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Non‐Parametric Estimation With Strongly Dependent Multivariate Time Series

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  • Javier Hidalgo

Abstract

Smooth non‐parametric kernel density and regression estimators are studied when the data are strongly dependent. In particular, we derive central (and non‐central) limit theorems for the kernel density estimator of a multivariate Gaussian process and an infinite‐order moving average of an independent identically distributed process, as well as the estimator's consistency for other types of data, such as non‐linear functions of a Gaussian process. We find that the kernel density estimator at two different points, under certain conditions, is not only perfectly correlated but may converge to the same random variable. Also, central (and non‐central) limit theorems of the non‐parametric kernel regression estimator are studied. One important and surprising characteristic found is that its asymptotic variance does not depend on the point at which the regression function is estimated and also that its asymptotic properties are the same whether or not regressors are strongly dependent. Finally, a Monte Carlo experiment is reported to assess the behaviour of the estimators in finite samples.

Suggested Citation

  • Javier Hidalgo, 1997. "Non‐Parametric Estimation With Strongly Dependent Multivariate Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 18(2), pages 95-122, March.
    • Handle: RePEc:bla:jtsera:v:18:y:1997:i:2:p:95-122
    DOI: 10.1111/1467-9892.00041
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    Cited by:

    1. Toshio Honda, 2000. "Nonparametric Density Estimation for a Long-Range Dependent Linear Process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 599-611, December.
    2. Linton, Oliver B. & Mammen, Enno, 2008. "Nonparametric transformation to white noise," Journal of Econometrics, Elsevier, vol. 142(1), pages 241-264, January.
    3. Hidalgo, J., 2008. "Specification testing for regression models with dependent data," Journal of Econometrics, Elsevier, vol. 143(1), pages 143-165, March.
    4. Toshio Honda, 2009. "Nonparametric density estimation for linear processes with infinite variance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(2), pages 413-439, June.
    5. Jens Klotsche & Andrew T. Gloster, 2012. "Estimating a Meaningful Point of Change," Journal of Educational and Behavioral Statistics, , vol. 37(5), pages 579-600, October.
    6. Masry, Elias & Mielniczuk, Jan, 1999. "Local linear regression estimation for time series with long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 82(2), pages 173-193, August.
    7. Benhenni, K. & Hedli-Griche, S. & Rachdi, M. & Vieu, P., 2008. "Consistency of the regression estimator with functional data under long memory conditions," Statistics & Probability Letters, Elsevier, vol. 78(8), pages 1043-1049, June.
    8. Youndjé, É. & Vieu, P., 2006. "A note on quantile estimation for long-range dependent stochastic processes," Statistics & Probability Letters, Elsevier, vol. 76(2), pages 109-116, January.
    9. Toshio Honda, 2010. "Nonparametric estimation of conditional medians for linear and related processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(6), pages 995-1021, December.
    10. Javier Hidalgo, 2007. "Specification Testing Forregression Models Withdependent Data," STICERD - Econometrics Paper Series 518, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    11. Trino-Manuel Ñíguez, 2003. "Volatility And Var Forecasting For The Ibex-35 Stock-Return Index Using Figarch-Type Processes And Different Evaluation Criteria," Working Papers. Serie AD 2003-33, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    12. Lihong Wang & Haiyan Cai, 2010. "Wavelet change‐point estimation for long memory non‐parametric random design models," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(2), pages 86-97, March.
    13. Hidalgo, Javier, 2007. "Specification testing for regression models with dependent data," LSE Research Online Documents on Economics 6799, London School of Economics and Political Science, LSE Library.

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