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Bayesian and Classical Approaches to Instrumental Variable Regression

  • Frank Kleibergen
  • Eric Zivot

We establish the relationships between certain Bayesian and classical approaches to instrumental variable regression. We determine the form of priors that lead to posteriors for structural parameters that have similar properties as classical 2SLS and LIML and in doing so provide some new insight to the small sample behavior of Bayesian and classical procedures in the limited information simultaneous equations model. Our approach is motivated by the relationship between Bayesian and classical procedures in linear regression models; i.e., Bayesian analysis with a diffuse prior leads to posteriors that are identical in form to the finite sample density of classical least squares estimators. We use the fact that the instrumental variables regression model can be obtained from a reduced rank restriction on a multivariate linear model to determine the priors that give rise to posteriors that have properties similar to classical 2SLS and LIML. As a by-product of this approach we provide a novel way to determine the exact finite sample density of the LIML estimator and the prior that corresponds with classical LIML. We show that the traditional Dreze (1976) and a new Bayesian Two Stage approach are similar to 2SLS whereas the approach based on the Jeffreys' prior corresponds to LIML.

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Paper provided by Department of Economics at the University of Washington in its series Discussion Papers in Economics at the University of Washington with number 0063.

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Date of creation: Aug 1998
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Handle: RePEc:fth:washer:0063
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  1. Park, Soo-Bin, 1982. "Some sampling properties of minimum expected loss (MELO) estimators of structural coefficients," Journal of Econometrics, Elsevier, vol. 18(3), pages 295-311, April.
  2. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
  3. Charles R. Nelson & Richard Startz & Eric Zivot, 1996. "Valid Confidence Intervals and Inference in the Presence of Weak Instruments," Econometrics 9612002, EconWPA.
  4. Chao, John C. & Phillips, Peter C. B., 2002. "Jeffreys prior analysis of the simultaneous equations model in the case with n+1 endogenous variables," Journal of Econometrics, Elsevier, vol. 111(2), pages 251-283, December.
  5. Kleibergen, F.R. & Paap, R., 1998. "Priors, posteriors and Bayes factors for a Bayesian analysis of cointegration," Econometric Institute Research Papers EI 9821, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  6. Anderson, T W & Sawa, Takamitsu, 1979. "Evaluation of the Distribution Function of the Two-Stage Least Squares Estimate," Econometrica, Econometric Society, vol. 47(1), pages 163-82, January.
  7. Sawa, Takamitsu, 1973. "The mean square error of a combined estimator and numerical comparison with the TSLS estimator," Journal of Econometrics, Elsevier, vol. 1(2), pages 115-132, June.
  8. John Shea, 1996. "Instrument Relevance in Multivariate Linear Models: A Simple Measure," NBER Technical Working Papers 0193, National Bureau of Economic Research, Inc.
  9. 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.
  10. Kleibergen, F.R. & van Dijk, H.K., 1997. "Bayesian Simultaneous Equations Analysis using Reduced Rank Structures," Econometric Institute Research Papers EI 9714/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  11. Anderson, T. W. & Kunitomo, Naoto & Morimune, Kimio, 1986. "Comparing Single-Equation Estimators in a Simultaneous Equation System," Econometric Theory, Cambridge University Press, vol. 2(01), pages 1-32, April.
  12. Charles Nelson & Richard Startz & Eric Zivot, 2000. "Improved Inference for the Instrumental Variables Estimator," Econometric Society World Congress 2000 Contributed Papers 1600, Econometric Society.
  13. Fuller, Wayne A, 1977. "Some Properties of a Modification of the Limited Information Estimator," Econometrica, Econometric Society, vol. 45(4), pages 939-53, May.
  14. Zellner, Arnold, 1978. "Estimation of functions of population means and regression coefficients including structural coefficients : A minimum expected loss (MELO) approach," Journal of Econometrics, Elsevier, vol. 8(2), pages 127-158, October.
  15. Zellner, Arnold & Tobias, Justin, 2001. "Further Results on Bayesian Method of Moments Analysis of the Multiple Regression Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 42(1), pages 121-40, February.
  16. Russell L. Lamb & Francis X. Diebold, 1996. "Why are estimates of agricultural supply response so variable?," Finance and Economics Discussion Series 96-8, Board of Governors of the Federal Reserve System (U.S.).
  17. Chuanming Gao & Kajal Lahiri, 2000. "A Comparison of Some Recent Bayesian and Classical Procedures for Simultaneous Equation Models with Weak Instruments," Econometric Society World Congress 2000 Contributed Papers 0230, Econometric Society.
  18. Maddala, G S, 1976. "Weak Priors and Sharp Posteriors in Simultaneous Equation Models," Econometrica, Econometric Society, vol. 44(2), pages 345-51, March.
  19. Forchini, Giovanni & Hillier, Grant, 2003. "Conditional Inference For Possibly Unidentified Structural Equations," Econometric Theory, Cambridge University Press, vol. 19(05), pages 707-743, October.
  20. Phillips, P.C.B., 1989. "Partially Identified Econometric Models," Econometric Theory, Cambridge University Press, vol. 5(02), pages 181-240, August.
  21. Zellner, Arnold, 1998. "The finite sample properties of simultaneous equations' estimates and estimators Bayesian and non-Bayesian approaches," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 185-212.
  22. Dreze, Jacques H, 1976. "Bayesian Limited Information Analysis of the Simultaneous Equations Model," Econometrica, Econometric Society, vol. 44(5), pages 1045-75, September.
  23. 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.
  24. Dreze, Jacques H., 1977. "Bayesian regression analysis using poly-t densities," Journal of Econometrics, Elsevier, vol. 6(3), pages 329-354, November.
  25. Jiahui Wang & Eric Zivot, 1998. "Inference on Structural Parameters in Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 66(6), pages 1389-1404, November.
  26. Anderson, T W, 1977. "Asymptotic Expansions of the Distributions of Estimates in Simultaneous Equations for Alternative Parameter Sequences," Econometrica, Econometric Society, vol. 45(2), pages 509-18, March.
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