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Download Bayesian Estimation of DSGE Models by Edward P. Herbst, Frank Schorfheide PDF

By Edward P. Herbst, Frank Schorfheide

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In the remainder of this section we provide a brief overview of a decision-theoretic approach to Bayesian inference. A textbook treatment is provided by Robert (1994). The posterior expected loss associated with a decision rule δ(Y ) is given by ρ δ(Y )|Y = Θ 1 L h(θ), δ(Y ) p(θ|Y )dθ. , L(y∗ , δ). 7 16:09:09 UTC 36 • Chapter 3 Note that in this calculation the observations Y are fixed and we are integrating over the unknown parameter θ under the posterior distribution. A Bayes decision is a decision that minimizes the posterior expected loss: δ ∗ (Y ) = argminδ∈D ρ δ|Y .

The proposed draw is always accepted if it raises the posterior density (relative to θ i−1 ) and it is sometimes accepted even if it lowers the posterior density. If the proposed draw is not accepted, then the chain does not move and θ i = θ i−1 . The indexacceptance probability is chosen to ensure that the distribution of the draws converges to the target posterior distribution. The algorithm takes the following form: Algorithm 5 (Generic MH Algorithm) For i = 1 to N: 1. Draw ϑ from a density q(ϑ|θ i−1 ).

Let Xt1 :t2 = {xt1 , xt1 +1 , . . , xt2 }. 14) T = t=1 p(yt |st , θ)p(st |st−1 , θ), where p(yt |st , θ) and p(st |st−1 , θ) represent the measurement and state-transition equations, respectively. 7), are equal to zero. If some elements of w2,t only depend on variables that can be measured in the data, this implication is most likely violated. To cope with this problem, one can either limit the number of observables included in yt , as we do in the New Keynesian model, or include so-called measurement errors as, for instance, in Sargent (1989), Altug (1989), and Ireland (2004).

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