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At this stage, we have three posterior simulation algorithms: For simplicity, we are using the same notation as in the previous chapters. The Best Books of A natural conjugate prior has the additional property that it has the same functional form as the likelihood function. We refer to such a prior as a relatively noninformative prior. The treat- ment of 9 as a random variable is controversial among some econometricians.

Flowever, if the units of measure- ment of X 2 were changed to hundreds of square feet, then the interpretation would be based on a statement of the form: S’, which can be averaged to produce estimates of posterior properties just as with Monte Carlo integration.

Wiley Higher Education Supplementary Website

The frequentist econometrician would often use p, the ordinary least squares estimate of p. The prop- erties of the Normal-Gamma distribution imply that the prior covariance matrix for p has the form: Computer programs for many of these diagnostics are available over the web. We introduce the concept of a Bayeian sampler, and show how it can be used to cany out posterior and predictive inference. If you have many models and are interested in comparing them, you can use posterior model probabilities 1.


Nevertheless, simulation methods allow for predictive inference to be carried out in a straightforward fashion.

In rare cases, 1.

Bayesian Econometrics

When facing a new model or reading a new chapter in the bookjust remember that Bayesian econometrics requires selection of a prior and a likelihood. These are very similar strategies, except for two important differences. The posterior means and standard deviations are similar to those in Table 3. We stress, however, that this similarity only holds far enough to provide some very crude intuition.

Hence, if we multiply 3. In the previous chapter we introduced the intuition that one bayesixn interpret an MCMC algorithm as wandering over the posterior, taking most draws in areas of high posterior probability and proportionately fewer in baydsian of low posterior probability.

In either case, there is a substantial probability that you are choosing the wrong model. But as we have stressed, posterior simulation is designed precisely to evaluate such quantities.

The Nonlinear Regression Model 97 In the case of the nonlinear regression model, maximizing the likelihood function posterior would involve writing a program which evaluates 5.

One measure of the magnitude of a matrix is its determinant. These algorithms will be used in later chapters. It is worth noting fconometrics there is a myriad of models for which Gibbs sampling can be done in a straightforward manner. However, when calculating posterior odds ratios, a noninformative path may not be acceptable.

Monte Carlo Integration 46 3. How sensitive is the posterior to changes in prior information? Particularly if 6 is high-dimensional it can be extremely difficult to find a good importance function.

Bayesian Econometrics All of these posterior features which the Bayesian may wish to calculate have the form: However, we also develop methods for these new priors to introduce some important concepts in Bayesian computation. If these different runs of the Gibbs sampler all yield essentially the same answer, the researcher is reassured that sufficient replications have been taken and enough burn-in replications discarded for the effect of the initial replication to vanish.


Bayesian analysis of the regression model subject to such restrictions is quite simple since we can simply impose them through the prior. V’ — P x i 2 i find it can be confirmed that matrix constructs of the form a’ a, where a is a vector, are sums of squares. As described in this pre- vious material, model comparison involving such inequality restrictions is quite easy and use of noninformative priors is not a problem.

Predictive inference can be carried out using 2. The key point to stress here is that an estimate of a 2 is available and can be calculated using the computer programs discussed above.

Using the properties of the expected value operator see Appendix B, Definition B. In such cases, development of posterior simu- lators is a more challenging task. How large does S have to be before you reproduce the results of Exercise 8 b to two decimal places?

The Savage-Dickey density ratio is, thus, a very powerful and widely-used tool for Bayes factor calculation. How sensitive is the Bayes factor? Secondly, the Bayesian inter- prets ft as a random variable, bajesian the frequentist interprets ft as a random variable.

For instance, if the posterior is multi-modal then the unimodal t-density will usually not work well.