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On the validity of Akaike’s identity for random fields

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  • Jentsch, Carsten
  • Meyer, Marco

Abstract

For univariate stationary and centered time series (Xt)t∈Z, Akaike’s identity links the inverse of the Yule–Walker matrix Γ(p)=E(XX′), where X=(Xt−1,…,Xt−p)′, to the corresponding finite predictor coefficients. It reads as a Cholesky-type factorization Γ(p)−1=L(p)′Σ(p)−1L(p), where L(p) is lower-triangular and Σ(p)−1 is diagonal. Whereas this Cholesky-type factorization exists whenever Γ(p) is positive definite, Akaike derived a meaningful interpretation of L(p) and Σ(p)−1 in terms of finite predictor coefficients. It is useful in many applications and is particularly crucial to derive asymptotic theory for Berk’s spectral density estimator.

Suggested Citation

  • Jentsch, Carsten & Meyer, Marco, 2021. "On the validity of Akaike’s identity for random fields," Journal of Econometrics, Elsevier, vol. 222(1), pages 676-687.
    • Handle: RePEc:eee:econom:v:222:y:2021:i:1:p:676-687
    DOI: 10.1016/j.jeconom.2020.04.044
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    3. Robinson, P.M. & Vidal Sanz, J., 2006. "Modified Whittle estimation of multilateral models on a lattice," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1090-1120, May.
    4. Gupta, Abhimanyu, 2018. "Autoregressive spatial spectral estimates," Journal of Econometrics, Elsevier, vol. 203(1), pages 80-95.
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    Cited by:

    1. Suhasini Subba Rao & Junho Yang, 2023. "A prediction perspective on the Wiener–Hopf equations for time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 44(1), pages 23-42, January.

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    More about this item

    Keywords

    Finite predictor coefficients; L2-projection; Random fields; Yule–Walker equations;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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