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Bayesian Simultaneous Equations Analysis Using Reduced Rank Structures

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Author Info
Kleibergen, Frank
van Dijk, Herman K.

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Abstract

Diffuse priors lead to pathological posterior behavior when used in Bayesian analyses of simultaneous equation models (SEM s). This results from the local nonidentification of certain parameters in SEM s. When this a priori known feature is not captured appropriately, it results in an a posteriori favoring of certain specific parameter values that is not the consequence of strong data information but of local nonidentification. We show that a proper consistent Bayesian analysis of a SEM explicitly has to consider the reduced form of the SEM as a standard linear model on which nonlinear (reduced rank) restrictions are imposed, which result from a singular value decomposition. The priors/posteriors of the parameters of the SEM are therefore proportional to the priors/posteriors of the parameters of the linear model under the condition that the restrictions hold. This leads to a framework for constructing priors and posteriors for the parameters of SEM s. The framework is used to construct priors and posteriors for one, two, and three structural equation SEM s. These examples together with a theorem, showing that the reduced forms of SEM s accord with sets of reduced rank restrictions on standard linear models, show how Bayesian analyses of generally specified SEM s can be conducted.

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Publisher Info
Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 14 (1998)
Issue (Month): 06 (December)
Pages: 701-743
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Handle: RePEc:cup:etheor:v:14:y:1998:i:06:p:701-743_14

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Maddala, G S, 1976. "Weak Priors and Sharp Posteriors in Simultaneous Equation Models," Econometrica, Econometric Society, vol. 44(2), pages 345-51, March. [Downloadable!] (restricted)
  2. Geweke, John, 1989. "Bayesian Inference in Econometric Models Using Monte Carlo Integration," Econometrica, Econometric Society, vol. 57(6), pages 1317-39, November. [Downloadable!] (restricted)
  3. Zellner, Arnold & Bauwens, Luc & Van Dijk, Herman K., 1988. "Bayesian specification analysis and estimation of simultaneous equation models using Monte Carlo methods," Journal of Econometrics, Elsevier, vol. 38(1-2), pages 39-72. [Downloadable!] (restricted)
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  4. Geweke, John, 1996. "Bayesian reduced rank regression in econometrics," Journal of Econometrics, Elsevier, vol. 75(1), pages 121-146, November. [Downloadable!] (restricted)
  5. Dreze, Jacques H. & Richard, Jean-Francois, 1983. "Bayesian analysis of simultaneous equation systems," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 1, chapter 9, pages 517-598 Elsevier. [Downloadable!] (restricted)
  6. Kloek, Tuen & van Dijk, Herman K, 1978. "Bayesian Estimates of Equation System Parameters: An Application of Integration by Monte Carlo," Econometrica, Econometric Society, vol. 46(1), pages 1-19, January. [Downloadable!] (restricted)
  7. Peter C.B. Phillips, 1987. "Partially Identified Econometric Models," Cowles Foundation Discussion Papers 845R, Cowles Foundation, Yale University, revised Aug 1988. [Downloadable!]
  8. Kleibergen, F.R. & Van Dijk, H.K., 1993. "On the Shape of the Likelyhood/Posterior in Cointegration Models," Papers 9315-a, Erasmus University of Rotterdam - Econometric Institute.
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  9. Kleibergen, Frank & Hoek, Henk, 1996. "Bayesian analysis of ARMA models using noninformative priors," Econometric Institute Report 39, Erasmus University Rotterdam, Econometric Institute. [Downloadable!]
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  10. Kleibergen, Frank, 1996. "Equality restricted random variables: densities and sampling algorithms," Econometric Institute Report 36, Erasmus University Rotterdam, Econometric Institute. [Downloadable!]
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  11. Dreze, Jacques H, 1976. "Bayesian Limited Information Analysis of the Simultaneous Equations Model," Econometrica, Econometric Society, vol. 44(5), pages 1045-75, September. [Downloadable!] (restricted)
  12. Frank Kleibergen & Richard Paap, 1997. "Priors, Posterior Odds and Lagrange Multiplier Statistics in Bayesian Analyses of Cointegration," Tinbergen Institute Discussion Papers 97-007/4, Tinbergen Institute.
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  13. Chao, J. C. & Phillips, P. C. B., 1998. "Posterior distributions in limited information analysis of the simultaneous equations model using the Jeffreys prior," Journal of Econometrics, Elsevier, vol. 87(1), pages 49-86, August. [Downloadable!] (restricted)
  14. Phillips, P.C.B., 1983. "Exact small sample theory in the simultaneous equations model," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 1, chapter 8, pages 449-516 Elsevier. [Downloadable!] (restricted)
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(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. L. Bauwens & C.S. Bos & H.K. van Dijk, 1999. "Adaptive Polar Sampling with an application to a Bayes measure of Value-at-Risk," Econometric Institute Report 167, Erasmus University Rotterdam, Econometric Institute. [Downloadable!]
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  2. Villani, Mattias, 2003. "Bayes Estimators of the Cointegration Space," Working Paper Series 150, Sveriges Riksbank (Central Bank of Sweden). [Downloadable!]
  3. Gael Martin, 2001. "Bayesian Analysis Of A Fractional Cointegration Model," Econometric Reviews, Taylor and Francis Journals, vol. 20(2), pages 217-234. [Downloadable!] (restricted)
  4. John C. Chao & Peter C.B. Phillips, 1996. "Bayesian Posterior Distributions in Limited Information Analysis of the Simultaneous Equations Model Using the Jeffreys Prior," Cowles Foundation Discussion Papers 1137, Cowles Foundation, Yale University. [Downloadable!]
  5. Andrea Carriero & George Kapetanios & Massimiliano Marcellino, 2009. "Forecasting Large Datasets with Bayesian Reduced Rank Multivariate Models," Economics Working Papers ECO2009/31, European University Institute. [Downloadable!]
  6. Antonio Ciccone & Marek Jarocinski, 2008. "Determinants of economic growth - will data tell?," Working Paper Series 852, European Central Bank. [Downloadable!]
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  7. HOOGERHEIDE, Lennart F. & KAASHOEK, Johan F. & VAN DIJK, Herman K., 2005. "On the shape of posterior densities and credible sets in instrumental variable regression models with reduced rank: An application of flexible sampling methods using neural networks," CORE Discussion Papers 2005029, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE). [Downloadable!]
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  8. Donald W.K. Andrews & James H. Stock, 2005. "Inference with Weak Instruments," Cowles Foundation Discussion Papers 1530, Cowles Foundation, Yale University. [Downloadable!]
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  9. Gael M. Martin, 2000. "US deficit sustainability: a new approach based on multiple endogenous breaks," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 15(1), pages 83-105. [Downloadable!]
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  10. Lennart F. Hoogerheide & Johan F. Kaashoek, 2004. "Functional Approximations to Likelihoods/Posterior Densities: A Neural Network Approach to Efficient Sampling," Computing in Economics and Finance 2004 74, Society for Computational Economics. [Downloadable!]
  11. Michiel D. de Pooter & René Segers & Herman K. van Dijk, 2006. "On the Practice of Bayesian Inference in Basic Economic Time Series Models using Gibbs Sampling," Tinbergen Institute Discussion Papers 06-076/4, Tinbergen Institute. [Downloadable!]
  12. John C. Chao & Peter C.B. Phillips, 1998. "Jeffreys Prior Analysis of the Simultaneous Equations Model in the Case with n+1 Endogenous Variables," Cowles Foundation Discussion Papers 1198, Cowles Foundation, Yale University. [Downloadable!]
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  13. Kleibergen, Frank & Paap, Richard, 1996. "Priors, posterior odds and Lagrange multiplier statistics in Bayesian analyses of cointegration," Econometric Institute Report 37, Erasmus University Rotterdam, Econometric Institute. [Downloadable!]
    Other versions:
  14. Dale J. Poirier & Gary Koop & Justin Tobias, 2005. "Semiparametric Bayesian inference in multiple equation models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 20(6), pages 723-747. [Downloadable!]
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  15. Stanislav Radchenko, 2004. "Limited Information Bayesian Analysis of a Simultaneous Equation with an Autocorrelated Error Term and its Application to the U.S. Gasoline Market," Econometrics 0408001, EconWPA. [Downloadable!]
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  16. Gary Koop & Roberto Leon-Gonzalez & Rodney Strachan, 2006. "Bayesian Inference in a Cointegrating Panel Data Model," Discussion Papers in Economics 06/2, Department of Economics, University of Leicester. [Downloadable!]
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  17. Lennart Hoogerheide & Herman K. van Dijk, 2008. "Possibly Ill-behaved Posteriors in Econometric Models," Tinbergen Institute Discussion Papers 08-036/4, Tinbergen Institute, revised 18 Apr 2008. [Downloadable!]
  18. H.K. Van Dijk, 2002. "On Bayesian structural inference in a simultaneous equation model," Econometric Institute Report 263, Erasmus University Rotterdam, Econometric Institute. [Downloadable!]
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