problem_set_2 - Problem Set 2 ECON 5280 Applied...

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Problem Set 2 ECON 5280: Applied Econometrics Instructor: Prof. Jin Seo Cho Preliminary Information The following is some information you need to answer the questions to follow: REGULARITY CONDITIONS: A (RCA) 1. ( Y t , X 0 t ) 0 is identically and independently distributed (IID) such that Y t = X 0 t β * + U t ; 2. E [ U t | X t ] = 0; 3. E [ U 2 t | X t ] = σ 2 * < ; 4. E [ X 2 jt ] < for j = 1 , 2 , . . . , k ; 5. E [ X t X 0 t ] is positive definite. REGULARITY CONDITIONS: B (RCB) 1. ( Y t , X 0 t ) 0 is identically and independently distributed (IID) such that Y t = X 0 t β * + U t ; 2. E [ U t | X t ] = 0; 3. E [ X 2 jt ] < for j = 1 , 2 , . . . , k ; 4. E [ U 2 t X 2 jt ] < for j = 1 , 2 , . . . , k ; 5. E [ X t X 0 t ] is positive definite; 6. E [ U 2 t X t X 0 t ] is positive definite. REGULARITY CONDITIONS: C (RCC) 1. ( Y t , X 0 t ) 0 is identically and independently distributed (IID) such that Y t = X 0 t β * + U t ; 1
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2. E [ U t | X t ] = 0; 3. E [ X 4 jt ] < for j = 1 , 2 , . . . , k ; 4. E [ | U t X 3 jt | ] < for j = 1 , 2 , . . . , k ; 5. E [ U 2 t X 2 jt ] < for j = 1 , 2 , . . . , k ; 6. E [ X t X 0 t ] is positive definite; 7. E [ U 2 t X t X 0 t ] is positive definite. 1 True or False Questions Write ‘T’ or ‘F’ to the next of question numbers if you think the following statements are correct or incorrect respectively. You don’t have to justify your answers. No partial credit will be granted for your justification. 1. If the normality assumption of the classical linear model (CLM) does not hold then the distribution of the ordinary least squares (OLS) estimator may not be normal. 2. tr( A - B ) = tr( A ) - tr( B ). 3. The ordinary least squares (OLS) estimator is linear with respect to the column vector of dependent variables. 4. The OLS estimator is unbiased even if the normality assumption of the CLM does not hold. 5. RCA assumes the conditional homoskedasticity. 6. RCB may provide suitable conditions for analyzing the ordinary least squares (OLS) estimator for β * under the presence of conditional homoskedasticity. 7. The OLS estimator for β * is unbiased under RCB. 8. The OLS estimator for β * under RCB is normally distributed with the covariance matrix identical to the OLS estimator under RCA condition. 9. When assuming the conditional homoskedasticity, hypotheses given in the form: H 0 : R β * = r versus H 1 : R β * 6 = r can be tested by the Wald test statistic obtained under the classical linear model assumption. 2
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10. When conditional heteroskedasticity is present, hypotheses given in the form: H 0 : R β * = r versus H 1 : R β * 6 = r can be tested using the standard Wald test statistic under RCB. 11. The formula of the Wald test statistic obtained under RCA is identical to the Wald test obtained under RCC. 12. The OLS estimator for β * is normally distributed under RCC condition. 13. The covariance matrix of the OLS estimator can be estimated by the conditions in RCB when the error is conditionally heterodskedastic. 14. RCA is more restrictive than RCB. 15. The OLS estimator is consistent under RCC.
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