【正文】 TTTTTTNWWiBwhe re Wan d B W?????? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ???y I y X βεε I ε u vyZ δ I i u vZ I y X δβISpatial Mixed Model Estimation ? TwoStage Estimation ? Sample moment functions are the same as in the spatial error AR(1) model. The efficient GMM estimator follows exactly the same as the spatial error AR(1) model. ? The transformed model which removes spatial error AR(1) correlation is estimated the same way as the spatial lag model using IV and GLS. Spatial Mixed Model Estimation Fixed Effects ? The Model: SPARAR(1,1) * * *()()()()()TTTTTTWWWWW?????? ? ? ? ?? ? ? ? ??? ? ? ???? ? ???? ? ? ?? ? ? ? ?y I y X βεy I y X βεε I ε eε I ε ve i u vy I y X βv**, . . . , ( )[ ( ) ] , [ ( ) ]TNT N T Nw h e r eWW??? ? ? ?? ? ? ? ? ?y Q y Q I J Iy I I y X I I XSpatial Mixed Model Estimation Fixed Effects ? Estimate b and ? iteratively: GMM/GLS ? IV/2SLS ? GMM ? GLS * * *? ?( ) ,? ? ? ? ?( , ) ( ) ( , )? ?? ? ?( ) ( ) ( ) ( ) ,TTTWWW??? ? ? ?? ? ? ? ?? ? ? ? ??? ? ? ???? ? ? ? ??y I y X β ε βε β I ε β vy I y X β v β**( ) [ ( ) ] , ( ) [ ( ) ]T N T Nwhe re W W? ? ? ?? ? ? ? ? ?y I I y X I I XSpatial Mixed Model Estimation Random Effects ? The Model: SPARAR(1,1) * * *()()()TTTTTWWW???? ? ? ?? ? ?? ? ?? ? ? ? ?? ?? ? ??y I y X βεε I ε ee i u vy I y X βee i u v**[ ( ) ] , [ ( ) ]T N T Nwhe re W W??? ? ? ? ? ?y I I y X I I XSpatial Mixed Model Estimation Random Effects ? Estimate b,? and ? iteratively: GMM/GLS ? IV/2SLS ? GMM ? GLS 22* * *22? ?( ) ,? ? ? ? ? ? ?( , ) ( ) ( , ) , ,? ?? ? ?( ) ( ) ( ) ( ) ,? ? ?( , )TT v uTvuWWWw i t h??? ? ? ? ? ?? ? ? ? ???? ? ? ? ??? ? ? ??? ? ? ? ? ???? ? ??y I y X β ε βε β I ε β ey I y X β e β**( ) [ ( ) ] , ( ) [ ( ) ]T N T Nwhe re W W? ? ? ?? ? ? ? ? ?y I I y X I I XExample: U. S. Productivity Baltagi (2020) [] ? Spatial Panel Data Model: GMM/GLS (Spatial Error) ln(GSP) = b0 + b1 ln(Public) + b2ln(Private) + b3ln(Labor) + b4(Unemp) + ?, ? =ρW ? + e, e = i?u + v Fixed Effects Random Effects b1 b2 * * b3 * * b4 * * b0 * ρ * * Example: U. S. Productivity Baltagi (2020) [] ? Spatial Panel Data Model: GMM/GLS (Spatial Mixed) ln(GSP) = b0 + b1 ln(Public) + b2ln(Private) + b3ln(Labor) + b4(Unemp) + λW ln(GSP) + ? , ? =ρW ? + e , e = i?u + v Fixed Effects Random Effects b1 b2 * * b3 * * b4 * * b0 * λ * * ρ * * Maximum Likelihood Estimation ? Error Components ? Assumptions ? Fixed Effects: ? Random Effects: 2~ ( , )v N TN ?v 0 I2 2 39。 ) / ( 1 ) ( 39。 / ( 1 ) , 39。 )V a r? ? ?????δ Z Ω ZZ Ω yδ Z Ω Z1? ( 39。 ) 39。 ) ]?[ 39。22239。? ?()?~ ( 1 )( 1 ) . ( 39。 )tr a c eE I w h e r enK?? ? ??MW M I X X X X39。,TI n o rm a l ii d E I V InW n NT????? ? ?ε W ε ε W εε εWI1()( ) , ( 39。 )tr a c eE I w h e r enK?? ? ??MW M I X X X X39。 2 22( ) [ ( ) ] [ ( ) ]( ) ( )( ) ( 2 )tr a c e tr a c e tr a c eV I E In K n K????? ? ?M W M W M W M W1? ?? ( 39。 )? ? ? ??, / ( 1 )TWL M Err orT t r ac e W W W WNT??????? ?????????? ? ? ?ε I εε yX β ε εLMError Test Statistic Random Effects Model ? LM Test Statistic for Spatial Error ? See Baltagi, B. H., S. H. Song, W. Koh (2020). 2222121441? ?39。39。 ]?39。 , ( ) ( 39。 ) 39。 /? ?