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Modeling covariance matrices via partial autocorrelations

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  • Daniels, M.J.
  • Pourahmadi, M.

Abstract

We study the role of partial autocorrelations in the reparameterization and parsimonious modeling of a covariance matrix. The work is motivated by and tries to mimic the phenomenal success of the partial autocorrelations function (PACF) in model formulation, removing the positive-definiteness constraint on the autocorrelation function of a stationary time series and in reparameterizing the stationarity-invertibility domain of ARMA models. It turns out that once an order is fixed among the variables of a general random vector, then the above properties continue to hold and follow from establishing a one-to-one correspondence between a correlation matrix and its associated matrix of partial autocorrelations. Connections between the latter and the parameters of the modified Cholesky decomposition of a covariance matrix are discussed. Graphical tools similar to partial correlograms for model formulation and various priors based on the partial autocorrelations are proposed. We develop frequentist/Bayesian procedures for modelling correlation matrices, illustrate them using a real dataset, and explore their properties via simulations.

Suggested Citation

  • Daniels, M.J. & Pourahmadi, M., 2009. "Modeling covariance matrices via partial autocorrelations," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2352-2363, November.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:10:p:2352-2363
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    5. Guney, Yesim & Arslan, Olcay & Yavuz, Fulya Gokalp, 2022. "Robust estimation in multivariate heteroscedastic regression models with autoregressive covariance structures using EM algorithm," Journal of Multivariate Analysis, Elsevier, vol. 191(C).
    6. Smith, Michael Stanley, 2015. "Copula modelling of dependence in multivariate time series," International Journal of Forecasting, Elsevier, vol. 31(3), pages 815-833.
    7. Hui Yao & Sungduk Kim & Ming-Hui Chen & Joseph G. Ibrahim & Arvind K. Shah & Jianxin Lin, 2015. "Bayesian Inference for Multivariate Meta-Regression With a Partially Observed Within-Study Sample Covariance Matrix," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 528-544, June.
    8. Li Su & Sarah Brown & Pulak Ghosh & Karl Taylor, 2012. "Modelling Household Debt and Financial Assets: A Bayesian Approach to a Bivariate Two-Part Model," Working Papers 2012009, The University of Sheffield, Department of Economics.
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    10. Lee, Keunbaik & Baek, Changryong & Daniels, Michael J., 2017. "ARMA Cholesky factor models for the covariance matrix of linear models," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 267-280.
    11. Peter J. Danaher & Michael S. Smith, 2011. "Rejoinder--Estimation Issues for Copulas Applied to Marketing Data," Marketing Science, INFORMS, vol. 30(1), pages 25-28, 01-02.
    12. Brown, Sarah & Ghosh, Pulak & Su, Li & Taylor, Karl, 2015. "Modelling household finances: A Bayesian approach to a multivariate two-part model," Journal of Empirical Finance, Elsevier, vol. 33(C), pages 190-207.
    13. Luigi Spezia, 2019. "Modelling covariance matrices by the trigonometric separation strategy with application to hidden Markov models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 399-422, June.
    14. Carter, Christopher K. & Wong, Frederick & Kohn, Robert, 2011. "Constructing priors based on model size for nondecomposable Gaussian graphical models: A simulation based approach," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 871-883, May.
    15. Daniel F. Schmidt & Enes Makalic, 2013. "Estimation of stationary autoregressive models with the Bayesian LASSO," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(5), pages 517-531, September.

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