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The information matrix of time-dependent models for vector time series

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  • Mélard, Guy

Abstract

The information matrix is essential for inference in time series models because it provides the asymptotic covariance matrix of maximum likelihood estimators. There are very good results for stationary and invertible scalar and vector autoregressive-moving average (VARMA) models, but the situation is different for VARMA models with time-dependent (td) coefficients or tdVARMA models, where a limit of an average is needed instead of a constant matrix. Here, (marginal) heteroscedasticity is also considered, so that the outer product of gradients W is required in addition to the Hessian V to form the so-called sandwich covariance matrix V−1WV−1. In this paper, the properties of V and W are established for different model specifications, homoscedastic or heteroscedastic, Gaussian process or not.

Suggested Citation

  • Mélard, Guy, 2025. "The information matrix of time-dependent models for vector time series," Statistics & Probability Letters, Elsevier, vol. 226(C).
    • Handle: RePEc:eee:stapro:v:226:y:2025:i:c:s0167715225001622
    DOI: 10.1016/j.spl.2025.110517
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    References listed on IDEAS

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    1. Abdelkamel Alj & Rajae Azrak & Guy Mélard, 2025. "General estimation results for tdVARMA array models," Journal of Time Series Analysis, Wiley Blackwell, vol. 46(1), pages 137-151, January.
    2. Abdelouahab Bibi & Christian Francq, 2003. "Consistent and asymptotically normal estimators for cyclically time-dependent linear models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(1), pages 41-68, March.
    3. David A. Harville, 1997. "Matrix Algebra From a Statistician’s Perspective," Springer Books, Springer, number 978-0-387-22677-4, December.
    4. Van Bellegem, Sebastien & von Sachs, Rainer, 2004. "Forecasting economic time series with unconditional time-varying variance," International Journal of Forecasting, Elsevier, vol. 20(4), pages 611-627.
    5. Abdelkamel Alj & Rajae Azrak & Christophe Ley & Guy Mélard, 2017. "Asymptotic Properties of QML Estimators for VARMA Models with Time-dependent Coefficients," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(3), pages 617-635, September.
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