A unit root test against globally stationary ESTAR models when local condition is non-stationary

•ESTAR model with the local-explosive characteristics is considered.•A modified Wald-type test is proposed to tackle a nonstandard testing problem.•The asymptotic distribution of our test statistic is derived.•Simulation results show that our test performs well.

This paper focuses on testing for the unit root hypothesis against local-explosive or local unit root but globally stationary ESTAR process. A modified Wald-type test for a joint hypothesis where one parameter is one-sided while the others are two-sided under the alternative is proposed. The asymptotic distribution of the test statistic is derived, which is shown to be a function of Brownian motions and does not depend on nuisance parameters. Critical values of the test are tabulated and some simulation results are reported. Results show that the modified Wald-type test performs well.