Econometrics-I-24

# For inference about a sample from the marginal p β

• Notes
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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

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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 0 1 Draw a random sample from bivariate normal  , 0 1 v u u (1) Direct approach:   where   are two v u u 1 0     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

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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.      Bu β i i i i t, y  gives the sign of y * . y * | , y  is a draw from      the truncated normal (above if y=0, below if y= 1)

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Part 24: Bayesian Estimation Gibbs Sampling for the Probit Model ™    13/34 i i (1) Choose an initial value for  (maybe the MLE)
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