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

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  • 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
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    References listed on IDEAS

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    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(05), pages 1291-1320, October.
    2. P. M. Robinson, 1989. "Hypothesis Testing in Semiparametric and Nonparametric Models for Econometric Time Series," Review of Economic Studies, Oxford University Press, vol. 56(4), pages 511-534.
    3. Kiefer, Nicholas M. & Vogelsang, Timothy J., 2005. "A New Asymptotic Theory For Heteroskedasticity-Autocorrelation Robust Tests," Econometric Theory, Cambridge University Press, vol. 21(06), pages 1130-1164, December.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Wu, Wei Biao & Shao, Xiaofeng, 2007. "A Limit Theorem For Quadratic Forms And Its Applications," Econometric Theory, Cambridge University Press, vol. 23(05), pages 930-951, October.
    9. 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.
    10. Stoker, Thomas M, 1986. "Consistent Estimation of Scaled Coefficients," Econometrica, Econometric Society, vol. 54(6), pages 1461-1481, November.
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    Cited by:

    1. Liu, Cheng & Sun, Yixiao, 2019. "A Simple and Trustworthy Asymptotic t Test in Difference-in-Differences Regressions," University of California at San Diego, Economics Working Paper Series qt0ck2109g, Department of Economics, UC San Diego.
    2. repec:eee:econom:v:210:y:2019:i:2:p:327-362 is not listed on IDEAS

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