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Journal of Econometric Methods

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Approximate p-Values of Certain Tests Involving Hypotheses About Multiple Breaks

Alastair Hall
  • Corresponding author
  • Economics, School of Social Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, UK
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/ Nikolaos Sakkas
Published Online: 2013-03-15 | DOI: https://doi.org/10.1515/jem-2012-0014

Abstract

We provide formulae for calculating approximate p-values for the non-standard asymptotic null distributions of a variety of tests used for detecting multiple structural change in a wide range of models. Our approximations are based on simulated quantiles obtained from 100,000 replications, and the latter are more accurate than the quantiles reported in the literature by increasing the number of replications by a factor of 10. The p-value response surfaces are approximated using a parametric method proposed by Hansen and their use is illustrated with an example. Using our p-value response surfaces, it is shown that the use of Bai and Perron’s response surfaces for the critical values of these tests can lead to misleading inferences, and thus should be used with extreme caution.

This article offers supplementary material which is provided at the end of the article.

Keywords: least squares; regression models; tests of parameter variation

References

  • Andrews, D. W. K. 1993. “Tests for Parameter Instability and Structural Change with Unknown Change Point.” Econometrica 61: 821–856.CrossrefGoogle Scholar

  • Andrews, D. W. K. 2003. “Tests for Parameter Instability and Structural Change with Unknown Change Point: A Corrigendum,” Econometrica 71: 395–398.CrossrefGoogle Scholar

  • Bai, J., H. Chen, T. Chong, and S. Wang. 2008. “Generic Consistency of the Breakpoint Estimators Under Specification Errors in a Multiple-Break Model.” Econometrics Journal 11: 287–307.CrossrefWeb of ScienceGoogle Scholar

  • Bai, J. and P. Perron. 1998. “Estimating and Testing Linear Models with Multiple Structural Changes.” Econometrica 66: 47–78.CrossrefWeb of ScienceGoogle Scholar

  • Bai, J. and P. Perron. 2003a. “Computation and Analysis of Multiple Structural Change Models.” Journal of Applied Econometrics 18: 1–22.CrossrefGoogle Scholar

  • Bai, J. and P. Perron. 2003b. “Critical Values for Multiple Structural Change Tests.” Econometrics Journal 6: 72–78.CrossrefGoogle Scholar

  • Boldea, O. and A. R. Hall. 2013. “Estimation and Inference in Unstable Nonlinear Least Squares Models.” Journal of Econometrics 172: 158–167.CrossrefWeb of ScienceGoogle Scholar

  • Ghysels, E., A. Guay, and A. R. Hall. 1997. “Predictive Test for Structural Change with Unknown Breakpoint.” Journal of Econometrics 82: 209–233.Google Scholar

  • Hall, A. R., S. Han, and O. Boldea. 2012. “Inference Regarding Multiple Structural Changes in Linear Models with Endogenous Regressors.” Journal of Econometrics 170: 281–302.PubMedCrossrefWeb of ScienceGoogle Scholar

  • Hansen, B. E. 1997. “Approximate Asymptotic p Values for Structural-Change Tests.” Journal of Business and Economic Statistics 15: 60–67.Google Scholar

  • Sen, A. 1999. “Approximate p-Values of Predictive Tests for Structural Stability.” Economics Letters 63: 245–253.CrossrefGoogle Scholar

  • Sen, A. and A. R. Hall. 1999. “Two Further Aspects of Some New Tests for Structural Stability.” Structural Change and Economic Dynamics 10: 431–443.Google Scholar

About the article

Corresponding author: Alastair Hall, Economics, School of Social Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, UK


Published Online: 2013-03-15

Published in Print: 2013-07-01


The error must also satisfy certain other conditions detailed in the sources mentioned below.

If p=0 then all parameters are regime specific and the model is said to exhibit pure structural change.

It should be noted that these limiting distributions only apply for the statistics based on FT(λ1,…,λk;q) if the regression error ut is homoscedastic and serially uncorrelated. If the errors are heteroscedastic and/or serially correlated then the same limiting distributions hold for the analogous functions of Wald statistics that satisfactorily account for the error structure; see Bai and Perron (1998) for further details.

Such large values of ε are reported because Andrews (1993) allows for asymmetric trimming.

This methodology has also been employed by Sen and Hall (1999) for the distributions of their overidentifying restrictions test (OT) and by Sen (1999) for the Ghysels, Guay, and Hall’s (1997) predictive test.

From the case of q=1, it can be seen that this case occurs for k=9. Further investigation reveals the main source of the distortion is Bai and Perron’s (2003b) response surface: the appropriate critical value reported in Bai and Perron (1998) is 5.20 which translates, using our response surfaces, to a p-value of 0.0758; the critical value predicted by Bai and Perron’s (2003b) response surface is 3.8023. In contrast, our simulations (based on an enhanced design) yield a critical value of 5.4210 for which our response surface gives a p-value of 0.05006.

In the original version of this paper, we restricted attention to q=1,5,10 but, given the nature of the results for those cases, a referee conjectured Bai and Perron’s (2003b) response surfaces would work well for cases q>10. While natural given results for q=1, 5, 10, this conjecture is erroneous as our results demonstrate. We therefore included these cases to avoid the reader making the same mistake. We attribute the inaccuracy of Bai and Perron’s (2003b) response surfaces to these cases to the fact that the surfaces were calculated using cases for which q≤10 only.

For completeness, we note that we performed similar calculations for ε=0.10,0.15,0.20,0.25 for q=1,5,10. The distortions are smaller than those reported for ε=0.05 but are nevertheless non-trivial in a number of circumstances. These results are available from the authors upon request.


Citation Information: Journal of Econometric Methods, Volume 2, Issue 1, Pages 53–67, ISSN (Online) 2156-6674, ISSN (Print) 2194-6345, DOI: https://doi.org/10.1515/jem-2012-0014.

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[1]
Alastair R. Hall, Denise R. Osborn, and Nikolaos Sakkas
The Manchester School, 2013, Volume 81, Page 54

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