【正文】 ? ? ? ?( ) , ( )TF E F E i ittvuu T v T N u T v T Nv v TN T N??? ? ? ???? ? ?? ? ? ?? ? ?? ? ? ? ? ??δ Z Z Z y v y Z δv v v vΩ J I I Ω J I ISpatial Error Model Estimation ? The Model: SPAR(1) ? Fixed Effects ? Random Effects ()TTWi???? ? ?? ? ?yX βεε I ε ee u v2 2 2 2 211(,( ) ,v u vT T N T NV a r T? ? ? ? ?? ? ? ?? ? ? ? ?11e ) Q I J I Q J I2( ) ( ) v NTV a r V a r ???e v I11( ) ( ) ( ) ( ) 39。 ) /( 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。 39。 ) / ( 1 )( 39。 , , /? ? ? ?? ?39。? ?( ) ( 39。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。 )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。22239。, ( 1 )TI n o rm a l ii d E I V InW n N T????? ? ? ?ε W ε ε W εε εWI1()( ) , ( 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。 ( ) ( )( 1 ) ( 39。2??~ ( 1 )? ? ? ??, / ( 1 ) ,? ? ?( ) ( ) / ( 1 ) . ( 39。[ ( 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。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。 ) / ( 1 ) 0vvE N TE N T trac e W W NE N T????????e Qee Q ee Qe2112111( 39。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????????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