【正文】 Random Effects ? Spatial Error Model: A=I or ?=0 ? Spatial Lag Model: B=I or ?=0 ? Panel Data Model: A=B=I 21( | , ) ( 39。 )vvvRL M ErrorBR Q W J WB T t rac e W W W W????????????? ? ? ???????? ? ? ?????ε εLMLag Test Statistic Pooled Model ? LM Test Statistic for Spatial Lag 239。 ) 39。~ ( ( ) , ( ) )? ??39。2??~ ( 1 )?? ? ? ?, / ,? ? ?( ) ( ) / . ( 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。 ) / ( 1 ) 0vvE N TE N T trac e W W NE N T????????e Qee Q ee Qe2112111( 39。??Z H H H H Z2( | ) 0 , ( , ) 0, TE C o vw h e r e W????? ? ???ε H Z HH X W X W X W ISpatial Lag Model Estimation Random Effects: IV/GLS ? Feasible Generalized Least Squares ? Estimate ?v2 and ?u2 from the fixed effects model: ? FGLS for random effects model: 1 1 1 1 1? ? ? ? ?? ? ?( 39。[ ( 39。 ( ) ( )( 1 ) ( 39。 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 W??? ( 39。, ( 1 )TI n o rm a l ii d E I V InW n N T????? ? ? ?ε W ε ε W εε εWI1()( ) , ( 39。 )TL M L a gBW N TB y y T t ra c e WW W W???????????????? ? ? ? ???? ? ???ε WyWI ε ε ε yX βW M WLMLag Test Statistic Fixed Effects Model ? LM Test Statistic for Spatial Lag 239。T W????? ? ?Z I y XδβyZ δεSpatial Lag Model Estimation ? Fixed Effects 2 2( ) ( ) ()Tv N T vV a r V a r V a r? ?? ? ? ? ?? ????????? ?? ?? ?yZ δ i u v yZ δvε v I vQ,()T T Nw h e r e? ? ?y = Q y Z = Q Z v = Q vQ I J ISpatial Lag Model Estimation Fixed Effects: IV or 2SLS ? Instrumental Variables ? TwoStage Least Squares 1 2 12? ? ?? ? ?( 39。 , , /? ? ? ?? ?39。 39。 ) /( 39。 ) 39。 ]?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。 )tr a c eE I w h e r enK?? ? ??MW M I X X X X39。 )tr a c eE I w h e r enK?? ? ??MW M I X X X X39。22239。 ) 39。 / ( 1 ) , 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**, . . . , ( )[ ( )