econ583lab7fall2009

# econ583lab7fall2009 - ML estimate for θ is ˆ θ mle = n...

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Econ 583 Lab 7 Eric Zivot Fall 2009 Due: Monday, November 30. 1 Reading 1. Hayashi, Chapter 7. 2. Hall, Chapter 3. 2 Hayahsi Exercises 1. Chapter 7, Page 502, Analytic Exercises 2 and 3. 2. Chapter 8, Page 510, Questions For Review 1 and 2 3. Chapter 8, Page 517, Questions For Review 1 and 2 4. Chapter 8, Page 531, Questions For Review 1. 3 Wald, LR and LM statistics based on maximum likelihood estimation Let X 1 ,...,X n be an iid sample of Bernoulli random variables; that is, each X i has density f ( x ; θ 0 )= θ x 0 (1 θ 0 ) 1 x . First, consider estimation of θ 0 using maximum likelihood (ML). Let x =( x 1 ,...,x n ) . 1. Derive the log-likelihood function, ln L ( θ | x ) , score function, S ( θ | x )= d ln L ( θ | x ) , Hessian function, H ( θ | x )= d 2 2 ln L ( θ | x ) , and information matrix, I ( θ 0 | x )= E [ H ( θ 0 | x )] = var( S ( θ 0 | x )) . Verify that E [ S ( θ 0 | x )] = 0 , and show that the

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Unformatted text preview: ML estimate for θ is ˆ θ mle = n − 1 P n i =1 x i . 2. Determine the asymptotic normal distribution for ˆ θ mle . 1 3. Derive the Wald, LR and LM statistics for testing the above hypothesis. These statistics have the form Wald = ( ˆ θ mle − θ ) 2 I ( ˆ θ mle | x ) LM = S ( θ | x ) 2 I ( θ | x ) − 1 LR = − 2[ln L ( θ | x ) − ln L ( ˆ θ mle | x )] Are these statistics numerically equivalent? For a 5% test, what is the decision rule to reject H for each statistic ? 4. Prove that the Wald and LM statistics have asymptotic χ 2 (1) distributions. 5. How do your results compare to those that you derived for GMM in the previous lab? 2...
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## This note was uploaded on 08/23/2010 for the course ECON 583 taught by Professor Zivot during the Fall '09 term at University of West Alabama-Livingston.

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econ583lab7fall2009 - ML estimate for θ is ˆ θ mle = n...

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