Abstract
This paper develops a uniform test of linearity against threshold effects in the quantile regression framework. The test is based on the supremum of the Wald process over the space of quantile and threshold parameters. We establish the limiting null distribution of the test statistic for stationary weakly dependent processes, and propose a simulation method to approximate the critical values. The proposed simulation method makes the test easy to implement. Monte Carlo experiments show that the proposed test has good size and reasonable power against non-linear threshold models.
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Notes
An important exception is Su and White (2012). This reference was noted by a referee.
In particular, Theorem 2 of Kato (2009) guarantees that there is no need to establish the uniform \(n^{-1/2}\) rate for \(\hat{\varvec{\beta }}(\tau ,\gamma )\) in the course of establishing the weak convergence of the process.
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Acknowledgments
The authors would like to express their appreciation to Roger Koenker, Zhongjun Qu, Zhijie Xiao, and to the participants at the 2009 North American Summer Meeting of the Econometrics Society and the 2009 Far East and South Asia Meeting of the Econometrics Society for helpful comments and discussions. Kato’s research was partially supported by Grant-in-Aid for Young Scientists (B) (22730179) from the JSPS. Jose Olmo acknowledges financial support from Ministerio de Economia y Competitividad ECO2011-22650 project.
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Galvao, A.F., Kato, K., Montes-Rojas, G. et al. Testing linearity against threshold effects: uniform inference in quantile regression. Ann Inst Stat Math 66, 413–439 (2014). https://doi.org/10.1007/s10463-013-0418-9
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DOI: https://doi.org/10.1007/s10463-013-0418-9