Lecture11-2011 - Outline Normal-Half Normal Model Maximum...

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Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Stochastic Production Functions II - Maximum Likelihood: Lecture XI Charles B. Moss 1 1 University of Florida September 29, 2011 Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec t u
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Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values 1 Normal-Half Normal Model 2 Maximum Likelihood Results Comparison 3 Two Applications of the Primal Basic Imputed Value Formulation Estimates Conclusions 4 Euler Equation and Land Values Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec t u
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Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Normal-Half Normal Model Assumptions about errors ν i N ( 0 , σ 2 ν ) u N * ( 0 , σ 2 u ) ν i and u i independent (1) The normal distribution ν i follows the standard normal (mean zero) formulation f ( ν ) = 1 2 πσ ν exp ± - ν 2 2 σ 2 ν ² (2) The half-normal distribution is represented by g ( u ) = 2 2 u exp ± - u 2 2 σ 2 u ² (3) Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec t u
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Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Assuming independence f ( u , ν ) = f ( ν ) g ( u ) = 2 2 πσ ν σ u exp ± - ν 2 2 σ 2 ν - u 2 2 σ 2 u ² (4) Since ± = ν - u , or by definition of the composed error term f ( u , ± ) = 2 2 u σ ν exp " - u 2 2 σ 2 u - ( ± + u ) 2 2 σ 2 ν # (5) Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec t u
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Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Integrating out u , we obtain the marginal distribution function for ± . From Weinstein x = ( X - μ X ) σ X y = ( Y - μ Y ) σ Y (6) x is distributed normal, while y is distributed half-normal f ( x ) = 1 2 π exp ± - x 2 2 ² g ( y ) = f ³ y σ ´ σ h 1 - F ³ - a a σ ´i , y ≥ - a 0 y < - a (7) Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec t u
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Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Continued Where σ = σ Y σ X F ( u ) = Z u -∞ f ( x ) dx (8) Note that if a = 0 g ( y ) = f ± y σ ² σ [1 - F (0)] = f ± y σ ² σ [1 - 0 . 50] = 2 f ± y σ ² σ = 2 2 πσ x σ exp - y 2 ( σσ X ) 2 ! (9) Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec t u
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Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values Substituting in σ = σ Y σ X g ( y ) = 2 2 πσ X σ Y σ X exp - y 2 ± σ Y σ X σ X ² 2 = 2 2 Y exp ± - y 2 σ 2 Y ² (10) Charles B. Moss Stochastic Production Functions II - Maximum Likelihood: Lec t u
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Outline Normal-Half Normal Model Maximum Likelihood Two Applications of the Primal Euler Equation and Land Values By integration Q ( t ) = Z t -∞ dz Z -∞ g ( z - x ) f ( x ) dx = 1 σ h 1 - F ± - a σ ²i
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This note was uploaded on 11/08/2011 for the course AEB 6184 taught by Professor Staff during the Spring '09 term at University of Florida.

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Lecture11-2011 - Outline Normal-Half Normal Model Maximum...

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