- 1 Consider The Following Statistic Panel Data Model Yi 20 A Va Where I 1 N T 1 T Assume We Observe A Ra 1 (75.27 KiB) Viewed 35 times
- 1 Consider The Following Statistic Panel Data Model Yi 20 A Va Where I 1 N T 1 T Assume We Observe A Ra 2 (73.14 KiB) Viewed 35 times
1. Consider the following statistic panel data model Yi+20+ a + va ... where i = 1, n, t= 1,...T. Assume we observe a random sample {(it. Vit. 2)}=1 over i= 1,...n. α₁ is the fixed effect and u,, is an unobserved variable with E(valar, 4₁,0) = 0. Let o=₁, and assume El(-a) (Ta-F)] is invertible and E(22) is invertible. (a) Answer true or false for the following statements. You do not need to explain your answer. i. Under the stated assumptions, the fixed effect estimator of B is consistent. ii. Under the stated assumptions, the fixed effect estimator of 0 is consistent. iii. If in addition, E(v..., T, 2₁, 0) <c< oo for some c> 0, then the fixed effect estimator of 3 is asymptotically normal. iv. Under the same condition as in (iii), the fixed effect estimator of 9 is asymptotically normal. v. Under the stated conditions, the fixed effect estimator of 3 is BLUE. (b) Answer true or false for the following statements. You do not need to ex- plain your answer. In addition to the stated assumptions, assume that a, and (ar, 2) are independent. Consider the OLS estimator, regressing it on it and 2₁. i. Under the stated assumptions, the OLS estimator of B is consistent. ii. Under the stated assumptions, the fixed effect estimator of 3 is consistent. iii. If in addition, for some, E(lra.....IT, 2, 0) < c < x for some e > 0, then the OLS estimator of 3 is asymptotically normal. iv. Under the same condition as in (iii), the fixed effect estimator of 3 is asymptotically normal. v. Under the stated conditions, the fixed effect estimator of 8 is BLUE. (c) i. Consider the residual term in (b) including the fixed effect. What is the variance-covariance matrix of the residual term of size nT x nT. Order T observations of the 1st individual as the first T elements, and then the T observations of the 2nd individual as the second T elements, and so on. ii. Define the GLS estimator. (d) Consider the following dynamic panel data model. Yit =B₁ya-1 + 2a-2+3+20+ a + V₁
for i=1...n, t= 1... T. and it = (yit-1, Yit-2, 4), B = (31, 32, 3). Assume s E(vula,..., T, 2₁, ₁) = 0. (*). Taking the first differenced model. Yit - Yit-1 =B₁ (it-1-Yit-2) + B₂ (Yit-2 - Yit-3) + (-1) + Vit - Vit-1 Answer true or false to the following statements. You do not need to explain your answer. i. Under the stated assumptions, OLS estimator of 3 of the first differenced model is a consistent estimator if E[(Tit - Tit-1)(Tit - Fit-1)] is invertible for t= 1,..., T. ii. Under the stated assumptions, GMM estimator 3 of the first differenced model using n T i=1 t=1 Witit - Yit-1-(-a-1)3] = 0, where wit = (yit-2+ Yit-2-Yit-3, Fit-Fit-1) as the moment condition yields a consistent estimator if T t=1 Ewa (-it-1)] has full rank using identity matrix as the weight matrix. iii. Efficiency of the estimator in (ii) can be improved. iv. Estimator in (ii) presumes that there is no serial correlation in Uit because assumption (*) precludes serial correction in Uit