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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 . Differences between Bayesian and Frequentist Approaches 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 29, 2007 Lecture 23 Reasons to be Bayesian Bayesian econometrics is based on two pieces: 1. A parametric model, giving a distribution, f ( Y T  θ ), for the data given parameters 2. A prior distribution for the parameters, p ( θ ) From these, we can form the joint distribution of the data and parameters, p ( Y T , θ ) = f ( Y T  θ ) p ( θ ) and the marginal distribution of the data p ( Y T ) = Z f ( Y T  θ ) p ( θ ) dθ Finally, using Baye’s rule, we can form the posterior distribution of the parameters given the data f ( Y T  θ ) p ( θ ) p ( θ Y T ) = p ( Y T ) One can make inferences based on the posterior distribution. For example, we can report the mode (or the mean) as a parameter estimate. Any set of posterior measure biggen than 1 α is called an 1 α credible set. Hypotheses can be tested based on posterior odds. Differences between Bayesian and Frequentist Approaches Frequentist • θ is fixed, but unkown • Uncertainty comes from sampling uncertainty. That is, from the fact that we can get different samples. • All probabilistic statements are statements about sampling uncertainty. For example, – ˆ E θ θ ( Y T ) = θ (unbiasedness) means in average over all possible repeated samples, one receives the true value – P θ { θ ∈ C ( Y T ) } = 1 α coverage probability of confidence sets is a statement about the ratio (in repeated samples) of sets containing θ . Once we observe a sample, θ is either in the set or not; there is no probability, after realization of a sample. The coverage probability is a statement about exante probability....
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