【正文】 V a r WBB?? ?? ? ?εXJI( | , ) 0EW ?εXSpatial Panel Data Models Example: U. S. Productivity (48 States, 17 Years) ? Panel Data Model ? ln(GSP) = b0 + b1 ln(Public) + b2ln(Private) + b3ln(Labor) + b4(Unemp) + e ? e ? i?u + v ? Spatial Lag Model ? ln(GSP) = b0 + b1 ln(Public) + b2ln(Private) + b3ln(Labor)+ b4(Unemp) + λW ln(GSP) + e ? e ? i?u + v ? Spatial Error Model ? ln(GSP) = b0 + b1 ln(Public) + b2ln(Private) + b3ln(Labor) + b4(Unemp) + e ? e ? ? We ? e , e ? i?u + v ? Spatial Mixed Model ? ln(GSP) = b0 + b1 ln(Public) + b2ln(Private) + b3ln(Labor) + b4(Unemp) + λW ln(GSP) + e ? e ? ? We ? e , e ? i?u + v Model Estimation ? Based on panel data models (pooled, fixed effects, random effects), we consider: ? Spatial Error Model ? Spatial Lag Model ? Spatial Mixed Model ? Model Estimation ? Generalized Least Squares (IV/GLS) ? Generalized Method of Moments (GMM/GLS) ? Maximum Likelihood Estimation Spatial Lag Model Estimation ? The Model: SPLAG(1) ? OLS is biased and inconsistent. ()( ( ) , ) 0TTTWCov W?? ? ? ?? ? ???y I y X βεε i u vIy ε? ?? ?()39。Spatial Econometric Analysis Using GAUSS 10 KuanPin Lin Portland State University Spatial Panel Data Models ? The General Model 11()()( ) [ ( ) ( ) ]( ) , ( )TTTT T TNNWWABw he re A W B W??????? ? ? ?? ? ? ? ?? ? ? ? ? ? ?? ? ? ?y I y X βεε I ε i u vy I X β I i u vIISpatial Panel Data Models ? Assumptions ? Fixed Effects ? Random Effects ? Spatial Error Model: A=I or ?=0 ? Spatial Lag Model: B=I or ?=0 ? Panel Data Model: A=B=I 21( | , ) ( 39。 )vTV a r W B B? ???ε X I2 2 1( | , )( ) ( 39。39。 ) 39。 )? ? ? ?? 39。 ) 39。 ) 39。 )V a r? ? ?????δ Z Ω ZZ Ω yδ Z Ω Z1? ( 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。 / ( 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 ) ( 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。 ) / ( 1 ) ( 39。 ) / ( 1 ) 0vvE N TE N T trac e W W NE N T????????e Qee Q ee Qe2112111( 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。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**, . .