### Abstract

We compare the asymptotic relative efficiency of the Exp, Mean, and Sup functionals of the Wald, LM and LR tests for structural change analyzed by Andrews [Andrews, D.W.K., 1993. Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821-856.] and Andrews and Ploberger [Andrews, D.W.K., Ploberger, W., 1994. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383-1414.]. We derive the approximate Bahadur slopes of these tests using large deviations techniques. These show that tests based on the Mean functional are inferior to those based on the Sup and Exp when using the same base statistic. Also, for a given functional, the Wald-based test dominates the LR-based test, which dominates the LM-based one. We show that the Sup- and Mean-type tests satisfy Wieand's [Wieand, H.S., 1976. A Condition under which the Pitman and Bahadur approaches to efficiency coincide. Annals of Statistics 4, 1003-1011.] condition so that their slopes yield the limiting (as the size tends to zero) asymptotic relative Pitman efficiency (whether this holds for the Exp-type tests still remains a conjecture). Using this measure of efficiency, the Mean-type tests are also inferior to the Sup. We also compare tests based on the Wald and LM statistics modified with a HAC estimator. In this case, the inferiority of the LM-based tests is especially pronounced. The relevance of our theoretical results in finite samples is assessed via simulations. Our results are in contrast to those obtained by the authors-in the second reference cited above-based on a local asymptotic framework, and our analysis thereby reveals its potential weaknesses in the context of structural change problems.

Original language | English |
---|---|

Pages (from-to) | 26-51 |

Number of pages | 26 |

Journal | Journal of Econometrics |

Volume | 149 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Apr 1 |

Externally published | Yes |

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### Keywords

- Change point
- Hypothesis testing
- Supremum statistics
- Unknown break date
- Wald tests

### ASJC Scopus subject areas

- History and Philosophy of Science
- Economics and Econometrics
- Applied Mathematics

### Cite this

**Assessing the relative power of structural break tests using a framework based on the approximate Bahadur slope.** / Kim, Dukpa; Perron, Pierre.

Research output: Contribution to journal › Article

*Journal of Econometrics*, vol. 149, no. 1, pp. 26-51. https://doi.org/10.1016/j.jeconom.2008.10.010

}

TY - JOUR

T1 - Assessing the relative power of structural break tests using a framework based on the approximate Bahadur slope

AU - Kim, Dukpa

AU - Perron, Pierre

PY - 2009/4/1

Y1 - 2009/4/1

N2 - We compare the asymptotic relative efficiency of the Exp, Mean, and Sup functionals of the Wald, LM and LR tests for structural change analyzed by Andrews [Andrews, D.W.K., 1993. Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821-856.] and Andrews and Ploberger [Andrews, D.W.K., Ploberger, W., 1994. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383-1414.]. We derive the approximate Bahadur slopes of these tests using large deviations techniques. These show that tests based on the Mean functional are inferior to those based on the Sup and Exp when using the same base statistic. Also, for a given functional, the Wald-based test dominates the LR-based test, which dominates the LM-based one. We show that the Sup- and Mean-type tests satisfy Wieand's [Wieand, H.S., 1976. A Condition under which the Pitman and Bahadur approaches to efficiency coincide. Annals of Statistics 4, 1003-1011.] condition so that their slopes yield the limiting (as the size tends to zero) asymptotic relative Pitman efficiency (whether this holds for the Exp-type tests still remains a conjecture). Using this measure of efficiency, the Mean-type tests are also inferior to the Sup. We also compare tests based on the Wald and LM statistics modified with a HAC estimator. In this case, the inferiority of the LM-based tests is especially pronounced. The relevance of our theoretical results in finite samples is assessed via simulations. Our results are in contrast to those obtained by the authors-in the second reference cited above-based on a local asymptotic framework, and our analysis thereby reveals its potential weaknesses in the context of structural change problems.

AB - We compare the asymptotic relative efficiency of the Exp, Mean, and Sup functionals of the Wald, LM and LR tests for structural change analyzed by Andrews [Andrews, D.W.K., 1993. Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821-856.] and Andrews and Ploberger [Andrews, D.W.K., Ploberger, W., 1994. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383-1414.]. We derive the approximate Bahadur slopes of these tests using large deviations techniques. These show that tests based on the Mean functional are inferior to those based on the Sup and Exp when using the same base statistic. Also, for a given functional, the Wald-based test dominates the LR-based test, which dominates the LM-based one. We show that the Sup- and Mean-type tests satisfy Wieand's [Wieand, H.S., 1976. A Condition under which the Pitman and Bahadur approaches to efficiency coincide. Annals of Statistics 4, 1003-1011.] condition so that their slopes yield the limiting (as the size tends to zero) asymptotic relative Pitman efficiency (whether this holds for the Exp-type tests still remains a conjecture). Using this measure of efficiency, the Mean-type tests are also inferior to the Sup. We also compare tests based on the Wald and LM statistics modified with a HAC estimator. In this case, the inferiority of the LM-based tests is especially pronounced. The relevance of our theoretical results in finite samples is assessed via simulations. Our results are in contrast to those obtained by the authors-in the second reference cited above-based on a local asymptotic framework, and our analysis thereby reveals its potential weaknesses in the context of structural change problems.

KW - Change point

KW - Hypothesis testing

KW - Supremum statistics

KW - Unknown break date

KW - Wald tests

UR - http://www.scopus.com/inward/record.url?scp=63149153838&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=63149153838&partnerID=8YFLogxK

U2 - 10.1016/j.jeconom.2008.10.010

DO - 10.1016/j.jeconom.2008.10.010

M3 - Article

VL - 149

SP - 26

EP - 51

JO - Journal of Econometrics

JF - Journal of Econometrics

SN - 0304-4076

IS - 1

ER -