Measuring the Effects of a Shock to Monetary Policy - Humboldt ...

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Bayesian FAVARs with Agnostic Identification 23 the idiosyncratic error term et. 4.3.3 Likelihood-Based Estimation The alternative to the approach discussed above is to use the joint estimation by likelihood- based Gibbs-Sampling techniques. The problem of such models with high dimensions is that it is very difficult get the joint marginal distribution by integration. As BBE [2005] state, the irregular nature of the likelihood function makes maximum likelihood estimation (henceforth MLE) infeasible in practice. In order to understand better why the Gibbs sampling is considered as a useful tool to estimate large DFMs and FAVARs, I will briefly discuss the technical background of the estimation procedure. The idea of this section is to give the interested reader an introduction to the Markov Chain Monte Carlo methods (MCMC in the following) afterward elaborate on the Gibbs sampler and on the multi move version of the Gibbs sampler in order to convey its usefulness for DFMs. 4.3.4 Markov Chain Monte Carlo The estimation of the parameter space (of LDFMs) is about integrating statistics, espe- cially in the Baye’s statistic or in Bayesian approaches obtaining the posterior distribution, which contains all relevant information on the unknown parameters given the observed data, requires the integration of high-dimensional functions. The statistical inference can be deduced from posterior distributions. The integration problem is one of the crucial ones and can be computationally very cumbersome. One can do the integration (evaluate the integrals) through approximation via numerical methods 23 but when the parameter space is multidimensional even numerical methods may fail. One can make use of stochas- tic algorithms such as the Monte Carlo Integration techniques. The MCMC methods use computer simulations of Markov chains in the parameter chain. For random variables of higher dimensions 24 , one has to solve multiple integrals 25 . The problem of such models 23 Such as the Simpson or Trapez method 24 as it is the case in FAVARs and (L)DFMs 25 In such cases the integration via numerical methods is very hard if solvable at all.
24 Bayesian FAVARs with Agnostic Identification with high dimensions is that it is very difficult get the joint marginal distribution by integration. As BBE (2005) state, the irregular nature of the likelihood function makes integration infeasible in practice. Bayesian analysis usually requires integration to get the marginal posterior distribution of the individual parameters from a joint posterior distribution of all unknown parameters of the model in which statistical practioneers are interested in. As already stated these integrals in a high dimensional case are very hard to solve. The joint posterior density itself is very difficult to derive, from which marginals are to be derived. The MCMC methods such as the Gibbs sampler serve the solution to such high dimensional problems, in that they allow to implement posterior simulation that allow the point wise evaluation of prior distribution and likelihood function. Markov chain refers to the sequence of random variables (Z0, ..., Zn) that are generated by a Markov process. The MCMC methods attempt to simulate direct draws from some complex distribution of interest. Here the Gibbs sampling methodology offers an easy way to solve the problem, given that conditional posterior distributions are readily available. It may be employed to obtain easily marginals of the parameters without integration and without having to know the joint density. 4.3.5 The Gibbs Sampler The Gibbs sampling methodology offers an easy way to to solve the dimensionality problem given that conditional posterior distribution are readily available. In order to exem- plify the Gibbs sampler, think of θ as the parameter space. Furthermore assume that p(θ | YT ) represents the joint probability density function where YT denotes the data. Following a cyclical iterative pattern, the Gibbs sampler generates the joint distribution of p(θ | YT ) which can be also referred to as the target distribution that one tries to approximate empirically via a Markov chain. The Gibbs sampler begins with a parti- tioning or blocking of the parameters in d subvectors θ ′ = (θ1, . . . , θd). In practice the blocking is chosen so that it is feasible to draw from each of the conditional pdf’s so
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