【正文】 Estimation Random Effects ? Moment Functions (Kapoor, Kelejian and Prucha, 2020) 22( 39。 ) / ( 1 )( 39。 ) / ( 1 ) ( 39。 ) /( 39。 ) / ( 1 ) 0vvE N TE N T trac e W W NE N T????????e Qee Q ee Qe2112111( 39。 ) /( 39。 ) / ( 39。 ) /( 39。 ) / 0 , ( )TENE N trac e W W NE N whe re W????? ? ?e Q ee Q ee Q e e I eSpatial Error Model Estimation Random Effects ? The Model: SPAR(1) ? Estimate b and ? iteratively: GMM/GLS ? OLS ? GMM ? GLS **()TTTW??? ?? ???? ? ? ???? ? ???? ? ? ?yX βεyX βeε I ε ee i u ve i u v**?? ? ?( ) ( ) ( )?? ?( ) ( )T W????? ? ??? ? ? ???? ? ? ?yX β ε βε β I ε β eyX β e β**[ ( ) ] , [ ( ) ]T N T Nwhe re W W??? ? ? ? ? ?y I I y X I I XSpatial Mixed Model Estimation ? The Model: SARAR(1,1) ? ? ? ?1()()( ) ( )( ) , 39。 39。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) + e, e =ρW e + 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) + e , e =ρW e + 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。~ ( , ) ,TNu T v T T T TN??? ? ? ? ? ?? ? ? ?e i u v 0 Ω Ω IJ I J i i22~ ( , ) , ~ ( , ) ,v N T u NN N t?? ?v 0 I u 0 IT??e i u + vMaximum Likelihood Estimation Fixed Effects ? LogLikelihood Function 222( , , , ) l n( 2 ) l n( )2239。l n | | l n | |2( ) , ( )( , , | , , ) ( ) ( ) ( )vvvNNT T TN T N TLT A T Bw here A I W B I WW I B I A I B? ? ? ? ??????? ? ?? ? ?? ? ? ?? ? ? ? ? ?βeeee β y X y X βMaximum Likelihood Estimation Fixed Effects ? LogLikelihood Function (Lee and Yu, 2020) ? Where z* is the transformation of z using the orthogonal eigenvector matrix of Q. 2239。* *2* * * *( 1 ) ( 1 )( , , , ) l n( 2 ) l n( )22( 1