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Asymptotic normality of residual density estimator in stationary and explosive autoregressive models

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  • Gao, Min
  • Yang, Wenzhi
  • Wu, Shipeng
  • Yu, Wei

Abstract

The error density estimator in the first-order autoregressive model is considered based on α-mixing errors. Since the errors are not observed, the residual kernel density estimator is provided. The asymptotic normality of the residual estimator is obtained when the autoregressive model is a stationary process or an explosive process. Moreover, some simulations such as the fitted curves, mean integrated square errors and histograms are illustrated to the residual kernel estimator and residual histogram estimator. It is shown that the residual kernel estimator with smooth kernel is smoother than the residual histogram estimator.

Suggested Citation

  • Gao, Min & Yang, Wenzhi & Wu, Shipeng & Yu, Wei, 2022. "Asymptotic normality of residual density estimator in stationary and explosive autoregressive models," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    • Handle: RePEc:eee:csdana:v:175:y:2022:i:c:s0167947322001293
    DOI: 10.1016/j.csda.2022.107549
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    1. Roussas, George G., 1990. "Nonparametric regression estimation under mixing conditions," Stochastic Processes and their Applications, Elsevier, vol. 36(1), pages 107-116, October.
    2. Liangjun Su & Xi Qu, 2017. "Specification Test for Spatial Autoregressive Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(4), pages 572-584, October.
    3. Cheng, Fuxia, 2015. "Strong consistency of the distribution estimator in the nonlinear autoregressive time series," Journal of Multivariate Analysis, Elsevier, vol. 142(C), pages 41-47.
    4. Liang, Han-Ying & Peng, Liang, 2010. "Asymptotic normality and Berry-Esseen results for conditional density estimator with censored and dependent data," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1043-1054, May.
    5. Aue, Alexander & Horváth, Lajos, 2007. "A Limit Theorem For Mildly Explosive Autoregression With Stable Errors," Econometric Theory, Cambridge University Press, vol. 23(2), pages 201-220, April.
    6. Yu, Dalei & Bai, Peng & Ding, Chang, 2015. "Adjusted quasi-maximum likelihood estimator for mixed regressive, spatial autoregressive model and its small sample bias," Computational Statistics & Data Analysis, Elsevier, vol. 87(C), pages 116-135.
    7. Sun, Yanqing & Zhang, Yuanqing & Huang, Jianhua Z., 2019. "Estimation of a semiparametric varying-coefficient mixed regressive spatial autoregressive model," Econometrics and Statistics, Elsevier, vol. 9(C), pages 140-155.
    8. Amiri, Aboubacar & Dabo-Niang, Sophie, 2018. "Density estimation over spatio-temporal data streams," Econometrics and Statistics, Elsevier, vol. 5(C), pages 148-170.
    9. Eckhard Liebscher, 1999. "Estimating the Density of the Residuals in Autoregressive Models," Statistical Inference for Stochastic Processes, Springer, vol. 2(2), pages 105-117, May.
    10. Cai, Zongwu & Roussas, George G., 1992. "Uniform strong estimation under [alpha]-mixing, with rates," Statistics & Probability Letters, Elsevier, vol. 15(1), pages 47-55, September.
    11. Zudi Lu, 2001. "Asymptotic Normality of Kernel Density Estimators under Dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 447-468, September.
    12. Cheng, Fuxia, 2018. "Glivenko–Cantelli Theorem for the kernel error distribution estimator in the first-order autoregressive model," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 95-102.
    13. Liang, Han-Ying & de Ua-lvarez, Jacobo, 2009. "A Berry-Esseen type bound in kernel density estimation for strong mixing censored samples," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1219-1231, July.
    14. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
    15. Wang, Xiaohu & Yu, Jun, 2015. "Limit theory for an explosive autoregressive process," Economics Letters, Elsevier, vol. 126(C), pages 176-180.
    16. Bachmann, Dirk & Dette, Holger, 2005. "A note on the Bickel-Rosenblatt test in autoregressive time series," Statistics & Probability Letters, Elsevier, vol. 74(3), pages 221-234, October.
    17. Funke, Benedikt & Hirukawa, Masayuki, 2019. "Nonparametric estimation and testing on discontinuity of positive supported densities: a kernel truncation approach," Econometrics and Statistics, Elsevier, vol. 9(C), pages 156-170.
    18. Lung-Fei Lee, 2004. "Asymptotic Distributions of Quasi-Maximum Likelihood Estimators for Spatial Autoregressive Models," Econometrica, Econometric Society, vol. 72(6), pages 1899-1925, November.
    19. Wenzhi Yang & Yiwei Wang & Shuhe Hu, 2017. "Some probability inequalities of least-squares estimator in non linear regression model with strong mixing errors," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(1), pages 165-175, January.
    20. Fuxia Cheng, 2010. "Global property of error density estimation in nonlinear autoregressive time series models," Statistical Inference for Stochastic Processes, Springer, vol. 13(1), pages 43-53, April.
    21. Laib, Naâmane & Louani, Djamal, 2010. "Nonparametric kernel regression estimation for functional stationary ergodic data: Asymptotic properties," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2266-2281, November.
    22. Ma, Yingying & Lan, Wei & Zhou, Fanying & Wang, Hansheng, 2020. "Approximate least squares estimation for spatial autoregressive models with covariates," Computational Statistics & Data Analysis, Elsevier, vol. 143(C).
    23. Roussas, George G., 2000. "Asymptotic normality of the kernel estimate of a probability density function under association," Statistics & Probability Letters, Elsevier, vol. 50(1), pages 1-12, October.
    24. Cheng, Fuxia & Sun, Shuxia, 2008. "A goodness-of-fit test of the errors in nonlinear autoregressive time series models," Statistics & Probability Letters, Elsevier, vol. 78(1), pages 50-59, January.
    25. Horváth, Lajos & Zitikis, Ricardas, 2004. "Asymptotics of the Lp-norms of density estimators in the first-order autoregressive models," Statistics & Probability Letters, Elsevier, vol. 66(2), pages 91-103, January.
    26. Chan, Ngai Hang & Li, Deyuan & Peng, Liang, 2012. "Toward A Unified Interval Estimation Of Autoregressions," Econometric Theory, Cambridge University Press, vol. 28(3), pages 705-717, June.
    27. Laïb, Naâmane & Louani, Djamal, 2019. "Asymptotic normality of kernel density function estimator from continuous time stationary and dependent processes," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 187-196.
    28. Wang, Qiying & Lin, Yan-Xia & Gulati, Chandra M., 2002. "The Invariance Principle For Linear Processes With Applications," Econometric Theory, Cambridge University Press, vol. 18(1), pages 119-139, February.
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