# New PDF release: Contemporary Bayesian Econometrics and Statistics

By John Geweke

ISBN-10: 0471679321

ISBN-13: 9780471679325

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Extra info for Contemporary Bayesian Econometrics and Statistics

Sample text

For the purposes of this problem, assume that no one is ever laid off or fired. The only way of leaving employment is by quitting. You can also assume that no one ever has more than one spell of employment with the company—if they quit, they never come back. In this problem we consider the case in which the consultant’s data consist entirely of “complete spells;” that is, for each individual t in the sample the consultant observes the length of time, yt , between hiring and quitting. (a) Express the joint density of the observables and find a sufficient statistic vector for θ.

44) may be interpreted as the combination of q independent components of information about β, in the same way that the covariates X provide T such independent components in the normal linear model. 44) the pdf of the random vector z = Rβ is p(z) = (2π)−q/2 |V|−1/2 exp[−(r − Rβ) V−1 (r − Rβ)/2]. 44) implies β | A ∼ N (β, H−1 ), with H = R V−1 R and β = (R V−1 R)−1 R V−1 r. 3. The following extension of the idea of a conjugate family of prior densities will prove useful in subsequent work. 2 In the data density p(YT | θ A , A) let θ A = (θ A1 , θ A2 ) and fix θ A2 = θ 0A2 .

50 (ν) of the chi-squared distribution with ν degrees of freedom. 2 SUFFICIENCY, ANCILLARITY, AND NUISANCE PARAMETERS The steps that are undertaken to derive the posterior distribution p(θ A | yo , A) or the marginal likelihood p(yo | A) depend on the relations between yo and θ A in these expressions. In particular circumstances these expressions can be simplified. Two of the most useful arise when the data can be reduced to a smaller set of statistics (called sufficient statistics) for the purpose of inference, and again when a subset of this set (called ancillary statistics) can be regarded as fixed for the same purpose.