Lecture+17+Bayesian+Analysis+of+Regression

Lecture+17+Bayesian+Analysis+of+Regression - Lecture 17,...

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Lecture 17, ECON 123A, Fall 2011 17-1 Dale J. Poirier Lecture 17 “The people who don’t know they are Bayesians are called non-Bayesians.” I. J. Good (1983) Notation changes: N denotes sample size; the regression coefficient vector 12 k is $ = [ $ , $ , ..., $ ] N . The linear regression model posits a linear relationship between the ii dependent variable y and a 1×k vector of explanatory variables, x , where i = 1, . . . , N indexes the relevant observational unit (e.g., individual, firm, time period, etc.). In matrix notation, the linear regression model can be written as
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Lecture 17, ECON 123A, Fall 2011 17-2 Dale J. Poirier
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Lecture 17, ECON 123A, Fall 2011 17-3 Dale J. Poirier
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Lecture 17, ECON 123A, Fall 2011 17-5 Dale J. Poirier Definition (Gamma Distribution): A positive continuous random variable h has a gamma distribution with mean m > 0 and degrees of freedom < > 0, denoted h - ( (m, < ), iff its p.d.f. is
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Lecture 17, ECON 123A, Fall 2011 17-6 Dale J. Poirier Definition (Normal-gamma Distribution): Let $ be a random vector and h a scalar random variable. If the conditional distribution of $ given h is normal and the marginal distribution for h is gamma then 2 = [ $ N , h] N is said to have a normal-gamma distribution . Formally, if $ |h, : , E - N( $ , h Q), -1 h * m, < - ( (m, < ), then 2 has a normal-gamma distribution denoted 2 - NG( $ , Q, m, < ). The NG corresponding p.d.f. is denoted f ( 2 * $ , Q, m, < ).
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Lecture 17, ECON 123A, Fall 2011 17-7 Dale J. Poirier
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Lecture 17, ECON 123A, Fall 2011 17-8 Dale J. Poirier
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Lecture 17, ECON 123A, Fall 2011 17-9 Dale J. Poirier
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Lecture 17, ECON 123A, Fall 2011 17-10 Dale J. Poirier
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Lecture 17, ECON 123A, Fall 2011 17-11 Dale J. Poirier
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Lecture 17, ECON 123A, Fall 2011 17-12 Dale J. Poirier
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Lecture+17+Bayesian+Analysis+of+Regression - Lecture 17,...

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