Bernoulli

  • Bernoulli
  • Volume 22, Number 3 (2016), 1301-1330.

Estimation of inverse autocovariance matrices for long memory processes

Ching-Kang Ing, Hai-Tang Chiou, and Meihui Guo

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Abstract

This work aims at estimating inverse autocovariance matrices of long memory processes admitting a linear representation. A modified Cholesky decomposition is used in conjunction with an increasing order autoregressive model to achieve this goal. The spectral norm consistency of the proposed estimate is established. We then extend this result to linear regression models with long-memory time series errors. In particular, we show that when the objective is to consistently estimate the inverse autocovariance matrix of the error process, the same approach still works well if the estimated (by least squares) errors are used in place of the unobservable ones. Applications of this result to estimating unknown parameters in the aforementioned regression model are also given. Finally, a simulation study is performed to illustrate our theoretical findings.

Article information

Source
Bernoulli, Volume 22, Number 3 (2016), 1301-1330.

Dates
Received: December 2013
Revised: November 2014
First available in Project Euclid: 16 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1458132983

Digital Object Identifier
doi:10.3150/14-BEJ692

Mathematical Reviews number (MathSciNet)
MR3474817

Zentralblatt MATH identifier
06579698

Keywords
inverse autocovariance matrix linear regression model long memory process modified Cholesky decomposition

Citation

Ing, Ching-Kang; Chiou, Hai-Tang; Guo, Meihui. Estimation of inverse autocovariance matrices for long memory processes. Bernoulli 22 (2016), no. 3, 1301--1330. doi:10.3150/14-BEJ692. https://projecteuclid.org/euclid.bj/1458132983


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References

  • [1] Berk, K.N. (1974). Consistent autoregressive spectral estimates. Ann. Statist. 2 489–502.
  • [2] Bickel, P.J. and Gel, Y.R. (2011). Banded regularization of autocovariance matrices in application to parameter estimation and forecasting of time series. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 711–728.
  • [3] Bickel, P.J. and Levina, E. (2008). Regularized estimation of large covariance matrices. Ann. Statist. 36 199–227.
  • [4] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [5] Cai, T.T., Ren, Z. and Zhou, H.H. (2013). Optimal rates of convergence for estimating Toeplitz covariance matrices. Probab. Theory Related Fields 156 101–143.
  • [6] Cheng, T.-C.F., Ing, C.-K. and Yu, S.-H. (2015). Toward optimal model averaging in regression models with time series errors. J. Econometrics. To appear.
  • [7] Findley, D.F. and Wei, C.Z. (1993). Moment bounds for deriving time series CLTs and model selection procedures. Statist. Sinica 3 453–480.
  • [8] Godet, F. (2010). Prediction of long memory processes on same-realisation. J. Statist. Plann. Inference 140 907–926.
  • [9] Ing, C.-K. and Wei, C.-Z. (2003). On same-realization prediction in an infinite-order autoregressive process. J. Multivariate Anal. 85 130–155.
  • [10] Ing, C.-K. and Wei, C.-Z. (2006). A maximal moment inequality for long range dependent time series with applications to estimation and model selection. Statist. Sinica 16 721–740.
  • [11] Inoue, A. and Kasahara, Y. (2006). Explicit representation of finite predictor coefficients and its applications. Ann. Statist. 34 973–993.
  • [12] Kokoszka, P.S. and Taqqu, M.S. (1995). Fractional ARIMA with stable innovations. Stochastic Process. Appl. 60 19–47.
  • [13] McMurry, T.L. and Politis, D.N. (2010). Banded and tapered estimates for autocovariance matrices and the linear process bootstrap. J. Time Series Anal. 31 471–482.
  • [14] Palma, W. and Pourahmadi, M. (2012). Banded regularization and prediction of long-memory time series. Working paper.
  • [15] Shibata, R. (1980). Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Ann. Statist. 8 147–164.
  • [16] Wei, C.Z. (1987). Adaptive prediction by least squares predictors in stochastic regression models with applications to time series. Ann. Statist. 15 1667–1682.
  • [17] Wu, W.B., Michailidis, G. and Zhang, D. (2004). Simulating sample paths of linear fractional stable motion. IEEE Trans. Inform. Theory 50 1086–1096.
  • [18] Wu, W.B. and Pourahmadi, M. (2003). Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika 90 831–844.
  • [19] Wu, W.B. and Pourahmadi, M. (2009). Banding sample autocovariance matrices of stationary processes. Statist. Sinica 19 1755–1768.
  • [20] Xiao, H. and Wu, W.B. (2012). Covariance matrix estimation for stationary time series. Ann. Statist. 40 466–493.
  • [21] Yajima, Y. (1988). On estimation of a regression model with long-memory stationary errors. Ann. Statist. 16 791–807.
  • [22] Yajima, Y. (1991). Asymptotic properties of the LSE in a regression model with long-memory stationary errors. Ann. Statist. 19 158–177.