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Limit Theory for an Explosive Autoregressive Process

Author

Listed:
  • Xiaohu Wang

    () (Chinese University of Hong Kong)

  • Jun Yu

    () (Singapore Management University, School of Economics)

Abstract

Large sample properties are studied for a rst-order autoregression (AR(1)) with a root greater than unity. It is shown that, contrary to the AR coe¢ cient, the least- squares (LS) estimator of the intercept and its t-statistic are asymptotically normal without requiring the Gaussian error distribution, and hence an invariance principle applies. While the invariance principle does not apply to the asymptotic distribution of the LS estimator of the AR coe¢ cient, we show explicitly how it depends on the initial condition and the intercept. Also established are the asymptotic independence between the LS estimators of the intercept and the AR coefficient and the asymptotic independence between their t-statistics. Asymptotic theory for explosive processes is compared to that for unit root AR(1) processes and stationary AR(1) processes. The coefficient based test and the t test have better power for testing the hypothesis of zero intercept in the explosive process than in the stationary process.

Suggested Citation

  • Xiaohu Wang & Jun Yu, 2013. "Limit Theory for an Explosive Autoregressive Process," Working Papers 08-2013, Singapore Management University, School of Economics.
  • Handle: RePEc:siu:wpaper:08-2013
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    References listed on IDEAS

    as
    1. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2014. "Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behaviour," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 76(3), pages 315-333, June.
    2. Magdalinos, Tassos, 2012. "Mildly explosive autoregression under weak and strong dependence," Journal of Econometrics, Elsevier, vol. 169(2), pages 179-187.
    3. Tom Engsted & Bent Nielsen, 2012. "Testing for rational bubbles in a coexplosive vector autoregression," Econometrics Journal, Royal Economic Society, vol. 15(2), pages 226-254, June.
    4. Peter C. B. Phillips & Yangru Wu & Jun Yu, 2011. "EXPLOSIVE BEHAVIOR IN THE 1990s NASDAQ: WHEN DID EXUBERANCE ESCALATE ASSET VALUES?," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 52(1), pages 201-226, February.
    5. Perron, Pierre, 1991. "A Continuous Time Approximation to the Unstable First-Order Autoregressive Process: The Case without an Intercept," Econometrica, Econometric Society, vol. 59(1), pages 211-236, January.
    6. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    7. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
    8. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    9. Nielsen, Bent, 2010. "Analysis Of Coexplosive Processes," Econometric Theory, Cambridge University Press, vol. 26(03), pages 882-915, June.
    10. Phillips, Peter C.B. & Magdalinos, Tassos, 2008. "Limit Theory For Explosively Cointegrated Systems," Econometric Theory, Cambridge University Press, vol. 24(04), pages 865-887, August.
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    Citations

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    Cited by:

    1. Wang, Xiaohu & Yu, Jun, 2016. "Double asymptotics for explosive continuous time models," Journal of Econometrics, Elsevier, vol. 193(1), pages 35-53.
    2. Christoph P. Kustosz & Anne Leucht & Christine H. MÜller, 2016. "Tests Based on Simplicial Depth for AR(1) Models With Explosion," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(6), pages 763-784, November.
    3. Norbert Christopeit & Michael Massmann, 2013. "Estimating Structural Parameters in Regression Models with Adaptive Learning," Tinbergen Institute Discussion Papers 13-111/III, Tinbergen Institute.
    4. Horváth, Lajos & Trapani, Lorenzo, 2016. "Statistical inference in a random coefficient panel model," Journal of Econometrics, Elsevier, vol. 193(1), pages 54-75.
    5. repec:eee:ecolet:v:163:y:2018:i:c:p:98-101 is not listed on IDEAS
    6. Xiao, Weilin & Yu, Jun, 2018. "Asymptotic Theory for Rough Fractional Vasicek Models," Economics and Statistics Working Papers 7-2018, Singapore Management University, School of Economics.
    7. repec:gam:jecnmx:v:5:y:2017:i:4:p:47-:d:115992 is not listed on IDEAS
    8. Andras Fulop & Jun Yu, 2017. "Bayesian Analysis of Bubbles in Asset Prices," Econometrics, MDPI, Open Access Journal, vol. 5(4), pages 1-23, October.

    More about this item

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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