autoregression

Testing for stationarity is an important issue in time series analysis. One approach for this is the unit root test in autoregression. For autoregressive models, a lot of statistics based on the least-squares estimator (LSE) of the coefficient have been used for the testing problem of unit root. In this paper, we develop an approach for this problem based on a generalized LSE (GLSE), which includes many important estimators as special cases. Then the asymptotics of some test statistics constructed by the GLSE is elucidated. Concretely, we derive their limiting distribution under both null and alternative hypotheses. Based on this result we evaluate their local power, and discuss their asymptotic optimality. Numerical studies for them are given.