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On The Partial Sums Of Residuals In Autoregressive And Moving Average Models

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  • Jushan Bai

Abstract

. The limiting process of partial sums of residuals in stationary and invertible autoregressive moving‐average models is studied. It is shown that the partial sums converge to a standard Brownian motion under the assumptions that estimators of unknown parameters are root‐n consistent and that innovations are independent and identically distributed random variables with zero mean and finite variance or, more generally, are martingale differences with moment restrictions specified in Theorem 1. Applications for goodness‐of‐fit and change‐point problems are considered. The use of residuals for constructing nonparametric density estimation is discussed.

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  • Jushan Bai, 1993. "On The Partial Sums Of Residuals In Autoregressive And Moving Average Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 14(3), pages 247-260, May.
    • Handle: RePEc:bla:jtsera:v:14:y:1993:i:3:p:247-260
    DOI: 10.1111/j.1467-9892.1993.tb00142.x
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    Cited by:

    1. Shiqing Ling & Michael McAleer, 2001. "On Adaptive Estimation in Nonstationary ARMA Models with GARCH Errors," ISER Discussion Paper 0548, Institute of Social and Economic Research, Osaka University.
    2. Hao Yu, 2003. "Hierarchical equilibria of branching populations," RePAd Working Paper Series lrsp-TRS391, Département des sciences administratives, UQO.
    3. Zhang, Rong-Mao & Sin, Chor-yiu (CY) & Ling, Shiqing, 2015. "On functional limits of short- and long-memory linear processes with GARCH(1,1) noises," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 482-512.
    4. Christopher Dienes & Alexander Aue, 2014. "On-Line Monitoring Of Pollution Concentrations With Autoregressive Moving Average Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(3), pages 239-261, May.
    5. Mo Li & QiQi Lu, 2022. "Changepoint detection in autocorrelated ordinal categorical time series," Environmetrics, John Wiley & Sons, Ltd., vol. 33(7), November.
    6. Shi, Xuesheng & Gallagher, Colin & Lund, Robert & Killick, Rebecca, 2022. "A comparison of single and multiple changepoint techniques for time series data," Computational Statistics & Data Analysis, Elsevier, vol. 170(C).
    7. Zdeněk Hlávka & Marie Hušková & Claudia Kirch & Simos Meintanis, 2012. "Monitoring changes in the error distribution of autoregressive models based on Fourier methods," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(4), pages 605-634, December.
    8. Park, Chul Gyu & Shin, Dong Wan, 1996. "On the asymptotics of residuals in autoregressive moving average processes with one autoregressive unit root," Statistics & Probability Letters, Elsevier, vol. 27(4), pages 341-346, May.
    9. Husková, M., 2003. "Serial rank statistics for detection of changes," Statistics & Probability Letters, Elsevier, vol. 61(2), pages 199-213, January.

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