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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Notes
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- 3.
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).
- 4.
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).
- 5.
The reader may notice that this equality is just a form of Bayes’ theorem.
- 6.
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).
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See Metropolis et al. [10] as a seminal reference.
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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.
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The Beta distribution, with parameters (a, b) has mathematical expectation a/(a + b) and variance ab/[(1 + a + b)(a + b)2].
- 10.
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.
- 11.
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.
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- 13.
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.
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Novales, A., Fernández, E., Ruiz, J. (2022). Empirical Methods: Bayesian Estimation. In: Economic Growth. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63982-5_11
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