【正文】 he 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。 ) / ( 39。 ) / ( 1 ) ( 39。 ) / ( 1 ) ( 39。 / ( 1 ) , 39。 , ( ) ( 39。 )V a r? ? ?????δ Z Ω ZZ Ω yδ Z Ω Z1? ( 39。 ) 39。 ) 39。 )vTV a r W B B? ???ε X I2 2 1( | , )( ) ( 39。 )u T v TV 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。 , ( ) ( 39。??Z H H H H Z2( | ) 0 , ( , ) 0, TE C o vw h e re W????? ? ???v H Z HH X W X W X W ISpatial Lag Model Estimation ? Random Effects 22( ) ( )Tu T v T NV ar ????? ? ?? ? ? ?yZ δεε i u vε Ω J I I2 2 2 2 211 ,( ) ,v u vT T N T NTw h e r e? ? ? ? ?? ? ? ?? ? ? ? ?11Ω Q I J I Q J ISpatial Lag Model Estimation Random Effects: IV/GLS ? Instrumental Variables ? TwoStage Generalized Least Squares 1 1 111? ? ?( 39。 ) 39。 )V a r? ? ? ? ???δ Z Ω ZZ Ω y δ Z Ω Z11222 2 1 2 2? ?? ? ? ? ?( 39。 /? ?? ? ? ?( ) , ( )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