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Published online by Cambridge University Press:  08 April 2021

Sébastien Laurent
Aix-Marseille University (Aix-Marseille School of Economics) CNRS & EHESS Aix-Marseille Graduate School of Management–IAE, France
Shuping Shi*
Macquarie University
Address correspondence to Shuping Shi, Department of Economics, Macquarie University, Sydney, NSW, Australia; e-mail:
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Deviations of asset prices from the random walk dynamic imply the predictability of asset returns and thus have important implications for portfolio construction and risk management. This paper proposes a real-time monitoring device for such deviations using intraday high-frequency data. The proposed procedures are based on unit root tests with in-fill asymptotics but extended to take the empirical features of high-frequency financial data (particularly jumps) into consideration. We derive the limiting distributions of the tests under both the null hypothesis of a random walk with jumps and the alternative of mean reversion/explosiveness with jumps. The limiting results show that ignoring the presence of jumps could potentially lead to severe size distortions of both the standard left-sided (against mean reversion) and right-sided (against explosiveness) unit root tests. The simulation results reveal satisfactory performance of the proposed tests even with data from a relatively short time span. As an illustration, we apply the procedure to the Nasdaq composite index at the 10-minute frequency over two periods: around the peak of the dot-com bubble and during the 2015–2106 stock market sell-off. We find strong evidence of explosiveness in asset prices in late 1999 and mean reversion in late 2015. We also show that accounting for jumps when testing the random walk hypothesis on intraday data is empirically relevant and that ignoring jumps can lead to different conclusions.

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© The Author(s), 2021. Published by Cambridge University Press


The authors gratefully acknowledge Peter C.B. Phillips, Jun Yu, Olivier Scaillet, Xiaohu Wang, and participants at the QFFE 2019 conference in Marseille for helpful discussions. We thank the coeditor Eric Renault and three anonymous referees for very useful comments. Shi acknowledges research support from the Australian Research Council under project No. DE190100840. Laurent acknowledges research support from the French National Research Agency Grant ANR-17-EURE-0020.



Ahn, C.M. & Thompson, H.E. (1988) Jump-diffusion processes and the term structure of interest rates. Journal of Finance 43, 155174.CrossRefGoogle Scholar
Aït-Sahalia, Y., Mykland, P., & Zhang, L. (2005) How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial Studies 18, 351416.CrossRefGoogle Scholar
Ait-Sahalia, Y. & Yu, J. (2009) High frequency market microstructure noise estimates and liquidity measures. Annals of Applied Statistics 3, 422457.CrossRefGoogle Scholar
Amsler, C. & Lee, J. (1995) An LM test for a unit root in the presence of a structural change. Econometric Theory 11, 359368.CrossRefGoogle Scholar
Andersen, T.G & Bollerslev, T. (1997) Intraday periodicity and volatility persistence in financial markets. Journal of Empirical Finance 4, 115158.CrossRefGoogle Scholar
Andersen, T.G. & Bollerslev, T. (1998a) Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39, 885905.CrossRefGoogle Scholar
Andersen, T.G. & Bollerslev, T. (1998b) Deutsch mark-dollar volatility: Intraday activity patterns, macroeconomic announcements, and longer run dependencies. Journal of Finance 53, 219265.CrossRefGoogle Scholar
Andersen, T.G, Bollerslev, T., & Diebold, F. (2007a) Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility. The Review of Economics and Statistics 89, 701720.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., & Dobrev, D. (2007b) No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects, jumps and iid noise: Theory and testable distributional implications. Journal of Econometrics 138, 125180.CrossRefGoogle Scholar
Bajgrowicz, P., Scaillet, O., & Treccani, A. (2015) Jumps in high-frequency data: Spurious detections, dynamics, and news. Management Science 62, 21982217.CrossRefGoogle Scholar
Balvers, R., Wu, Y., & Gilliland, E. (2000) Mean reversion across national stock markets and parametric contrarian investment strategies. The Journal of Finance 55, 745772.CrossRefGoogle Scholar
Banerjee, A., Lumsdaine, R.L., & Stock, J.H. (1992) Recursive and sequential tests of the unit-root and trend-break hypotheses: Theory and international evidence. Journal of Business & Economic Statistics 10, 271287.Google Scholar
Bauwens, L., Hafner, C., & Laurent, S. (2012) Handbook of Volatility Models and Their Applications, vol. 3. Wiley.CrossRefGoogle Scholar
Blanchard, O.J. & Watson, M.W. (1982) Bubbles, Rational Expectations and Financial Markets. NBER Working paper 945, National Bureau of Economic Research.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307327.CrossRefGoogle Scholar
Boswijk, H.P. & Zu, Y. (2018) Adaptive wild bootstrap tests for a unit root with nonstationary volatility. The Econometrics Journal 21, 87113.CrossRefGoogle Scholar
Boudt, K., Croux, C., & Laurent, S. (2011) Robust estimation of intraweek periodicity in volatility and jump detection. Journal of Empirical Finance 18, 353367.CrossRefGoogle Scholar
Brooks, C. & Katsaris, A. (2005) A three-regime model of speculative behaviour: Modelling the evolution of the S&P 500 composite index. The Economic Journal 115, 767797.CrossRefGoogle Scholar
Chaudhuri, K. & Wu, Y. (2003) Random walk versus breaking trend in stock prices: Evidence from emerging markets. Journal of Banking & Finance 27, 575592.CrossRefGoogle Scholar
Chong, T.T.L. (2001) Structural change in AR (1) models. Econometric Theory 17, 87155.CrossRefGoogle Scholar
Clemente, J., Montañés, A., & Reyes, M. (1998) Testing for a unit root in variables with a double change in the mean. Economics Letters 59, 175182.CrossRefGoogle Scholar
Diba, B.T. & Grossman, H.I. (1988) Explosive rational bubbles in stock prices? The American Economic Review 78, 520530.Google Scholar
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427431.Google Scholar
Dickey, D.A. & Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 10571072.CrossRefGoogle Scholar
Enders, W. & Lee, J. (2012) A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics 74, 574599.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 9871007.CrossRefGoogle Scholar
Etienne, X.L., Irwin, S.H., & Garcia, P. (2013) Bubbles in food commodity markets: Four decades of evidence. Journal of International Money and Finance 97, 6587.Google Scholar
Evans, G.W. (1991) Pitfalls in testing for explosive bubbles in asset prices. The American Economic Review 81, 922930.Google Scholar
Fama, E.F. & French, K.R. (1988) Permanent and temporary components of stock prices. Journal of Political Economy 96, 246273.CrossRefGoogle Scholar
Fantazzini, D. (2016) The oil price crash in 2014/15: Was there a (negative) financial bubble? Energy Policy 96, 383396.CrossRefGoogle Scholar
Gatev, E., Goetzmann, W.N., & Rouwenhorst, K.G. (2006) Pairs trading: Performance of a relative-value arbitrage rule. Review of Financial Studies 19, 797827.CrossRefGoogle Scholar
Guenster, N. & Kole, E. (2009) Bubbles and Investment Horizons. Available at SSRN: Scholar
Gutierrez, L. (2012) Speculative bubbles in agricultural commodity markets. European Review of Agricultural Economics 40, 217238.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.CrossRefGoogle Scholar
Harvey, D., Leybourne, S., & Zu, Y. (2019) Testing explosive bubbles with time-varying volatility. Econometric Review 38, 11311151.CrossRefGoogle Scholar
Harvey, D.I., Leybourne, S.J., & Newbold, P. (2001) Innovational outlier unit root tests with an endogenously determined break in level. Oxford Bulletin of Economics and Statistics 63, 559575.CrossRefGoogle Scholar
Homm, U. & Breitung, J. (2012) Testing for speculative bubbles in stock markets: A comparison of alternative methods. Journal of Financial Econometrics 10, 198231.CrossRefGoogle Scholar
Hu, Y. & Oxley, L. (2018) Do 18th century bubbles survive the scrutiny of 21st century time series econometrics? Economics Letters 162, 131134.CrossRefGoogle Scholar
Jiang, L., Wang, X., & Yu, J. (2018) New distribution theory for the estimation of structural break point in mean. Journal of Econometrics 205, 156176.CrossRefGoogle Scholar
Jiang, L., Wang, X., & Yu, J. (2020) In-fill asymptotic theory for structural break point in autoregression: A unified theory. Econometric Reviews, Scholar
Kim, J. & Park, J.Y. (2019) Unit Root, Mean Reversion and Nonstationarity in Financial Time Series, Working paper.Google Scholar
Kim, M.J., Nelson, C.R., & Startz, R. (1991) Mean reversion in stock prices? A reappraisal of the empirical evidence. The Review of Economic Studies 58, 515528.CrossRefGoogle Scholar
Kim, T.H., Leybourne, S., & Newbold, P. (2002) Unit root tests with a break in innovation variance. Journal of Econometrics 109, 365387.CrossRefGoogle Scholar
Kou, S.G. (2002) A jump-diffusion model for option pricing. Management Science 48, 10861101.CrossRefGoogle Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54, 159178.CrossRefGoogle Scholar
Laurent, S. & Shi, S. (2020) Volatility estimation and jump detection for drift–diffusion processes. Journal of Econometrics 217, 259290.CrossRefGoogle Scholar
Lee, J. & Strazicich, M.C. (2001) Break point estimation and spurious rejections with endogenous unit root tests. Oxford Bulletin of Economics and Statistics 63, 535558.CrossRefGoogle Scholar
Lee, J. & Strazicich, M.C. (2003) Minimum Lagrange multiplier unit root test with two structural breaks. Review of Economics and Statistics 85, 10821089.CrossRefGoogle Scholar
Lee, S.S. (2012) Jumps and information flow in financial markets. Review of Financial Studies 25, 439479.CrossRefGoogle Scholar
Lee, S.S. & Mykland, P.A. (2008) Jumps in financial markets: A new nonparametric test and jump dynamics. The Review of Financial Studies 21, 25352563.CrossRefGoogle Scholar
Lo, A.W. & MacKinlay, A.C. (1988) Stock market prices do not follow random walks: Evidence from a simple specification test. Review of Financial Studies 1, 4166.CrossRefGoogle Scholar
Lumsdaine, R.L. & Papell, D.H. (1997) Multiple trend breaks and the unit-root hypothesis. Review of Economics and Statistics 79, 212218.CrossRefGoogle Scholar
Mancini, C. (2011) Jumps. In Bauwens, L., Hafner, C., & Laurent, S. (Eds.), Wiley Handbook in Financial Engineering and Econometrics: Volatility Models and Their Applications. Wiley, 427474.Google Scholar
McQueen, G. (1992) Long-horizon mean-reverting stock prices revisited. Journal of Financial and Quantitative Analysis 27, 118.CrossRefGoogle Scholar
Merton, R.C. (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3, 125144.CrossRefGoogle Scholar
Merton, R.C. (1980) On estimating the expected return on the market: An exploratory investigation. Journal of Financial Economics 8, 323361.CrossRefGoogle Scholar
Miller, M.H., Muthuswamy, J., & Whaley, R.E. (1994) Mean reversion of Standard & Poor's 500 index basis changes: Arbitrage-induced or statistical illusion? The Journal of Finance 49, 479513.CrossRefGoogle Scholar
Milunovich, G., Shi, S., & Tan, D. (2019) Bubble detection and sector trading in real time. Quantitative Finance 19, 247263.CrossRefGoogle Scholar
Narayan, P.K., Sharma, S.S., Phan, D.H.B. (2016) Asset price bubbles and economic welfare. International Review of Financial Analysis 44, 139148.CrossRefGoogle Scholar
Nelson, C.R. & Plosser, C.R. (1982) Trends and random walks in macroeconmic time series: Some evidence and implications. Journal of Monetary Economics 10, 139162.CrossRefGoogle Scholar
Nelson, D.B. (1991) ARCH models as diffusion approximations. Journal of Econometrics 45, 738.CrossRefGoogle Scholar
Pavlidis, E. et al. (2016) Episodes of exuberance in housing markets: In search of the smoking gun. The Journal of Real Estate Finance and Economics 53, 419449.CrossRefGoogle Scholar
Perron, P. (1989) The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 13611401.CrossRefGoogle Scholar
Perron, P. (1990) Testing for a unit root in a time series with a changing mean. Journal of Business & Economic Statistics 8, 153162.Google Scholar
Perron, P. (1991) A continuous time approximation to the unstable first-order autoregressive process: The case without an intercept. Econometrica 59, 211236.CrossRefGoogle Scholar
Perron, P. (1997) Further evidence on breaking trend functions in macroeconomic variables. Journal of Econometrics 80, 355385.CrossRefGoogle Scholar
Phillips, P.C.B. (1987a) Time series regression with a unit root. Econometrica 55, 277301.CrossRefGoogle Scholar
Phillips, P.C.B. (1987b) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535547.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335346.CrossRefGoogle Scholar
Phillips, P.C.B. & Shi, S. (2018) Financial bubble implosion and reverse regression. Econometric Theory 34, 705753.CrossRefGoogle Scholar
Phillips, P.C.B. & Shi, S. (2019) Detecting financial collapse and ballooning sovereign risk. Oxford Bulletin of Economics and Statistics 81, 13361361.CrossRefGoogle Scholar
Phillips, P.C.B., Shi, S., & Yu, J. (2015a) Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P 500. International Economic Review 56, 10431078.CrossRefGoogle Scholar
Phillips, P.C.B., Shi, S., & Yu, J. (2015b) Testing for multiple bubbles: Limit theory of real-time detectors. International Economic Review 56, 10791134.CrossRefGoogle Scholar
Phillips, P.C.B., Wu, Y., & Yu, J. (2011) Explosive behavior in the 1990s NASDAQ: When did exuberance escalate asset values? International Economic Review 52, 201226.CrossRefGoogle Scholar
Phillips, P.C.B. & Yu, J. (2011) Dating the timeline of financial bubbles during the subprime crisis. Quantitative Economics 2, 455491.CrossRefGoogle Scholar
Phillips, P.C.B. & Yu, J. (2013) Bubble or roller coaster in world stock markets. The Business Times. June 28.Google Scholar
Poterba, J.M. & Summers, L.H. (1988) Mean reversion in stock prices: Evidence and implications. Journal of Financial Economics 22, 2759.CrossRefGoogle Scholar
Richards, A.J. (1997) Winner-loser reversals in national stock market indices: Can they be explained? The Journal of Finance 52, 21292144.CrossRefGoogle Scholar
Richardson, M. (1993) Temporary components of stock prices: A skeptic's view. Journal of Business & Economic Statistics 11, 199207.Google Scholar
Said, S.E. & Dickey, D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, 599607.CrossRefGoogle Scholar
Saikkonen, P. & Lütkepohl, H. (2002) Testing for a unit root in a time series with a level shift at unknown time. Econometric Theory 18, 313348.CrossRefGoogle Scholar
Schmidt, P. & Phillips, P.C.B. (1992) LM tests for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54, 257287.CrossRefGoogle Scholar
Serban, A.F. (2010) Combining mean reversion and momentum trading strategies in foreign exchange markets. Journal of Banking & Finance 34, 27202727.CrossRefGoogle Scholar
Shi, S. (2017) Speculative bubbles or market fundamentals? An investigation of US regional housing markets. Economic Modelling 66, 101111.CrossRefGoogle Scholar
Shi, S. & Song, Y. (2016) Identifying speculative bubbles using an infinite hidden Markov model. Journal of Financial Econometrics 14, 159184.Google Scholar
Shiller, R.J. & Perron, P. (1985) Testing the random walk hypothesis: Power versus frequency of observation. Economics Letters 18, 381386.CrossRefGoogle Scholar
Tao, Y., Phillips, P.C.B., & Yu, J. (2019) Random coefficient continuous systems: Testing for extreme sample path behavior. Journal of Econometrics 209, 208237.CrossRefGoogle Scholar
Taylor, S. & Xu, X. (1997) The incremental volatility information in one million foreign exchange quotations. Journal of Empirical Finance 4, 317340.CrossRefGoogle Scholar
Taylor, S.J. (1994) Modeling stochastic volatility: A review and comparative study. Mathematical Finance 4, 183204.CrossRefGoogle Scholar
Vogelsang, T.J. & Perron, P. (1998) Additional tests for a unit root allowing for a break in the trend function at an unknown time. International Economic Review 39, 10731100.CrossRefGoogle Scholar
Wang, X. & Yu, J. (2016) Double asymptotics for explosive continuous time models. Journal of Econometrics 193, 3553.CrossRefGoogle Scholar
Yu, J. (2014) Econometric analysis of continuous time models: A survey of Peter Phillips's work and some new results. Econometric Theory 30, 737774.CrossRefGoogle Scholar
Zhou, Q. & Yu, J. (2015) Asymptotic theory for linear diffusions under alternative sampling schemes. Economics Letters 128, 15.CrossRefGoogle Scholar
Zivot, E. & Andrews, D.W.K. (2002) Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business & Economic Statistics 20, 2544.CrossRefGoogle Scholar
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