PanelDataNotes-5

# PanelDataNotes-5 - Econometric Analysis of Panel Data...

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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 5. 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[ ε 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 ε t = + c + , T observations in group i = + + , note (c ,c ,...,c ) = + + , T observations in the sample c= ( , ,... ) , ′ ′ = Σ ′ ′ ′ ′ x β y X i β ε X c c β ε y X c β ε c c c N i= 1 i T by 1 vector Σ 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 ε E[ | ] ε E[ | ] ε σ E[u | ] E[u | ] σ = + + u for T observations ε ′ = = = = = it i x b X X X X y X i β ε 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 ] ε ε ε σ σ σ σ σ σ σ σ σ σ σ σ + + = + i ε L L M M O M K Notation 1 1 1 2 2 2 N N N i u T observations u T observations u T observations = + + T observations = + = + + Σ 1 1 1 2 2 2 N N N N i=1 y X i ε y X i ε β y X i ε X u β ε X w β I n all that fo M M M M M ′ ′ i it it llows, except where explicitly noted, X, X and x contain a constant term as the first element. To avoid notational clutter, in those cases, x etc. will simply denote the counterpart without the constant term. Use of the symbol K for the number of variables will thus be context specific but will usually include the constant term. 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 ε I ii I ii Ω Ω Ω w X Ω L L M M O M K L L M M O M K ε ε ε ε ε σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ i (Note these differ only in the dimension T) Regression Model-Orthogonality = = = = = ′ ′ Σ = Σ = Σ Σ ′ ′ Σ Σ Σ ′ ′ Σ Σ = Σ i N N i 1 i i 1 i i i i N i 1 i i i i i i i i N i i i 1 i 1 plim # observations 1 1 plim plim ( + u ) T T 1 plim T + Tu T T T T plim f + f u , 0 < f < 1 T T T pli N N i=1 i i i=1 i i N N i i i i i=1 i=1 N N i i i i i=1 i=1 X'w X w X i ε X X i ε X X i ε ′ Σ Σ = = 2200...
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PanelDataNotes-5 - Econometric Analysis of Panel Data...

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