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2022 | OriginalPaper | Buchkapitel

11. Empirical Methods: Bayesian Estimation

verfasst von : Alfonso Novales, Esther Fernández, Jesús Ruiz

Erschienen in: Economic Growth

Verlag: Springer Berlin Heidelberg

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Abstract

The chapter starts with an introduction to Bayesian inference, and two applications examples in the context of regression models. After that, we introduce Markov Chain Monte Carlo Methods and provide a theoretical discussion of two families of such methods: Gibbs-sampling and Metropolis-Hastings algorithms. We estimate the parameters of a linear regression model using the Gibbs-sampling algorithm. Three applications of the Metropolis-Hastings algorithm are considered: random number generation from a Cauchy distribution; estimation of a GARCH(1,1) model, and estimation of a DSGE model which has been already estimated in Chap. 10 under a frequentist approach, so that the reader can compare the two different methodologies for the estimation of Growth models.

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Vorheriges Kapitel Empirical Methods: Frequentist Estimation

Nächstes Kapitel Mathematical Appendix

Classical references in Bayesian statistics and econometrics are Zellner [14], Tierney [13], Poirier [12], Geweke [6, 7], and Koop [9].

Chib and Greenberg [2, 3] are good references for the reader interested in MCMC simulators and, in particular, in the Metropolis-Hastings algorithm.

Weak law of large numbers for a random sample: If {Y _T} is a sequence of independent and identically distributed (i.i.d.) random variables with mean μ and variance σ ², we will have \( {\overline{Y}}_T\underset{p}{\to}\mu \). Central Limit Theorem for a random sample: Let {Y _T} be a sequence of independent and identically distributed (i.i.d.) random variables with mean μ and variance σ ². Let us define a sequence of i.i.d. random variables {Z _T}, where \( {Z}_T=\frac{\sqrt{T}\left({\overline{Y}}_T-\mu \right)}{\sigma } \). Then, we have:\( {Z}_T\underset{D}{\to }Z \) where Z ∼ N(0, 1).

The Gamma distribution is a two-parameter family of distributions. Its density function is:

\( y=f\left(x|a,b\right)=\frac{1}{b^a\kern0.24em \gamma (a)}{x}^{a-1}{e}^{-x/b} \), where: \( \gamma (a)={\int}_0^{\infty }{x}^{\beta -1}{e}^{-x} dx \). The mathematical expectation and variance are: E(y) = ab; V(y) = ab ², respectively. The χ ² distribution is a Gamma distribution with parameters (a, 2), while the exponential distribution is a Gamma distribution with parameters (1, b).

The reader may notice that this equality is just a form of Bayes’ theorem.

We can easily extend this Bayesian estimation procedure to vector autoregression (VAR) models insofar as they can be formulated as a multi-equation linear regression model. In that case, the conjugate prior for the covariance matrix of the disturbance term must be an inverse Wishart distribution (this distribution is nothing more than a multivariate Gamma distribution).

See Metropolis et al. [10] as a seminal reference.

The Gamma distribution, with parameters (a, b) has mathematical expectation ab and variance ab ². It is important to bear that in mind when choosing values for these two parameters.

The Beta distribution, with parameters (a, b) has mathematical expectation a/(a + b) and variance ab/[(1 + a + b)(a + b)²].

The Dirichlet distribution for two variables, with parameters (a, b, c) has mathematical expectation a/(a + b + c) for the first variable and b/(a + b + c) for the second variable. Their variances are a(b + c)/[(1 + a + b + c)(a + b + c)²] and b(a + c)/[(1 + a + b + c)(a + b + c)²], respectively.

In frequentist estimation, we have already used this idea of transforming the parameters to simplify the numerical maximization of the log-likelihood function under restrictions on parameter values, into an unrestricted optimization problem, much easier to solve.

Canova [1], DeJong and Dave [4], Del Negro and Schorfheide [5] and Miao [11] are excellent references to advance learning about Bayesian estimation of DSGE models.

When the reader executes this file, the estimates obtained could differ slightly from those that appear in Table 11.4, because the sampling will not be identical to that carried out in the writing of this chapter, but they will be statistically not different.

Canova, F. (2007). Methods for applied macroeconomic research. Princeton University Press.CrossRef

Chib, S., & Greenberg, E. (1995). Understanding the Metropolis-Hastings Algorithm. The American Statistician, 49(4), 327–335.

Chib, S., & Greenberg, E. (1996). Markov Chain Monte Carlo simulation methods in econometrics. Econometric Theory, 12(4), 409–431.CrossRef

DeJong, D. N., & Dave, C. (2011). Structural macroeconometrics (2nd ed.). Princeton University Press.

Del Negro, M., & Schorfheide, F. (2011). Bayesian macroeconometrics. In J. Geweke, G. Koop, & H. Van Dijk (Eds.), The Oxford handbook of Bayesian econometrics. Oxford University Press.

Geweke, J. (1999). Using simulation methods for Bayesian econometric models: Inference, development, and communication. Econometric Reviews, 18, 1–126.CrossRef

Geweke, J. (2005). Contemporary Bayesian econometrics and statistics. Wiley.CrossRef

Hansen, G.D. (1985). Indivisible labor and the business cycle. Journal of Monetary Economics, 16(3), 309–327.

Koop, G. (2003). Bayesian econometrics. Wiley.

10.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equations of state calculations by fast computing machine. Journal of Chemistry and Physics, 21, 1087–1091.CrossRef

11.

Miao, J. (2020). Economic dynamics in discrete time (2nd ed.). MIT Press.

12.

Poirier, D. J. (1995). Intermediate statistics and econometrics: A comparative approach. MIT Press.

13.

Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Annals of Statistics, 22, 1701–1762.

14.

Zellner, A. (1971). An introduction to Bayesian inference in econometrics. Wiley.

Titel: Empirical Methods: Bayesian Estimation
verfasst von: Alfonso Novales
Esther Fernández
Jesús Ruiz
Verlag: Springer Berlin Heidelberg
Buch: Economic Growth
Print ISBN: 978-3-662-63981-8

Electronic ISBN: 978-3-662-63982-5

Copyright-Jahr: 2022
DOI: https://doi.org/10.1007/978-3-662-63982-5_11

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