On Trend, Breaks and Initial Condition in Unit Root Testing

Recent approaches in unit root testing that take into account the influences of the initial condition, trend, and breaks in the data using pre-testing and performing the union of rejection testing strategies based on the information obtained. This allows for the use of more powerful tests, if there is uncertainty about some of the parameters in the model. This paper proposes the extension of the Harvey et al. (2012b) approach to the case of uncertainty over the initial condition. It has been shown that the procedures of Harvey et al. (2012b) have low power under a large initial condition because they include GLS-based tests. Therefore, the efficiency of some ADF-type unit root tests with breaks under various magnitudes of initial condition will be investigated, and the decision rule based on pre-testing for a magnitude of the initial condition and simultaneous use of tests based on both GLS and OLS detrending is proposed.Additionally, the modification of the proposed algorithm using pre-testing for the trend co-efficient will be analyzed. Analysis of a situation with the possible presence of multiple structural breaks in trend will also be conducted in the paper. Two algorithms are proposed: the first involves only pre-testing the initial condition, while the second involves pre-testing the number of breaks based on the Kejriwal and Perron (2010) test. The asymptotic behavior of all tests is analyzed under both a local-to-unity representation of the autoregressive root and a local-to-zero representation of trend and breaks magnitudes. The proposed modification save the high power for small initial conditions/trend/breaks and at the same time lead to the power close to one of the effective tests for large initial condition/trend/breaks