econ583finalfall2009

# econ583finalfall2009 - Econ 583 Take Home Final Exam Fall...

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Econ 583 Take Home Final Exam Fall 2009 Eric Zivot December 17, 2009 1 Asymptotics Consider the simple AR(1) model =  1 +   (0  2 )  =1  | | 1  0 is f xed. 1. Is { } covariance stationary and ergodic? What are [ ] and var( )? 2. Consider the sample mean ¯ = 1 P =1 Show that ¯ is an unbiased and consistent estimator for [ ] For the consistency result, be sure to state the appropriate LLN. 3. What is the asymptotic distribution of ¯ ? Be sure to state the appropriate CLT to justify your result. 4. How would you estimate the asymptotic variance of ¯ ? 5. The least squares estimator of is ˆ = ³ P =1 2 1 ´ 1 P =2 1 Is ˆ an unbiased estimator of ? Brie F y explain. 6. Show that ˆ is a consistent estimator of  Be sure to state the appropriate LLN to justify this result. 7. Let = 1 and = { 1  0 } Show that {  } is a MDS. 8. What is the asymptotic distribution of )? Be sure to state the appro- priate CLT to justify your result. 9. How would you estimate the asymptotic variance of )? 1

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2 Single Equation Linear GMM Consider the regression model with a lagged dependent variable and autocorrelated errors = 0 + 1 1 + 2 1 + = z 0 δ + =  1 +   (0  2 ) z =( 1  1  1 ) 0 δ 0  1 2 ) 0 | 1 | 1 | | 1 Assume that 1 is ergodic-stationary and [ 1 ]=0 for all values of and  1. Show that [ z ] 6 =0 and use this result to show that the least squares estimate of δ is inconsistent. 2. GMM can be used to get consistent and asymptotically normal estimates of δ given suitable instruments x What instruments would you use to identify and estimate δ ? Brie f y justify your choice of instruments by showing that your choice of x satis F es [ x ]= 0 and [ x 0 Σ  has full rank =3 3. For your choice of instruments x is g ( δ )= x an ergodic-stationary MDS or is it a serially correlated process? What is the asymptotic variance of g ( δ ³ 1 ´ P x ? That is, what is S =avar ³ g ( δ ) ´ ?
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econ583finalfall2009 - Econ 583 Take Home Final Exam Fall...

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