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Bayesian econometrics

Bayesian Inference, Monte Carlo methods, Markov chain and MCMC algorithms

Bayesian ingredients: Prior, posterior, and predictive distributions, sequential Bayes.  Conjugate analysis: normal model and normal prior. Example from Box and Tiao (1973).

• Exchangeability
• Principles of data reduction
•  More on estimators
• Decision theory

Bayesian inference via Monte Carlo (MC) methods: Chronological advances, MCMC: a bit of history, simple Monte Carlo integration, Monte Carlo integration via importance function, rejection method and weighted resampling

•  Example 0. Approximating pi.
•  Example i. Monte Carlo integration.
•  Example ii. Monte Carlo integration via importance sampling.
•  Example iii. Rejection method and sampling importance resampling.
• A simple, practical and realistic example
  • Logit regression (Rcode)
  • Logit regression, 3 link functions (R code)
  • Logit regression, 3 link functions + 3 prior specifications (R code)

Bayesian inference via Markov chain Monte Carlo (MCMC) methods

• Markov chain: a briefreview
• MCMC algorithms: Historical background, Metropolis-Hastings algorithms, Simulated annealing, Gibbs sampler
  • Example iv. Rejection method, SIRand Metropolis-Hastings algorithms.
  • Example v. Rejection method, SIR and Metropolis-Hastings algorithms.
  • Example vi. Random walk MH and independent MHalgorithms.
  • Example vii. Simulated annealing.
  • Example viii. Gibbs sampler. Raoblackwellization.
  • Example ix. Random effects model.

Additional Bayesian links

• International Society for BayesianAnalysis (ISBA)
• Section on Bayesian Statistical Science (SBSS)
• Bayesian Analysis
• ISBA Bulletin

An incomplete list (in chronological order) of books on Bayesian Econometrics

• Zellner (1971) An Introduction to Bayesian Inference and Econometrics, Wiley.
• Berry, Chaloner and Geweke (1996) Bayesian Analysis in Statistics and Econometrics, Essays in Honor of Arnold Zellner, Wiley, New York.
• West and Harrison (1997) Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag.
• Bauwens, Lubrano and Richard (1999) Bayesian Inference in Dynamic Econometric Models, Oxford UniversityPress, Oxford.
• Congdon (2001) Bayesian Statistical Modelling,Wiley, New York.
• Koop (2003) BayesianEconometrics, Wiley, New York.
• Press (2003) Subjective and Objective Bayesian Statistics: Principles, Models, and Applications,2nd edn, Wiley, New York.
• Lancaster (2004) AnIntroduction to Modern Bayesian Econometrics, Blackwell Publishing.
• Geweke (2005) Contemporary Bayesian Econometrics andStatistics, Wiley, New York.
• Rossi, Allenby and McCulloch (2005) Bayesian Statistics and Marketing, Wiley, New York.

A few references on the Monte Carlo method (including several seminal papers)
· Metropolisand Ulam (1949) The Monte Carlo method. JASA, 44,335-341.
· Metropolis, Rosenbluth, Rosenbluth,Teller and Teller (1953) Equation of state calculations by fast computingmachines. Journal of Chemical Physics, 21, 2087-1092.
· Hastings(1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97-109.
· Peskun (1973) Optimum Monte Carlo sampling using Markovchains. Biometrika, 60, 607-612.
· Besag (1974) Spatial Interaction and the StatisticalAnalysis of Lattice Systems. JRSS-B, 36, 192-236.
Kloek, and van Dijk (1978) Bayesian estimates of equation system parameters: anapplication of integration by Monte Carlo. Econometrica, 46, 1-19.
· Kirkpatrick, Gelatt and Vecchi (1983)Optimization by Simulated Annealing. Science, 220 (4598), 671-680.
· Geman and Geman (1984) Stochasticrelaxation, Gibbs distributions, and the Bayesian restoration of images. IEEETrans. Pattern Analysis and Machine Intelligence, 6, 721-741.
· Tannerand Wong (1987) The Calculation of Posterior Distributions by DataAugmentation. JASA, 82, 528-540.
· Eckhardt, R.(1987) Stan Ulam, John von Neumann, and theMonte Carlo Method. Los Alamos Science, Number 15, pp131-143.
· Metropolis, N. (1987) The Beginning of the Monte Carlo Method. LosAlamos Science, Number 15, 125-130 (special issue in honor of S. Ulam).
· Casellaand George (1992) Explaining the Gibbs Sampler. The American Statistician, 46,167-174.
· Pearl(1987) Evidential reasoning using stochastic simulation of causal models.Artificial intelligence, 32, 245-257.
· Geweke (1989) Bayesian Inference in Econometric ModelsUsing Monte Carlo Integration. Econometrica, 57,1317-1339.
· Gelfand and Smith (1990) Sampling-Based Approaches toCalculating Marginal Densities. JASA, 85, 398-409.
· Gilks and Wild (1992) Adaptive Rejection Sampling for GibbsSampling. Applied Statistics, 41, 337-348.
· Smithand Gelfand (1992) Bayesian Statistics without Tears:A Sampling-Resampling Perspective. The AmericanStatistician, 46, 84-88.
· Chib and Greenberg (1995) Understanding the Metropolis-Hastings algorithm. The American Statistician, 49, 327-335.
· Chib and Greenberg (1996) Markov Chain Monte CarloSimulation Methods in Econometrics, EconometricTheory, 12, 409-431.
· Gamerman, D. andLopes, H. F. (2006) Markov ChainMonte Carlo: Stochastic Simulation for Bayesian Inference (2^nd edition). Boca Raton: Chapman& Hall/CRC Press.

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