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Using difference‐based methods for inference in nonparametric regression with time series errors

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  • Peter Hall
  • Ingrid Van Keilegom

Abstract

Summary. We show that difference‐based methods can be used to construct simple and explicit estimators of error covariance and autoregressive parameters in nonparametric regression with time series errors. When the error process is Gaussian our estimators are efficient, but they are available well beyond the Gaussian case. As an illustration of their usefulness we show that difference‐based estimators can be used to produce a simplified version of time series cross‐validation. This new approach produces a bandwidth selector that is equivalent, to both first and second orders, to that given by the full time series cross‐validation algorithm. Other applications of difference‐based methods are to variance estimation and construction of confidence bands in nonparametric regression.

Suggested Citation

  • Peter Hall & Ingrid Van Keilegom, 2003. "Using difference‐based methods for inference in nonparametric regression with time series errors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 443-456, May.
    • Handle: RePEc:bla:jorssb:v:65:y:2003:i:2:p:443-456
    DOI: 10.1111/1467-9868.00395
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    Cited by:

    1. Huang, Lei & Jiang, Hui & Wang, Huixia, 2019. "A novel partial-linear single-index model for time series data," Computational Statistics & Data Analysis, Elsevier, vol. 134(C), pages 110-122.
    2. T. Subba Rao & Gyorgy Terdik, 2017. "A New Covariance Function and Spatio-Temporal Prediction (Kriging) for A Stationary Spatio-Temporal Random Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(6), pages 936-959, November.
    3. Kim, Tae Yoon & Park, Byeong U. & Moon, Myung Sang & Kim, Chiho, 2009. "Using bimodal kernel for inference in nonparametric regression with correlated errors," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1487-1497, August.
    4. Yuejin Zhou & Yebin Cheng & Wenlin Dai & Tiejun Tong, 2018. "Optimal difference-based estimation for partially linear models," Computational Statistics, Springer, vol. 33(2), pages 863-885, June.
    5. Krivobokova, Tatyana & Serra, Paulo & Rosales, Francisco & Klockmann, Karolina, 2022. "Joint non-parametric estimation of mean and auto-covariances for Gaussian processes," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    6. Huan Wang & Mary C. Meyer & Jean D. Opsomer, 2013. "Constrained spline regression in the presence of AR(p) errors," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(4), pages 809-827, December.
    7. A. Pérez-González & J. Vilar-Fernández & W. González-Manteiga, 2009. "Asymptotic properties of local polynomial regression with missing data and correlated errors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(1), pages 85-109, March.
    8. Wei, Honglei & Zhang, Hongfan & Jiang, Hui & Huang, Lei, 2022. "On the semi-varying coefficient dynamic panel data model with autocorrelated errors," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    9. Inder Tecuapetla-Gómez & Axel Munk, 2017. "Autocovariance Estimation in Regression with a Discontinuous Signal and m-Dependent Errors: A Difference-Based Approach," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(2), pages 346-368, June.
    10. Zhong, Chen, 2024. "Oracle-efficient estimation and trend inference in non-stationary time series with trend and heteroscedastic ARMA error," Computational Statistics & Data Analysis, Elsevier, vol. 193(C).
    11. Lyubchich, Vyacheslav & Gel, Yulia R., 2016. "A local factor nonparametric test for trend synchronism in multiple time series," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 91-104.
    12. K De Brabanter & F Cao & I Gijbels & J Opsomer, 2018. "Local polynomial regression with correlated errors in random design and unknown correlation structure," Biometrika, Biometrika Trust, vol. 105(3), pages 681-690.
    13. Qiu, D. & Shao, Q. & Yang, L., 2013. "Efficient inference for autoregressive coefficients in the presence of trends," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 40-53.
    14. Dalla, Violetta & Giraitis, Liudas & Robinson, Peter M., 2020. "Asymptotic theory for time series with changing mean and variance," Journal of Econometrics, Elsevier, vol. 219(2), pages 281-313.
    15. Q. Shao, 2023. "Simultaneous Confidence Band Approach for Comparison of COVID-19 Case Counts," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 15(2), pages 372-383, July.
    16. Qin Shao & Lijian Yang, 2017. "Oracally efficient estimation and consistent model selection for auto-regressive moving average time series with trend," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 507-524, March.

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