Econometrics-I-24

For inference about a sample from the marginal

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Unformatted text preview: For inference about , a sample from the marginal posterior, p( |data) would suffice. For inference about , a sample from the marginal p β β β- σ σ σ σ ′ Σ- σ × = σ 2 2 2 2 1 2 2 i i osterior of , p( |data) would suffice. Can we deduce these? For this problem, we do have conditionals: p( | ,data) = N[ , ( ) ] (y ) p( | ,data) = K a gamma distributio i 2 b X'X β x β β σ 2 n Can we use this information to sample from p( |data) and p( |data)? β Part 24: Bayesian Estimation The Gibbs Sampler p Target: Sample from marginals of f(x1, x2) = joint distribution p Joint distribution is unknown or it is not possible to sample from the joint distribution. p Assumed: f(x1|x2) and f(x2|x1) both known and samples can be drawn from both. p Gibbs sampling: Obtain one draw from x1,x2 by many cycles between x1| x2 and x2|x1. n Start x1,0 anywhere in the right range. n Draw x2,0 from x2|x1,0. n Return to x1,1 from x1|x2,0 and so on. n Several thousand cycles produces the draws n Discard the first several thousand to avoid initial conditions. (Burn in) p Average the draws to estimate the marginal means. ˜˜™™™ ™ 9/34 Part 24: Bayesian Estimation Bivariate Normal Sampling ˜˜™™™ ™ 10/34 ρ ÷ ÷ ρ = Γ ÷ ÷ ÷ Γ θ θ 1 1 1 2 2 2 r r 1 2 1 Draw a random sample from bivariate normal , 1 v u u (1) Direct approach: where are two v u u 1 independent standard normal draws (easy) and = ÷ ρ ΓΓ θ = ρ θ =- ρ ÷ ρ ρ- ρ ρ- ρ 2 1 2 2 1 2 2 2 2 1 1 1 such that '= . , 1 . 1 (2) Gibbs sampler: v | v ~ N v , 1 v | v ~ N v , 1 Part 24: Bayesian Estimation Gibbs Sampling for the Linear Regression Model ˜˜™™™ ™ 11/34- σ σ ′ Σ- σ × σ = i 2 b X'X β x β β 2 2 1 2 2 i i p( | ,data) = N[ , ( ) ] (y ) p( | ,data) = K a gamma distribution Iterate back and forth between these two distributions Part 24: Bayesian Estimation Application – the Probit Model ˜˜™™ ™ 12/34 ′ = ε ε = i i x + β β i i i i i i (a) y * ~ N[0,1] (b) y 1 if y * > 0, 0 otherwise Consider estimation of and y * (data augmentation) (1) If y* were observed, this would be a linear regression (y would not be useful ′ i | β β x β i i i i since it is just sgn(y * ).) We saw in the linear model before, p( y * , y ) (2) If (only) were observed, y * would be a draw from the normal distribution with mean and variance 1....
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For inference about a sample from the marginal posterior p...

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