By Edward P. Herbst, Frank Schorfheide
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Additional resources for Bayesian Estimation of DSGE Models
5) with density p(θ) = (2πτ 2 )−p/2 exp − 1 θθ . 6) The hyperparameter τ controls the variance of the prior distribution and can be used to illustrate the effect of the prior variance on the posterior distribution. According to Bayes Theorem the posterior distribution of θ is proportional (∝) to the product of prior density and likelihood function p(θ|Y ) ∝ p(θ)p(Y |θ). 7) To simplify the notation we dropped Y1−p:0 from the conditioning set and we replaced y1:t by the matrix Y . 8) 2 −θ X Xθ − τ −2 θ θ] .
We divide this variance by Vπ [h]/N so that noted by V[h it is on the same scale as the asymptotic inefficiency factor: InEffN = ¯N ] V[h . 56) In general, the asymptotic approximation is very accurate. 2: IS Approximations of Eπ [θ] and Eπ [θ 2 ]. Panel (i) depicts the posterior density π(θ) (solid) as well as two importance sampling densities (‘‘concentrated’’ (dashed) and ‘‘diffuse’’ (dotted)) g(θ). Panels (ii) and (iii) depict large sample inefficiency factors InEff∞ = Ω(h)/Vπ [h] (dashed) as well as their small sample approximations (solid) based on Nrun = 1, 000.
Now suppose neither player knows h(θ) and both players minimize their posterior expected loss under the same distribution p(h(θ)|Y ). 23) = (δu − δl ) + max− λ P(δl ≤ h(θ) ≤ δu |Y ) − (1 − α) . 23) is the shortest connected interval with coverage probability 1−α. The shortest connected interval can be computed based on equally weighted draws as follows: sort the draws h(θi ) in ascending order to obtain the sequence h(i) ; for i = 1 to N α minimize h( N (1−α) +i) − h(i) with respect to i. To allow for disjoint credible intervals, the difference δu −δl in the above loss function has to be replaced by the sum of the lengths of the disjoint intervals.
Bayesian Estimation of DSGE Models by Edward P. Herbst, Frank Schorfheide