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Unformatted text preview: Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business Econometric Analysis of Panel Data 6. Maximum Likelihood Estimation of the Random Effects Linear Model The Random Effects Model The random effects model c i is uncorrelated with x it for all t; E[c i  X i ] = 0 E[ it  X i ,c i ]=0 it it i it i i i i i i i i i i i i N i= 1 i 1 2 N y = + c + , observation for person i at time t = + c + , T observations in group i = + + , note (c ,c ,...,c ) = + + , T observations in the sample c= ( , ,... ) , = x y X i X c c y X c c c c N i= 1 i T by 1 vector Error Components Model Generalized Regression Model it it i it i 2 2 it i i i 2 2 i i u i i i i y + + u E[  ] E[  ] E[u  ] E[u  ] = + + u for T observations = = = = = it i x b X X X X y X i 2 2 2 2 u u u 2 2 2 2 u u u i i 2 2 2 2 u u u Var[ +u ] + + = + i L L M M O M K Notation 2 2 2 2 u u u 2 2 2 2 u u u i i 2 2 2 2 u u u 2 2 u i i 2 2 u i 1 2 N Var[ + u ] = T T = = Var[  ] + + = + + + = i i T T i I ii I ii w X L L M M O M K L L M M O M K i (Note these differ only in the dimension T) Maximum Likelihood + + i it i i1 i2 iT i i i i 2 2 u Assuming normality of and u. Treat T joint observations on [( , ,... ),u ] as one T variate observation. The mean vector of u is zero and the covariance matrix is = I . The j i i ii  = =  + + i T / 2 1/ 2 1 2 N i=1 i 2 2 i u i i oint density for ( ) is f( ) (2 )   exp ( ) ( ) logL= logL where1 logL ( , ) = T log2 log  ( ) ( ) 21 = T l 2 i i i1 i i i i i i i1 i i i i i i y  X y  X y  X , y  X y  X + + og2 log  1 i i i i Panel Data Algebra (3) = + = + = + i 2 T 2 2 2 i u 2 i i i i i i 2 2 2 i u 2 2 i i i i 2 2 2 i u 1 I T So, 1 T (T ) 1 T1 i1 i ii ii Panel Data Algebra (3, cont.) = + + i 2 2 2 2 2 u T 2 t 1 2 2 = = [ ]=  =( ) , = a characteristic root of Roots are (real since is symmetric) solutions to...
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 Fall '11
 WillamGreene
 Maximum likelihood, Likelihood function, TI, log likelihood, logL, Tijk ρu

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