IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v124y2014i1p646-677.html
   My bibliography  Save this article

A martingale decomposition for quadratic forms of Markov chains (with applications)

Author

Listed:
  • Atchadé, Yves F.
  • Cattaneo, Matias D.

Abstract

We develop a martingale-based decomposition for a general class of quadratic forms of Markov chains, which resembles the well-known Hoeffding decomposition of U-statistics of i.i.d. data up to a reminder term. To illustrate the applicability of our results, we discuss how this decomposition may be used to studying the large-sample properties of certain statistics in two problems: (i) we examine the asymptotic behavior of lag-window estimators in time series, and (ii) we derive an asymptotic linear representation and limiting distribution of U-statistics with varying kernels in time series. We also discuss simplified examples of interest in statistics and econometrics.

Suggested Citation

  • Atchadé, Yves F. & Cattaneo, Matias D., 2014. "A martingale decomposition for quadratic forms of Markov chains (with applications)," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 646-677.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:646-677
    DOI: 10.1016/j.spa.2013.09.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414913002378
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2013.09.001?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. Meitz, Mika & Saikkonen, Pentti, 2008. "Ergodicity, Mixing, And Existence Of Moments Of A Class Of Markov Models With Applications To Garch And Acd Models," Econometric Theory, Cambridge University Press, vol. 24(5), pages 1291-1320, October.
    2. Stoker, Thomas M, 1986. "Consistent Estimation of Scaled Coefficients," Econometrica, Econometric Society, vol. 54(6), pages 1461-1481, November.
    3. Kiefer, Nicholas M. & Vogelsang, Timothy J., 2005. "A New Asymptotic Theory For Heteroskedasticity-Autocorrelation Robust Tests," Econometric Theory, Cambridge University Press, vol. 21(6), pages 1130-1164, December.
    4. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, vol. 57(6), pages 1403-1430, November.
    5. Wu, Wei Biao & Shao, Xiaofeng, 2007. "A Limit Theorem For Quadratic Forms And Its Applications," Econometric Theory, Cambridge University Press, vol. 23(5), pages 930-951, October.
    6. Bhansali, R.J. & Giraitis, L. & Kokoszka, P.S., 2007. "Approximations and limit theory for quadratic forms of linear processes," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 71-95, January.
    7. P. M. Robinson, 1989. "Hypothesis Testing in Semiparametric and Nonparametric Models for Econometric Time Series," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 56(4), pages 511-534.
    8. Cheng, Bing & Robinson, P. M., 1994. "Semiparametric estimation from time series with long-range dependence," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 335-353.
    9. Cattaneo, Matias D. & Crump, Richard K. & Jansson, Michael, 2010. "Robust Data-Driven Inference for Density-Weighted Average Derivatives," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1070-1083.
    10. Dehling, Herold & Wendler, Martin, 2010. "Central limit theorem and the bootstrap for U-statistics of strongly mixing data," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 126-137, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Liu, Cheng & Sun, Yixiao, 2019. "A simple and trustworthy asymptotic t test in difference-in-differences regressions," Journal of Econometrics, Elsevier, vol. 210(2), pages 327-362.

    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. Nishiyama, Y., 2004. "Minimum normal approximation error bandwidth selection for averaged derivatives," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 64(1), pages 53-61.
    2. Kaido, Hiroaki, 2017. "Asymptotically Efficient Estimation Of Weighted Average Derivatives With An Interval Censored Variable," Econometric Theory, Cambridge University Press, vol. 33(5), pages 1218-1241, October.
    3. Yukitoshi Matsushita & Taisuke Otsu, 2017. "Likelihood inference on semiparametric models: Average derivative and treatment effect," STICERD - Econometrics Paper Series 592, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    4. Cattaneo, Matias D. & Crump, Richard K. & Jansson, Michael, 2014. "Small Bandwidth Asymptotics For Density-Weighted Average Derivatives," Econometric Theory, Cambridge University Press, vol. 30(1), pages 176-200, February.
    5. Ichimura, Hidehiko & Todd, Petra E., 2007. "Implementing Nonparametric and Semiparametric Estimators," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 74, Elsevier.
    6. Temel, Tugrul T., 2001. "A Nonparametric Hypothesis Test Via The Bootstrap Resampling," 2001 Annual meeting, August 5-8, Chicago, IL 20600, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).
    7. Linton, Oliver, 1995. "Second Order Approximation in the Partially Linear Regression Model," Econometrica, Econometric Society, vol. 63(5), pages 1079-1112, September.
    8. Lahiri, Kajal & Yang, Liu, 2013. "Forecasting Binary Outcomes," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 2, chapter 0, pages 1025-1106, Elsevier.
    9. Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," Econometrica, Econometric Society, vol. 63(4), pages 767-804, July.
    10. van Delft, Anne, 2020. "A note on quadratic forms of stationary functional time series under mild conditions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4206-4251.
    11. Halbert White & Karim Chalak, 2013. "Identification and Identification Failure for Treatment Effects Using Structural Systems," Econometric Reviews, Taylor & Francis Journals, vol. 32(3), pages 273-317, November.
    12. Coppejans, Mark, 2001. "Estimation of the binary response model using a mixture of distributions estimator (MOD)," Journal of Econometrics, Elsevier, vol. 102(2), pages 231-269, June.
    13. Jeong, Kiho & Härdle, Wolfgang Karl, 2008. "A consistent nonparametric test for causality in quantile," SFB 649 Discussion Papers 2008-007, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    14. Escanciano, Juan Carlos & Song, Kyungchul, 2010. "Testing single-index restrictions with a focus on average derivatives," Journal of Econometrics, Elsevier, vol. 156(2), pages 377-391, June.
    15. Yu, Ping & Phillips, Peter C.B., 2018. "Threshold regression with endogeneity," Journal of Econometrics, Elsevier, vol. 203(1), pages 50-68.
    16. Dong, Hao & Taylor, Luke, 2022. "Nonparametric Significance Testing In Measurement Error Models," Econometric Theory, Cambridge University Press, vol. 38(3), pages 454-496, June.
    17. Pavel Yaskov, 2018. "LLN for Quadratic Forms of Long Memory Time Series and Its Applications in Random Matrix Theory," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2032-2055, December.
    18. Joseph G. Altonji & Hidehiko Ichimura & Taisuke Otsu, 2012. "Estimating Derivatives in Nonseparable Models With Limited Dependent Variables," Econometrica, Econometric Society, vol. 80(4), pages 1701-1719, July.
    19. Prakasa Rao, B. L. S., 1995. "Consistent estimation of density-weighted average derivative by orthogonal series method," Statistics & Probability Letters, Elsevier, vol. 22(3), pages 205-212, February.
    20. Chalak, Karim, 2024. "Nonparametric Gini-Frisch bounds," Journal of Econometrics, Elsevier, vol. 238(1).

    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:spapps:v:124:y:2014:i:1:p:646-677. 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/wps/find/journaldescription.cws_home/505572/description#description .

    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.