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By Edward P. Herbst, Frank Schorfheide

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Extra resources for Bayesian Estimation of DSGE Models

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5. 25] which has a large probability under the posterior distribution. The accuracy of the importance sampling approximations are illustrated in Panels (ii) and (iii) as a function of the number of draws N . 53) as well as a simulation-based inefficiency factor InEffN in which we replace the asymptotic variance of the importance sampler by a finite sample estimate computed from multiple runs of the algorithm. More specifically, we run the importance sampling algorithm Nrun = 1, 000 times and compute the variance of the Monte Carlo approximations of Eπ [θ] and Eπ [θ 2 ] across the runs, de¯ N ].

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. The credible interval that minimizes the posterior risk under the loss function that penalizes the total length of the disjoint segments has the property that p(δl |Y ) = p(δu |Y ) = κ. It is called the highestposterior-density (HPD) set because the density of all values of θ that are included in this set exceeds the threshold κ. The HPD set is formally defined as CSHP D = h |p(h|Y ) ≥ κ .

For this model, the posterior distribution can be characterized analytically and closed-form expressions for its moments are readily available. Draws from the posterior distribution can be easily generated using a direct sampling algorithm. 2 discusses how to turn posterior distributions—or draws from posterior distributions—into point estimates, interval estimates, forecasts, and how to solve general decision problems. 3 we modify the parameterization of an AR(1) model to introduce some identification problems.

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