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Variance–covariance from a metropolis chain on a curved, singular manifold

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  • Gallant, A. Ronald

Abstract

We consider estimation of variance and covariance from a point cloud that are draws from a posterior distribution that lie on a curved, singular manifold. The motivating application is Bayesian inference regarding a likelihood subject to overidentified moment equations using MCMC (Markov Chain Monte Carlo). The MCMC draws lie on a singular manifold that typically is curved. Variance and covariance are Euclidean concepts. A curved, singular manifold is not typically a Euclidean space. We explore some suggestions on how to adapt a Euclidean concept to a non-Euclidean space then build on them to propose and illustrate appropriate methods.

Suggested Citation

  • Gallant, A. Ronald, 2023. "Variance–covariance from a metropolis chain on a curved, singular manifold," Journal of Econometrics, Elsevier, vol. 235(2), pages 843-861.
    • Handle: RePEc:eee:econom:v:235:y:2023:i:2:p:843-861
    DOI: 10.1016/j.jeconom.2022.08.002
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    References listed on IDEAS

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    1. Simon Byrne & Mark Girolami, 2013. "Geodesic Monte Carlo on Embedded Manifolds," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 825-845, December.
    2. Gallant, Ronald & Tauchen, George, 1989. "Seminonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications," Econometrica, Econometric Society, vol. 57(5), pages 1091-1120, September.
    3. Gallant, A. Ronald, 2022. "Nonparametric Bayes subject to overidentified moment conditions," Journal of Econometrics, Elsevier, vol. 228(1), pages 27-38.
    4. Gallant, A Ronald & Nychka, Douglas W, 1987. "Semi-nonparametric Maximum Likelihood Estimation," Econometrica, Econometric Society, vol. 55(2), pages 363-390, March.
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    More about this item

    Keywords

    Method of moments; Bayesian inference; Simultaneously valid credibility intervals; Point cloud; Curved; Singular manifold;
    All these keywords.

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C36 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Instrumental Variables (IV) Estimation
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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