IDEAS home Printed from https://ideas.repec.org/a/eee/econom/v179y2014i1p66-82.html
   My bibliography  Save this article

Testing stationarity of functional time series

Author

Listed:
  • Horváth, Lajos
  • Kokoszka, Piotr
  • Rice, Gregory

Abstract

Economic and financial data often take the form of a collection of curves observed consecutively over time. Examples include, intraday price curves, yield and term structure curves, and intraday volatility curves. Such curves can be viewed as a time series of functions. A fundamental issue that must be addressed, before an attempt is made to statistically model such data, is whether these curves, perhaps suitably transformed, form a stationary functional time series. This paper formalizes the assumption of stationarity in the context of functional time series and proposes several procedures to test the null hypothesis of stationarity. The tests are nontrivial extensions of the broadly used tests in the KPSS family. The properties of the tests under several alternatives, including change-point and I(1), are studied, and new insights, present only in the functional setting are uncovered. The theory is illustrated by a small simulation study and an application to intraday price curves.

Suggested Citation

  • Horváth, Lajos & Kokoszka, Piotr & Rice, Gregory, 2014. "Testing stationarity of functional time series," Journal of Econometrics, Elsevier, vol. 179(1), pages 66-82.
    • Handle: RePEc:eee:econom:v:179:y:2014:i:1:p:66-82
    DOI: 10.1016/j.jeconom.2013.11.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304407613002327
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jeconom.2013.11.002?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. István Berkes & Robertas Gabrys & Lajos Horváth & Piotr Kokoszka, 2009. "Detecting changes in the mean of functional observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 927-946, November.
    2. Lajos Horváth & Piotr Kokoszka & Ron Reeder, 2013. "Estimation of the mean of functional time series and a two-sample problem," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(1), pages 103-122, January.
    3. Ole E. Barndorff-Nielsen & Neil Shephard, 2004. "Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics," Econometrica, Econometric Society, vol. 72(3), pages 885-925, May.
    4. Panaretos, Victor M. & Tavakoli, Shahin, 2013. "Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2779-2807.
    5. Kwiatkowski, Denis & Phillips, Peter C. B. & Schmidt, Peter & Shin, Yongcheol, 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, Elsevier, vol. 54(1-3), pages 159-178.
    6. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-1072, June.
    7. Anestis Antoniadis & Efstathios Paparoditis & Theofanis Sapatinas, 2006. "A functional wavelet–kernel approach for time series prediction," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(5), pages 837-857, November.
    8. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    9. Lo, Andrew W, 1991. "Long-Term Memory in Stock Market Prices," Econometrica, Econometric Society, vol. 59(5), pages 1279-1313, September.
    10. Politis, Dimitris N., 2011. "Higher-Order Accurate, Positive Semidefinite Estimation Of Large-Sample Covariance And Spectral Density Matrices," Econometric Theory, Cambridge University Press, vol. 27(4), pages 703-744, August.
    11. Giraitis, Liudas & Kokoszka, Piotr & Leipus, Remigijus & Teyssiere, Gilles, 2003. "Rescaled variance and related tests for long memory in volatility and levels," Journal of Econometrics, Elsevier, vol. 112(2), pages 265-294, February.
    12. Kristian Jönsson, 2011. "Testing Stationarity in Small‐ and Medium‐Sized Samples when Disturbances are Serially Correlated," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 73(5), pages 669-690, October.
    13. Gabrys, Robertas & Horváth, Lajos & Kokoszka, Piotr, 2010. "Tests for Error Correlation in the Functional Linear Model," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1113-1125.
    14. Andrews, Donald W K & Monahan, J Christopher, 1992. "An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator," Econometrica, Econometric Society, vol. 60(4), pages 953-966, July.
    15. GIRAITIS, Liudas & KOKOSZKA, Piotr & LEIPUS, Remigijus & TEYSSIÈRE, Gilles, 2003. "Rescaled variance and related tests for long memory in volatility and levels," LIDAM Reprints CORE 1594, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    16. Müller, Hans-Georg & Sen, Rituparna & Stadtmüller, Ulrich, 2011. "Functional data analysis for volatility," Journal of Econometrics, Elsevier, vol. 165(2), pages 233-245.
    17. Pelagatti, Matteo M. & Sen, Pranab K., 2013. "Rank tests for short memory stationarity," Journal of Econometrics, Elsevier, vol. 172(1), pages 90-105.
    18. de Jong, Robert M. & Amsler, Christine & Schmidt, Peter, 2007. "A robust version of the KPSS test based on indicators," Journal of Econometrics, Elsevier, vol. 137(2), pages 311-333, April.
    19. Kargin, V. & Onatski, A., 2008. "Curve forecasting by functional autoregression," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2508-2526, November.
    20. Hörmann, Siegfried & Horváth, Lajos & Reeder, Ron, 2013. "A Functional Version Of The Arch Model," Econometric Theory, Cambridge University Press, vol. 29(2), pages 267-288, April.
    21. Lee, Dongin & Schmidt, Peter, 1996. "On the power of the KPSS test of stationarity against fractionally-integrated alternatives," Journal of Econometrics, Elsevier, vol. 73(1), pages 285-302, July.
    22. Horváth, Lajos & Husková, Marie & Kokoszka, Piotr, 2010. "Testing the stability of the functional autoregressive process," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 352-367, February.
    23. Siegfried Hörmann & Łukasz Kidziński & Marc Hallin, 2015. "Dynamic functional principal components," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(2), pages 319-348, March.
    24. repec:eca:wpaper:2013/131191 is not listed on IDEAS
    25. Antoniadis, Anestis & Sapatinas, Theofanis, 2003. "Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 133-158, October.
    26. Berkes, István & Horváth, Lajos & Rice, Gregory, 2013. "Weak invariance principles for sums of dependent random functions," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 385-403.
    27. Yogesh Dwivedi & Suhasini Subba Rao, 2011. "A test for second‐order stationarity of a time series based on the discrete Fourier transform," Journal of Time Series Analysis, Wiley Blackwell, vol. 32(1), pages 68-91, January.
    28. Piotr Kokoszka & Matthew Reimherr, 2013. "Predictability of shapes of intraday price curves," Econometrics Journal, Royal Economic Society, vol. 16(3), pages 285-308, October.
    29. Jönsson, Kristian, 2006. "Finite-Sample Stability of the KPSS Test," Working Papers 2006:23, Lund University, Department of Economics.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Salish, Nazarii & Gleim, Alexander, 2019. "A moment-based notion of time dependence for functional time series," Journal of Econometrics, Elsevier, vol. 212(2), pages 377-392.
    2. Chen, Yichao & Pun, Chi Seng, 2019. "A bootstrap-based KPSS test for functional time series," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    3. 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.
    4. Zhang, Xianyang, 2016. "White noise testing and model diagnostic checking for functional time series," Journal of Econometrics, Elsevier, vol. 194(1), pages 76-95.
    5. Guglielmo Caporale & Luis Gil-Alana, 2013. "Long memory in US real output per capita," Empirical Economics, Springer, vol. 44(2), pages 591-611, April.
    6. Lujia Bai & Weichi Wu, 2021. "Detecting long-range dependence for time-varying linear models," Papers 2110.08089, arXiv.org, revised Mar 2023.
    7. Berkes, István & Horváth, Lajos & Rice, Gregory, 2016. "On the asymptotic normality of kernel estimators of the long run covariance of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 150-175.
    8. Jiang, Bibo & Lu, Ye & Park, Joon Y., 2020. "Testing for Stationarity at High Frequency," Journal of Econometrics, Elsevier, vol. 215(2), pages 341-374.
    9. Horváth, Lajos & Hušková, Marie & Rice, Gregory, 2013. "Test of independence for functional data," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 100-119.
    10. van Delft, Anne & Eichler, Michael, 2017. "Locally Stationary Functional Time Series," LIDAM Discussion Papers ISBA 2017023, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Doyle, John R. & Chen, Catherine Huirong, 2012. "A multidimensional classification of market anomalies: Evidence from 76 price indices," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 22(5), pages 1237-1257.
    12. Axel Bücher & Holger Dette & Florian Heinrichs, 2020. "Detecting deviations from second-order stationarity in locally stationary functional time series," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(4), pages 1055-1094, August.
    13. Assaf, Ata, 2016. "MENA stock market volatility persistence: Evidence before and after the financial crisis of 2008," Research in International Business and Finance, Elsevier, vol. 36(C), pages 222-240.
    14. Ata Assaf, 2006. "Canadian REITs and Stock Prices: Fractional Cointegration and Long Memory," Review of Pacific Basin Financial Markets and Policies (RPBFMP), World Scientific Publishing Co. Pte. Ltd., vol. 9(03), pages 441-462.
    15. Shao, Xiaofeng & Wu, Wei Biao, 2007. "Local asymptotic powers of nonparametric and semiparametric tests for fractional integration," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 251-261, February.
    16. Assaf, Ata, 2015. "Long memory and level shifts in REITs returns and volatility," International Review of Financial Analysis, Elsevier, vol. 42(C), pages 172-182.
    17. Rice, Gregory & Wirjanto, Tony & Zhao, Yuqian, 2019. "Tests for conditional heteroscedasticity with functional data and goodness-of-fit tests for FGARCH models," MPRA Paper 93048, University Library of Munich, Germany.
    18. Gao, Yuan & Shang, Han Lin & Yang, Yanrong, 2019. "High-dimensional functional time series forecasting: An application to age-specific mortality rates," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 232-243.
    19. Francis Ahking, 2010. "Non-parametric tests of real exchange rates in the post-Bretton Woods era," Empirical Economics, Springer, vol. 39(2), pages 439-456, October.
    20. Assaf, A., 2006. "Dependence and mean reversion in stock prices: The case of the MENA region," Research in International Business and Finance, Elsevier, vol. 20(3), pages 286-304, September.

    More about this item

    Keywords

    Change point; Functional data; Integrated time series; Intraday price curves; Test of stationarity;
    All these keywords.

    JEL classification:

    • 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
    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:econom:v:179:y:2014:i:1:p:66-82. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jeconom .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.