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Estimating wold matrices and vector moving average processes

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  • Jonas Krampe
  • Timothy L. McMurry

Abstract

The Wold decomposition gives a moving average (MA) representation of a purely non‐deterministic stationary process. In this article, we derive estimates of the Wold matrices for a d‐dimensional process by using a Cholesky decomposition of a banded and tapered version of the sample autocovariance matrix, and we derive convergence rates for the estimation error of the (possibly infinite) sequence of Wold matrices. By analogy to lag‐window estimates of the spectral density, this method can be used to obtain finite vector MA models with an adaptive lag‐order. We additionally show how these results can be applied to impulse response analysis and to derive a bootstrap procedure. Our theoretical results are complemented by simulations which investigate the finite sample performance of the estimator.

Suggested Citation

  • Jonas Krampe & Timothy L. McMurry, 2021. "Estimating wold matrices and vector moving average processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(2), pages 201-221, March.
    • Handle: RePEc:bla:jtsera:v:42:y:2021:i:2:p:201-221
    DOI: 10.1111/jtsa.12562
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    References listed on IDEAS

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    1. Timothy L. McMurry & Dimitris N. Politis, 2018. "Estimating MA Parameters through Factorization of the Autocovariance Matrix and an MA†Sieve Bootstrap," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(3), pages 433-446, May.
    2. 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.
    3. Heather Mitchell & Peter Brockwell, 1997. "Estimation Of The Coefficients Of A Multivariate Linear Filter Using The Innovations Algorithm," Journal of Time Series Analysis, Wiley Blackwell, vol. 18(2), pages 157-179, March.
    4. Takemura, Akimichi, 2016. "Exponential decay rate of partial autocorrelation coefficients of ARMA and short-memory processes," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 207-210.
    5. Kilian,Lutz & Lütkepohl,Helmut, 2018. "Structural Vector Autoregressive Analysis," Cambridge Books, Cambridge University Press, number 9781107196575, September.
    6. Jonas Krampe & Jens‐Peter Kreiss & Efstathios Paparoditis, 2018. "Estimated Wold representation and spectral‐density‐driven bootstrap for time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(4), pages 703-726, September.
    7. 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.
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