【正文】 . , ( )[ ( ) ] , [ ( ) ]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。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。12 2 2 2( , , , , )1l n( 2 ) l n | | ( ) l n | | l n | |2 2 2( , )( ) , ( )( , , | , , ) ( ) ( ) ( )uvNT v u v uNNT T TLN T NI T A T Bw he re I JA I W B I WW I B I A I B? ? ? ??? ? ? ???????? ? ? ? ? ? ? ?? ? ? ? ?? ? ? ?? ? ? ? ? ?βeeee β y X y X βExample: U. S. Productivity Baltagi (2020) [] ? Spatial Panel Data Model: QML (Spatial Lag) ln(GSP) = b0 + b1 ln(Public) + b2ln(Private) + b3ln(Labor) + b4(Unemp) + λW ln(GSP) + e , e = i?u + v Fixed Effects Random Effects b1 b2 * * b3 * * b4 * * b0 * λ * * Example: U. S. Productivity Baltagi (2020) [] ? Spatial Panel Data Model: QML (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: QML (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 * * References ? Elhorst, J. P. (2020). Specification and estimation of spatial panel data models, International Regional Science Review 26, 244268. ? Kapoor M., Kelejian, H. and I. R. Prucha, “Panel Data Models with Spatially Correlated Error Components,” Journal of Econometrics, 140, 2020: 97130. ? Lee, L. F., and J. Yu, “Estimation of Spatial Autoregressive Panel Data Models with Fixed Effects,” Journal of Econometrics 154, 2020: 165185. Spatial Econometric Analysis Using GAUSS 2 KuanPin Lin Portland State Univerisity GAUSS Mathematical and Statistical System ? Windows Interface ? Windows ? Command, Error, Log, … ? Menu ? File, Edit, Run, …, Help ? Operation ? Interactive Mode