This is the seminal work by Andrews (1993) about unknown structural break tests. You might incorporate this test in unite root test. For example, the persistence parameter in an AR process might have changed over time which can induce deceptively high value of autoregressive root.
Bai and Perron (1998;2003) and some others are simply extensions of this single break tests to multiple ones. However, one has to specify maximum break numbers. Another disadvantage of Bai-Perron algorithm is that the tests do not provide p-values. Andrews (1993) and Andrews and Ploberger (1994) are the most elegant ones. In particular, the exponential Wald statistic and Average Wald statistic are optimal.
I have some experience of this, and Gauss code is available when requested from 四海为家。
Kindly regards.
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