Slides07-2010

Slides07-2010 - Outline Maximum Likelihood Estimating with...

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Outline Maximum Likelihood Estimating with Production Functions and Trends Empirical Example Nonlinear Solutions Maximum Likelihood and Examples: Lecture VII Charles B. Moss September 2, 2010 Charles B. Moss Maximum Likelihood

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Outline Maximum Likelihood Estimating with Production Functions and Trends Empirical Example Nonlinear Solutions Maximum Likelihood Estimating with Production Functions and Trends Time Trend Production Function Empirical Example Bera-Jarque Test Nonlinear Solutions Charles B. Moss Maximum Likelihood
Outline Maximum Likelihood Estimating with Production Functions and Trends Empirical Example Nonlinear Solutions Maximum Likelihood I An alternative objective approach to estimating the parameters of a distribution function is by maximum likelihood. I The argument behind maximum likelihood is to choose those parameters that maximize the likelihood or relative probability of drawing a particular sample. I The likelihood function (or the probability of a particular sample) can then be written as L = N Y i =1 f ( x i | μ, σ 2 ) = ( 2 πσ 2 ) N 2 exp " 1 2 σ 2 N X i =1 ( x i μ ) 2 # (1) Charles B. Moss Maximum Likelihood

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Outline Maximum Likelihood Estimating with Production Functions and Trends Empirical Example Nonlinear Solutions I Maximizing this function with respect to the parameters μ and σ 2 implies max μ,σ 2 L = ( 2 πσ 2 ) N 2 exp " 1 2 σ 2 N X i =1 ( x i μ ) 2 # (2) I Taking the Frst-order conditions with respect to μ Frst L ∂μ = ( 2 2 ) N 2 exp " 1 2 σ 2 N X i =1 ( x 1 μ ) 2 # × 1 2 σ 2 N X i =1 [ 2( x i μ )] ! =0 1 σ 2 N X i =1 x i N μ ! ˆ μ = 1 N N X i =1 x i (3) Charles B. Moss Maximum Likelihood
Outline Maximum Likelihood Estimating with Production Functions and Trends Empirical Example Nonlinear Solutions I In order to solve for the Frst-order conditions with respect to the variance, we treat σ 2 as a single variable L ∂σ 2 = N 2 (2 π ) N 2 ( σ 2 ) N 2 1 exp " 1 2 σ 2 N X i =1 ( x i μ ) 2 # + ( 2 πσ 2 ) N 2 ( σ 2 ) 2 2 N X i =1 ( x i μ ) 2 ! exp " 1 2 σ 2 N X i =1 ( x i μ ) 2 # =0 = N ( σ 2 ) 1 2 + ( σ 2 ) 2 2 N X i =1 ( x i μ ) 2 = N σ 2 N X i =1 ( x i μ ) 2 ˆ σ 2 = 1 N N X i =1 ( x i μ ) 2 (4) Charles B. Moss Maximum Likelihood

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Outline Maximum Likelihood Estimating with Production Functions and Trends Empirical Example Nonlinear Solutions I The derivation of the maximum likelihood estimates can be simpliFed by maximizing the logarithm of the likelihood function. ln ( L )= N 2 ln ( σ 2 ) 1 2 σ 2 N X i =1 ( x i μ ) 2 ln ( L ) ∂μ = 1 2 σ 2 N X i =1 [ 2( x i μ )] = 0 N X i =1 x i N μ =0 ˆ μ = 1 N N X i =1 x i ln ( L ) ∂σ 2 = N 2 1 σ 2 + 1 2 ( σ 2 ) 2 N X i =1 ⇒− N σ 2 + N X ( x i μ ) 2 ˆ σ 2 = 1 N N X ( x i μ ) 2 (5) Charles B. Moss Maximum Likelihood
Outline Maximum Likelihood Estimating with Production Functions and Trends

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This note was uploaded on 07/15/2011 for the course AEB 6182 taught by Professor Weldon during the Fall '08 term at University of Florida.

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Slides07-2010 - Outline Maximum Likelihood Estimating with...

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