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Unformatted text preview: Concentrated Likelihood Functions, Normal Equations, and Properties of Maximum Likelihood: Lecture XX Charles B. Moss October 19, 2010 I. Concentrated Likelihood Functions A. In the last lecture I introduced the concept of maximum likelihood using a known variance normal distribution of unknown mean. Undoubtedly this framework is fictional. Even if we know that the distribution is normal, it would be a rare event to know the distribution of the variance. B. What I would like to do now is to demonstrate how maximum likelihood functions may be applied by demonstrating how the variance can be concentrated out of the standard normal estima- tion. 1. The more general form of the normal likelihood function can be written as L X | μ, σ 2 = n Y i =1 1 √ 2 πσ 2 exp " − ( X i − μ ) 2 2 σ 2 # (1) 2. Ignoring the constants, the natural logarithm of the likelihood function can be written as ln ( L ) = − n 2 ln σ 2 − 1 2 σ 2 n X i =1 ( X i − μ ) 2 (2) 3. This expression can be solved for the optimal choice of σ 2 by differentiating with respect to σ 2 1 AEB 6571 Econometric Methods I...
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.
- Spring '10