Econometrics-I-24

For inference about a sample from the marginal p β

Info icon This preview shows pages 9–15. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 10
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
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 12
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)
Image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Part 24: Bayesian Estimation Gibbs Sampling for the Probit Model ™    13/34 i i (1) Choose an initial value for  (maybe the MLE)
Image of page 14
Image of page 15
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern