The Annals of Statistics

Estimation of semivarying coefficient time series models with ARMA errors

Huang Lei, Yingcun Xia, and Xu Qin

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Abstract

Serial correlation in the residuals of time series models can cause bias in both model estimation and prediction. However, models with such serially correlated residuals are difficult to estimate, especially when the regression function is nonlinear. Existing estimation methods require strong assumption for the relation between the residuals and the regressors, which excludes the commonly used autoregressive models in time series analysis. By extending the Whittle likelihood estimation, this paper investigates in details a semi-parametric autoregressive model with ARMA sequence of residuals. Asymptotic normality of the estimators is established, and a model selection procedure is proposed. Numerical examples are employed to illustrate the performance of the proposed estimation method and the necessity of incorporating the serial correlation in the residuals.

Article information

Source
Ann. Statist., Volume 44, Number 4 (2016), 1618-1660.

Dates
Received: August 2015
Revised: December 2015
First available in Project Euclid: 7 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aos/1467894710

Digital Object Identifier
doi:10.1214/15-AOS1430

Mathematical Reviews number (MathSciNet)
MR3519935

Zentralblatt MATH identifier
1346.60020

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
ARMA process B-spline correlated errors semi-varying coefficient model spectral density function Whittle likelihood estimation

Citation

Lei, Huang; Xia, Yingcun; Qin, Xu. Estimation of semivarying coefficient time series models with ARMA errors. Ann. Statist. 44 (2016), no. 4, 1618--1660. doi:10.1214/15-AOS1430. https://projecteuclid.org/euclid.aos/1467894710


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References

  • Adams, R., Chen, C., McCarl, B. and Weiher, R. (1999). The economic consequences of ENSO events for agriculture. Clim. Res. 3 165–172.
  • Barrow, D. L. and Smith, P. W. (1978/79). Asymptotic properties of best $L_{2}[0,1]$ approximation by splines with variable knots. Quart. Appl. Math. 36 293–304.
  • Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • Cai, Z. (2007). Trending time-varying coefficient time series models with serially correlated errors. J. Econometrics 136 163–188.
  • Cai, Z., Fan, J. and Yao, Q. (2000). Functional-coefficient regression models for nonlinear time series. J. Amer. Statist. Assoc. 95 941–956.
  • Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477–489.
  • Chen, Z., Li, R. and Li, Y. (2015). Varying coefficient models for data with auto-correlated error process. Statist. Sinica 25 709–723.
  • Chen, R. and Tsay, R. S. (1993). Functional-coefficient autoregressive models. J. Amer. Statist. Assoc. 88 298–308.
  • Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.
  • Gao, J. (2007). Nonlinear Time Series: Semiparametric and Nonparametric Methods. Monographs on Statistics and Applied Probability 108. Chapman & Hall/CRC, Boca Raton, FL.
  • Giraitis, L. and Robinson, P. M. (2001). Whittle estimation of ARCH models. Econometric Theory 17 608–631.
  • Glantz, M. (2001). Currents of Change: Impacts of El Niño and La Niña on Climate and Society. Cambridge Univ Press, Cambridge.
  • Hall, P. and Van Keilegom, I. (2003). Using difference-based methods for inference in nonparametric regression with time series errors. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 443–456.
  • Hannan, E. J. (1973). The asymptotic theory of linear time-series models. J. Appl. Probab. 10 130–145, corrections, ibid. 10 (1973), 913.
  • Hart, J. D. (1991). Kernel regression estimation with time series errors. J. Roy. Statist. Soc. Ser. B 53 173–187.
  • Hastie, T. J. and Tibshirani, R. J. (1993). Varying-coefficient models (with discussion). J. Roy. Statist. Soc. Ser. B 55 757–796.
  • Huang, J. Z. and Yang, L. (2004). Identification of non-linear additive autoregressive models. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 463–477.
  • Li, Q., Huang, C. J., Li, D. and Fu, T. (2002). Semiparametric smooth coefficient models. J. Bus. Econom. Statist. 20 412–422.
  • Liu, J. M., Chen, R. and Yao, Q. (2010). Nonparametric transfer function models. J. Econometrics 157 151–164.
  • Ljung, G. M. and Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika 65 297–303.
  • Ma, S. and Yang, L. (2011). Spline-backfitted kernel smoothing of partially linear additive model. J. Statist. Plann. Inference 141 204–219.
  • Mahdi, E. and McLeod, A. I. (2012). Improved multivariate portmanteau test. J. Time Series Anal. 33 211–222.
  • McLeod, A. I. and Li, W. K. (1983). Diagnostic checking ARMA time series models using squared-residual autocorrelations. J. Time Series Anal. 4 269–273.
  • Opsomer, J., Wang, Y. and Yang, Y. (2001). Nonparametric regression with correlated errors. Statist. Sci. 16 134–153.
  • Pierce, D. A. (1971). Least squares estimation in the regression model with autoregressive-moving average errors. Biometrika 58 299–312.
  • Ray, B. K. and Tsay, R. S. (1997). Bandwidth selection for kernel regression with long-range dependent errors. Biometrika 84 791–802.
  • Severini, T. A. and Wong, W. H. (1992). Profile likelihood and conditionally parametric models. Ann. Statist. 20 1768–1802.
  • Su, L. and Ullah, A. (2006). More efficient estimation in nonparametric regression with nonparametric autocorrelated errors. Econometric Theory 22 98–126.
  • Tjøstheim, D. and Auestad, B. H. (1994). Nonparametric identification of nonlinear time series: Projections. J. Amer. Statist. Assoc. 89 1398–1409.
  • Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford Statistical Science Series 6. The Clarendon Press, New York.
  • Trenberth, K. and Stepaniak, D. (2001). Indices of El Niño evolution. J. Climate 14 1697–1701.
  • Ubilava, D. and Helmers, C. G. (2013). Forecasting ENSO with a smooth transition autoregressive model. Environ. Model. Softw. 40 181–190.
  • Wang, T. and Xia, Y. (2014). Whittle likelihood estimation of nonlinear autoregressive models with moving average residuals. J. Amer. Statist. Assoc. 110 1083–1099.
  • Whittle, P. (1953). The analysis of multiple stationary time series. J. Roy. Statist. Soc. Ser. B. 15 125–139.
  • Xiao, H. and Wu, W. B. (2012). Covariance matrix estimation for stationary time series. Ann. Statist. 40 466–493.
  • Xiao, Z., Linton, O. B., Carroll, R. J. and Mammen, E. (2003). More efficient local polynomial estimation in nonparametric regression with autocorrelated errors. J. Amer. Statist. Assoc. 98 980–992.
  • Xue, L. and Yang, L. (2006). Additive coefficient modeling via polynomial spline. Statist. Sinica 16 1423–1446.
  • Yao, Q. and Brockwell, P. J. (2006). Gaussian maximum likelihood estimation for ARMA models. I. Time series. J. Time Series Anal. 27 857–875.
  • Yu, B. (1994). Rates of convergence for empirical processes of stationary mixing sequences. Ann. Probab. 22 94–116.
  • Zhang, W., Lee, S. and Song, X. (2002). Local polynomial fitting in semivarying coefficient model. J. Multivariate Anal. 82 166–188.
  • Zhou, S., Shen, X. and Wolfe, D. A. (1998). Local asymptotics for regression splines and confidence regions. Ann. Statist. 26 1760–1782.
  • Zygmund, A. (1959). Trigonometric Series. Cambridge Univ. Press, Cambridge, UK.