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A Fixed-b Perspective on the Phillips-Perron Unit Root Tests

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  • Vogelsang, Timothy J.

    (Department of Economics, Michigan State University, East Lansing, USA)

  • Wagner, Martin

    (Department of Economics and Finance, Institute for Advanced Studies, Vienna, Austria, and Frisch Centre for Economic Research, Oslo, Norway)

Abstract

We extend fixed-b asymptotic theory to the nonparametric Phillips-Perron (PP) unit root tests. We show that the fixed-b limits depend on nuisance parameters in a complicated way. These non-pivotal limits provide an alternative theoretical explanation for the well known finite sample problems of PP tests. We also show that the fixed-b limits depend on whether deterministic trends are removed using one-step or two-step approaches, contrasting the asymptotic equivalence of the one- and two-step approaches under a consistency approximation for the long run variance estimator. Based on these results we introduce modified PP tests that allow for fixed-b inference. The theoretical analysis is cast in the framework of near-integrated processes which allows to study the asymptotic behavior both under the unit root null hypothesis as well as for local alternatives. The performance of the original and modified tests is compared by means of local asymptotic power and a small simulation study.

Suggested Citation

  • Vogelsang, Timothy J. & Wagner, Martin, 2011. "A Fixed-b Perspective on the Phillips-Perron Unit Root Tests," Economics Series 272, Institute for Advanced Studies.
    • Handle: RePEc:ihs:ihsesp:272
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    File URL: https://irihs.ihs.ac.at/id/eprint/2081
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    References listed on IDEAS

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    1. Pierre Perron & Serena Ng, 1996. "Useful Modifications to some Unit Root Tests with Dependent Errors and their Local Asymptotic Properties," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 63(3), pages 435-463.
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    Cited by:

    1. Hanck, Christoph & Demetrescu, Matei & Kruse, Robinson, 2015. "Fixed-b Asymptotics for t-Statistics in the Presence of Time-Varying Volatility," VfS Annual Conference 2015 (Muenster): Economic Development - Theory and Policy 112916, Verein für Socialpolitik / German Economic Association.
    2. Pedroni, Peter L. & Vogelsang, Timothy J. & Wagner, Martin & Westerlund, Joakim, 2015. "Nonparametric rank tests for non-stationary panels," Journal of Econometrics, Elsevier, vol. 185(2), pages 378-391.
    3. Demetrescu, Matei & Kruse, Robinson, 2015. "Testing heteroskedastic time series for normality," VfS Annual Conference 2015 (Muenster): Economic Development - Theory and Policy 113221, Verein für Socialpolitik / German Economic Association.
    4. Demetrescu, Matei & Kruse-Becher, Robinson, 2025. "Is U.S. real output growth non-normal? A tale of time-varying location and scale," Journal of Economic Dynamics and Control, Elsevier, vol. 171(C).
    5. Ahmed, Muhammad Ashfaq & Nawaz, Nasreen, 2023. "A Sufficient Statistical Test for Dynamic Stability," MPRA Paper 116684, University Library of Munich, Germany.
    6. Cho, Cheol-Keun & Amsler, Christine & Schmidt, Peter, 2015. "A test of the null of integer integration against the alternative of fractional integration," Journal of Econometrics, Elsevier, vol. 187(1), pages 217-237.
    7. Christis Katsouris, 2023. "Limit Theory under Network Dependence and Nonstationarity," Papers 2308.01418, arXiv.org, revised Aug 2023.
    8. Casini, Alessandro, 2024. "The fixed-b limiting distribution and the ERP of HAR tests under nonstationarity," Journal of Econometrics, Elsevier, vol. 238(2).
    9. Cui, Yu & Khan, Sufyan Ullah & Sauer, Johannes & Kipperberg, Gorm & Zhao, Minjuan, 2023. "Agricultural carbon footprint, energy utilization and economic quality: What causes what, and where?," Energy, Elsevier, vol. 278(PA).
    10. Matei Demetrescu & Christoph Hanck & Robinson Kruse, 2016. "Fixed-b Inference in the Presence of Time-Varying Volatility," CREATES Research Papers 2016-01, Department of Economics and Business Economics, Aarhus University.
    11. Stypka, Oliver & Wagner, Martin, 2019. "The Phillips unit root tests for polynomials of integrated processes revisited," Economics Letters, Elsevier, vol. 176(C), pages 109-113.
    12. Anca-Gabriela Turtureanu & Rodica Pripoaie & Carmen-Mihaela Cretu & Carmen-Gabriela Sirbu & Emanuel Ştefan Marinescu & Laurentiu-Gabriel Talaghir & Florentina Chițu, 2022. "A Projection Approach of Tourist Circulation under Conditions of Uncertainty," Sustainability, MDPI, vol. 14(4), pages 1-21, February.
    13. Laurini, Márcio Poletti & Ohashi, Alberto, 2015. "A noisy principal component analysis for forward rate curves," European Journal of Operational Research, Elsevier, vol. 246(1), pages 140-153.
    14. Hirukawa, Masayuki, 2023. "Robust Covariance Matrix Estimation in Time Series: A Review," Econometrics and Statistics, Elsevier, vol. 27(C), pages 36-61.
    15. Wagner, Martin & Grabarczyk, Peter & Hong, Seung Hyun, 2020. "Fully modified OLS estimation and inference for seemingly unrelated cointegrating polynomial regressions and the environmental Kuznets curve for carbon dioxide emissions," Journal of Econometrics, Elsevier, vol. 214(1), pages 216-255.
    16. Frühwirth, Manfred & Sögner, Leopold, 2015. "Weather and SAD related mood effects on the financial market," The Quarterly Review of Economics and Finance, Elsevier, vol. 57(C), pages 11-31.
    17. Wookjae Heo & Eunchan Kim, 2025. "Smoothing the Subjective Financial Risk Tolerance: Volatility and Market Implications," Mathematics, MDPI, vol. 13(4), pages 1-34, February.
    18. Vogelsang, Timothy J. & Wagner, Martin, 2014. "Integrated modified OLS estimation and fixed-b inference for cointegrating regressions," Journal of Econometrics, Elsevier, vol. 178(2), pages 741-760.
    19. Zhenxin Wang & Shaoping Wang & Yayi Yan, 2024. "Sieve Bootstrap for Fixed-b Phillips–Perron Unit Root Test," Computational Economics, Springer;Society for Computational Economics, vol. 64(6), pages 3181-3205, December.
    20. Mehdi Hosseinkouchack & Uwe Hassler, 2016. "Powerful Unit Root Tests Free of Nuisance Parameters," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(4), pages 533-554, July.

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    More about this item

    Keywords

    Nonparametric kernel estimator; long run variance; detrending; one-step; two-step;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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