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

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
Outline Maximum Likelihood Estimating with Production Functions and Trends
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 21

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

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online