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Asymptotic Behavior of Delay Times of Bubble Monitoring Tests

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  • Eiji Kurozumi

Abstract

We investigate the asymptotic properties of the stopping times based on the monitoring tests for a bubble, that is, ADF‐type and CUSUM‐type tests. We derive the stochastic orders of these stopping times and find that they depend on the relative position of the emerging date of a bubble. We show that when a bubble emerges early in the monitoring period, the diverging rate of the stopping time based on the CUSUM‐type detector is slower than that based on the ADF‐type detector, which implies that the former tends to find a bubble earlier than the latter, whereas this advantage disappears for the late emerging bubble. We also derive the limiting distribution of the stopping time based on the CUSUM detector. The finite sample simulations support the theoretical result.

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  • Eiji Kurozumi, 2021. "Asymptotic Behavior of Delay Times of Bubble Monitoring Tests," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(3), pages 314-337, May.
    • Handle: RePEc:bla:jtsera:v:42:y:2021:i:3:p:314-337
    DOI: 10.1111/jtsa.12569
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    References listed on IDEAS

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    1. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2015. "Testing For Multiple Bubbles: Historical Episodes Of Exuberance And Collapse In The S&P 500," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56(4), pages 1043-1078, November.
    2. Aue, Alexander & Horváth, Lajos & Reimherr, Matthew L., 2009. "Delay times of sequential procedures for multiple time series regression models," Journal of Econometrics, Elsevier, vol. 149(2), pages 174-190, April.
    3. David I. Harvey & Stephen J. Leybourne & Robert Sollis, 2015. "Recursive Right-Tailed Unit Root Tests for an Explosive Asset Price Bubble," Journal of Financial Econometrics, Oxford University Press, vol. 13(1), pages 166-187.
    4. Alexander Aue & Lajos Horváth & Marie Hušková & Piotr Kokoszka, 2006. "Change-point monitoring in linear models," Econometrics Journal, Royal Economic Society, vol. 9(3), pages 373-403, November.
    5. Harvey, David I. & Leybourne, Stephen J. & Sollis, Robert & Taylor, A.M. Robert, 2016. "Tests for explosive financial bubbles in the presence of non-stationary volatility," Journal of Empirical Finance, Elsevier, vol. 38(PB), pages 548-574.
    6. Sam Astill & David I. Harvey & Stephen J. Leybourne & Robert Sollis & A. M. Robert Taylor, 2018. "Real‐Time Monitoring for Explosive Financial Bubbles," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(6), pages 863-891, November.
    7. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2015. "Testing For Multiple Bubbles: Limit Theory Of Real‐Time Detectors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56(4), pages 1079-1134, November.
    8. Diba, Behzad T & Grossman, Herschel I, 1988. "Explosive Rational Bubbles in Stock Prices?," American Economic Review, American Economic Association, vol. 78(3), pages 520-530, June.
    9. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2015. "Testing For Multiple Bubbles: Limit Theory Of Real‐Time Detectors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56, pages 1079-1134, November.
    10. Phillips, Peter C.B. & Shi, Shu-Ping, 2018. "Financial Bubble Implosion And Reverse Regression," Econometric Theory, Cambridge University Press, vol. 34(4), pages 705-753, August.
    11. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2015. "Testing For Multiple Bubbles: Historical Episodes Of Exuberance And Collapse In The S&P 500," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56, pages 1043-1078, November.
    12. Aue, Alexander & Horváth, Lajos, 2004. "Delay time in sequential detection of change," Statistics & Probability Letters, Elsevier, vol. 67(3), pages 221-231, April.
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    Cited by:

    1. Yang Hu, 2023. "A review of Phillips‐type right‐tailed unit root bubble detection tests," Journal of Economic Surveys, Wiley Blackwell, vol. 37(1), pages 141-158, February.
    2. Esteve Vicente & Prats Maria A., 2021. "Structural Breaks and Explosive Behavior in the Long-Run: The Case of Australian Real House Prices, 1870–2020," Economics - The Open-Access, Open-Assessment Journal, De Gruyter, vol. 15(1), pages 72-84, January.
    3. Emilio Congregado & Carmen Díaz-Roldán & Vicente Esteve, 2023. "Deficit sustainability and fiscal theory of price level: the case of Italy, 1861–2020," Empirica, Springer;Austrian Institute for Economic Research;Austrian Economic Association, vol. 50(3), pages 755-782, August.
    4. Vicente Esteve & María A. Prats, 2021. "Financial bubbles and sustainability of public debt: The case of Spain," Working Papers 2111, Department of Applied Economics II, Universidad de Valencia.
    5. Vicente Esteve & María A. Prats, 2021. "Testing for rational bubbles in Australian housing market from a long-term perspective," Working Papers 2113, Department of Applied Economics II, Universidad de Valencia.
    6. Skrobotov Anton, 2023. "Testing for explosive bubbles: a review," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-26, January.

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