Econometrics Lecture Notes Hayashi, Chapter 5c Unbalanced Panels Missing Observations Assuming common coefficients, for each equation m (=1,...,M) and for each observation i (=1,...,n), yim = zim + i + im Typically, there are missing observations. Assumption: No Selection Bias Whether an observation stays in the sample does not depend on the error term. Missing Observations Let dim = 1 if observation m is in the sample; dim = 0 otherwise. Let Mi be the number of observations from i. d i1 M di = M M i = di' di = , d im m =1 iM d The Model yi = Fi + di(bi' ) + dii + i (i=1,...,n) ' i1fi1 d d d i1y i1 i1i1 M F = M yi = , , i = M (M i# ) (M i n1) (M i 1) f' iM y iM i iM iM d d d iM iM The Transformed Model Let Qi = IMi di(di'di)-1di' = IMi didi'/Mi Qiyi = QiFi + Qidi(bi' ) + Qidii + Qi i Qiyi = QiFi + Qi i since Qidi = 0 yi* = Fi* + i* y* = F* + * With "zeroing out" missing observation, the transformed model remains the same as without missing observations. Therefore, the formula for the fixed-effects estimator remains the same. Fixed-Effects Estimator The fixed-effects estimator of is the pooled OLS estimator, which is consistent and asymptotically normal: FE = (F*'F*) 1F*'y FE- = (F*'F*) 1F*' * Est(Avar( FE)) = 1/n (F*'F*) 1(F*'VF*) (F*'F*) 1 where V = (y-F* FE)(y-F* FE)' Fixed-Effects Estimator If the assumption of spherical distribution is assumed: E( i i) = 2IMi, so that E( i* i*) = 2Qi Est(Avar( FE)) = n s2 (F*'F*) 1 s2 is the estimator of 2: (y* - F*FE )' (y* - F*FE ) s2 = n Mi - n - # i =1 No Selection Bias Assumption Recall the Orthogonality Assumption for the transf...