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Threshold regression asymptotics: From the compound Poisson process to two-sided Brownian motion

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  • Yu, Ping
  • Phillips, Peter C.B.

Abstract

The asymptotic distribution of the least squares estimator in threshold regression is expressed in terms of a compound Poisson process when the threshold effect is fixed and as a functional of two-sided Brownian motion when the threshold effect shrinks to zero. This paper explains the relationship between this dual limit theory by showing how the asymptotic forms are linked in terms of joint and sequential limits. In one case, joint asymptotics apply when both the sample size diverges and the threshold effect shrinks to zero, whereas sequential asymptotics operate in the other case in which the sample size diverges first and the threshold effect shrinks subsequently. The two operations lead to the same limit distribution, thereby linking the two different cases. The proofs make use of ideas involving limit theory for sums of a random number of summands.

Suggested Citation

  • Yu, Ping & Phillips, Peter C.B., 2018. "Threshold regression asymptotics: From the compound Poisson process to two-sided Brownian motion," Economics Letters, Elsevier, vol. 172(C), pages 123-126.
    • Handle: RePEc:eee:ecolet:v:172:y:2018:i:c:p:123-126
    DOI: 10.1016/j.econlet.2018.08.039
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    References listed on IDEAS

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    1. Bruce E. Hansen, 2000. "Sample Splitting and Threshold Estimation," Econometrica, Econometric Society, vol. 68(3), pages 575-604, May.
    2. Jushan Bai, 1997. "Estimation Of A Change Point In Multiple Regression Models," The Review of Economics and Statistics, MIT Press, vol. 79(4), pages 551-563, November.
    3. Yu, Ping, 2012. "Likelihood estimation and inference in threshold regression," Journal of Econometrics, Elsevier, vol. 167(1), pages 274-294.
    4. Yu, Ping, 2014. "The Bootstrap In Threshold Regression," Econometric Theory, Cambridge University Press, vol. 30(3), pages 676-714, June.
    5. Peter C. B. Phillips & Hyungsik R. Moon, 1999. "Linear Regression Limit Theory for Nonstationary Panel Data," Econometrica, Econometric Society, vol. 67(5), pages 1057-1112, September.
    6. Yu, Ping, 2015. "Adaptive estimation of the threshold point in threshold regression," Journal of Econometrics, Elsevier, vol. 189(1), pages 83-100.
    7. James Heckman, 1997. "Instrumental Variables: A Study of Implicit Behavioral Assumptions Used in Making Program Evaluations," Journal of Human Resources, University of Wisconsin Press, vol. 32(3), pages 441-462.
    8. Anna Bykhovskaya & Peter C. B. Phillips, 2018. "Boundary Limit Theory for Functional Local to Unity Regression," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(4), pages 523-562, July.
    9. Ping Yu & Yongqiang Zhao, 2013. "Asymptotics for threshold regression under general conditions," Econometrics Journal, Royal Economic Society, vol. 16(3), pages 430-462, October.
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    Cited by:

    1. Chaoyi Chen & Thanasis Stengos, 2022. "Estimation and Inference for the Threshold Model with Hybrid Stochastic Local Unit Root Regressors," JRFM, MDPI, vol. 15(6), pages 1-15, May.
    2. Zhang, Xinyu & Li, Dong & Tong, Howell, 2023. "On the least squares estimation of multiple-threshold-variable autoregressive models," LSE Research Online Documents on Economics 118377, London School of Economics and Political Science, LSE Library.

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    More about this item

    Keywords

    Threshold regression; Sequential asymptotics; Doob’s martingale inequality; Compound Poisson process; Brownian motion;
    All these keywords.

    JEL classification:

    • C24 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Truncated and Censored Models; Switching Regression Models; Threshold Regression Models

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