This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . AcceptanceRejection Method (AR) 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 29, 2007 Lecture 25 MCMC: Metropolis Hastings Algorithm A good reference is Chib and Greenberg ( The American Statistician 1995). Recall that the key object in Bayesian econometrics is the posterior distribution: f ( Y T  θ ) p ( θ ) p ( θ Y T ) = R f ˜ ( Y T  θ ) dθ ˜ It is often difficult to compute this distribution. In particular, the integral in the denominator is difficult. So far, we have gotten around this by using conjugate priors – classes of distributions for which we know the form of the posterior. Generally, it’s easy to compute the numerator, f ( Y T  θ ) p ( θ ), but it is hard to compute ˜ ˜ the normalizing constant, the integral in the denominator, f ( Y T  θ ) dθ . One approach is to try to compute this integral in some clever way. Another, more common approac R h is Markov Chain MonteCarlo (MCMC). The goal here is to generate a random sample θ 1 , .., θ N from p ( θ Y T ). We can then use moments from this sample to approximate moments of the posterior distribution. For example, 1 E ( θ Y T ) ≈ θ N X n There are a number of methods for generating random samples from an arbitrary distribution. AcceptanceRejection Method (AR) The goal is to simulate ξ ∼ π ( x ). We can calculate for each the value of a function, f ( x ), such that ) ( x ) = f ( x π h k . The constant k is unknown. We have some candidate pdf ( x ) that we can simulate draws from, and there is a known constant c such that f ( x ) ≤ ch ( x ) We simulate draws from π ( x ) as follows: 1. Draw z ∼ h ( x ), u ∼ U [0 , 1] ) 2. If ≤ f ( z u z ch , ξ ( z then = . Otherwise repeat (1)....
View
Full
Document
This note was uploaded on 03/20/2012 for the course 14 14.02 at MIT.
 '09
 Geurrieri

Click to edit the document details