【正文】 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。 ) / 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。 ) / ( 1 ) 0vvE N TE N T trac e W W NE N T????????vvvvvv****, ( )[ ( ) ] , [ ( ) ]TT N T Nw h e r e WWW??? ? ?? ? ? ? ? ?v y X β v I vy I I y X I I XSpatial Error Model Estimation Fixed Effects ? The Model: SPAR(1) ? Estimate b and ? iteratively: GMM/GLS ? OLS ? GMM ? GLS **()T W????? ? ??? ? ? ?yX βε yX βvε I ε v**?? ? ?( ) ( ) ( )?? ?( ) ( )T W????? ? ??? ? ? ???? ? ??yX β ε βε β I ε β vyX β v βSpatial Error Model Estimation Random Effects ? Moment Functions (Kapoor, Kelejian and Prucha, 2020) 22( 39。TTNV a r B V a r Bw h e r e B W???? ? ???ε I e IISpatial Error Model Estimation Fixed Effects ? Moment Functions 22( 39。 ) 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。 )? ? ? ?? 39。39。[ ( 39。[ ( 39。2??~ ( 1 )? ? ? ??, / ( 1 ) ,? ? ?( ) ( ) / ( 1 ) . ( 39。2??~ ( 1 )?? ? ? ?, / ,? ? ?( ) ( ) / . ( 39。 ( ) ( )( 1 ) ( 39。 )?? ? ? ?,/TWL M ErrorT t rac e W W W WNT??????? ????????? ? ?ε I εε yX β ε εLMError Test Statistic Fixed Effects Model ? LM Test Statistic for Spatial Error 239。 ) 39。~ ( ( ) , ( ) )? ??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。39。~ ( ( ) , ( ) )? ? ?39。 ) 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。????ε yX ββ X X X yLMError Test Statistic Pooled Model ? LM Test Statistic for Spatial Error 239。22239。 )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。 )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。 )TL M L agBW N TB y y T t ra c e WW W W???????????????? ? ? ? ? ???? ? ? ???ε WyWI ε ε ε yX βW M WLMLag Test Statistic Random Effects Model ? LM Test Statistic for Spatial Lag