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A Durbin-Levinson Regularized Estimator of High Dimensional Autocovariance Matrices

Author

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  • Tommaso Proietti

    (Università di Roma "Tor Vergata" and CREATES)

  • Alessandro Giovannelli

    (Università di Roma "Tor Vergata")

Abstract

We consider the problem of estimating the high-dimensional autocovariance matrix of a stationary random process, with the purpose of out of sample prediction and feature extraction. This problem has received several solutions. In the nonparametric framework, the literature has concentrated on banding and tapering the sample autocovariance matrix. This paper proposes and evaluates an alternative approach, based on regularizing the sample partial autocorrelation function, via a modified Durbin-Levinson algorithm that receives as input the banded and tapered partial autocorrelations and returns a sample autocovariance sequence which is positive definite. We show that the regularized estimator of the autocovariance matrix is consistent and its convergence rates is established. We then focus on constructing the optimal linear predictor and we assess its properties. The computational complexity of the estimator is of the order of the square of the banding parameter, which renders our method scalable for high-dimensional time series. The performance of the autocovariance estimator and the corresponding linear predictor is evaluated by simulation and empirical applications.

Suggested Citation

  • Tommaso Proietti & Alessandro Giovannelli, 2017. "A Durbin-Levinson Regularized Estimator of High Dimensional Autocovariance Matrices," CREATES Research Papers 2017-20, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2017-20
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    References listed on IDEAS

    as
    1. Stefano, Fasani, 2016. "Long-run Unemployment and Macroeconomic Volatility," Working Papers 352, University of Milano-Bicocca, Department of Economics, revised 18 Oct 2016.
    2. Lutkepohl, Helmut & Saikkonen, Pentti, 1997. "Impulse response analysis in infinite order cointegrated vector autoregressive processes," Journal of Econometrics, Elsevier, vol. 81(1), pages 127-157, November.
    3. Silvia Gonçalves & Lutz Kilian, 2003. "Asymptotic and Bootstrap Inference for AR( Infinite ) Processes with Conditional Heteroskedasticity," CIRANO Working Papers 2003s-28, CIRANO.
    4. Lewis, Richard & Reinsel, Gregory C., 1985. "Prediction of multivariate time series by autoregressive model fitting," Journal of Multivariate Analysis, Elsevier, vol. 16(3), pages 393-411, June.
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    8. Datta Gupta, Syamantak & Mazumdar, Ravi R. & Glynn, Peter, 2013. "On the convergence of the spectrum of finite order approximations of stationary time series," Journal of Multivariate Analysis, Elsevier, vol. 121(C), pages 1-21.
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    10. McMurry, Timothy L & Politis, D N, 2010. "Banded and Tapered Estimates for Autocovariance Matrices and the Linear Process Bootstrap," University of California at San Diego, Economics Working Paper Series qt5h9259mb, Department of Economics, UC San Diego.
    11. Timothy L. McMurry & Dimitris N. Politis, 2010. "Banded and tapered estimates for autocovariance matrices and the linear process bootstrap," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(6), pages 471-482, November.
    12. Saikkonen, Pentti & Lütkepohl, HELMUT, 1996. "Infinite-Order Cointegrated Vector Autoregressive Processes," Econometric Theory, Cambridge University Press, vol. 12(5), pages 814-844, December.
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    Cited by:

    1. Serge B. Provost & John N. Haddad, 2019. "A recursive approach for determining matrix inverses as applied to causal time series processes," METRON, Springer;Sapienza Università di Roma, vol. 77(1), pages 53-62, April.
    2. Proietti, Tommaso & Maddanu, Federico, 2024. "Modelling cycles in climate series: The fractional sinusoidal waveform process," Journal of Econometrics, Elsevier, vol. 239(1).

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    More about this item

    Keywords

    Toeplitz systems; Optimal linear prediction; Partial autocorrelation function;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • C55 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Large Data Sets: Modeling and Analysis

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