【正文】 tial Weights Matrix ? Anselin (1988) [] ? China 30 Provinces [, ] ? Ertur and Kosh (2020) [] ? Homework ? . 48 Lower States [] ? . 3109 Counties [] [] Spatial Contiguity Weights Matrix Anselin (1988): W1, W2, W3 use gpe2。 w6=denseSubmat(sparseOnes(spwpower(spw(w1),6),n,n),0,0)。 d=3963*arccos(sin(y39。), 1 , 2 , .. .,? ?? ( ) ( 39。 ) 39。 )nni j i j i jij kVaree??????????X X x xβ X X X X X Xm a x( / ) ( / )ij ij ij ijk K d d or k K d d??ij ijdk? ? ?Time Series HAC Estimator General Heteroscedasticity and Autocorrelation ? NeweyWest Estimator 239。11 , 2 , .. .,ni j jj wWin?????????? xXW? ? ?yX β X γ ε2( | , ) 0( | , ) ( 39。( 39。 ) 39。111? ?39。 DistanceBased Spatial Weights Ertur and Kosh (2020) ? Kernel Weight Function ? Parzen Kernel ? Bartlett Kernel (Tricubic Kernel) ? TurkeyHanning Kernel ? Guassian or Exponeial Kernel 00: [ 1 , 1 ]( ) 0 | || | ( ) 0 | |KRE it he r K z if z z for som e zO r z K z as z????? ? ?( ) 1 , ( ) 0 , | ( ) |()K z dz z K z dz K z dzz K z dz k w he r e k i s a c ons t ant? ? ? ??? ? ??Kernel Weights Spatial Matrix An Example ? Negative Exponential Distance ? Negative Gaussian Distance ? ?( / ) e x p 2 /ij ij ijk K d d d d? ? ?? ?2m a x m a x( / ) e x p ( / )ij ij ijk K d d d d? ? ?1iiijijkw W Kk i j??? ? ? ????IGaussian Distance Weights Matrix Ertur and Kosh (2020) Spatial HAC Estimator ? The Classical Model 11垐垐39。 x=pi*xc/180。 w4=denseSubmat(sparseOnes(spwpower(spw(w1),4),n,n),0,0)。 do model estimation ? // variables y, x are generated earlier ? /* ? ** Retrieve output control variables for model evaluation and analysis ? */ ? /* ? ** Set more input control variables if needed, for model prediction ? ** (. _b for estimated parameters) ? */ ? call forecast(y,x)。 y = y~qtoa1(x[.,i])。)。x)。s numerical calculation? This example addresses this important problem. Suppose e is a known small positive number, and the 5x4 matrix X is defined as follows: Verify that the eigenvalues of X39。 do until ... endo。(transpose) ? Useful Algrbra and Matrix Operations exp ln log abs sqrt pi sin cos inv invpd(inverse) det(determinant) ? Example ? Least Squares: b=y/x GAUSS Programming Useful GAUSS Functions ? System Functions: use, load, output ? Data Generating Functions: ones, zeros, eye, seqa, seqm, rndu, rndn ? Data Conversion Functions: reshape, selif, delif, vec, vech, xpnd, submat, diag, diagrv ? Basic Matrix Functions: ? Matrix Description: rows, cols, maxc, minc, meanc, median, stdc ? Matrix Operations: sumc,cumsumc,prodc,cumprodc,sortc,sorthc,sortind ? Matrix Computation: det,inv,invpd,solpd,vcx,corrx,cond,rank,eig,eigh ? Probability and Statistical Functions: pdfn, cdfn, cdftc, cdffc, cdfchic, dstat, ols ? Calculus Functions: gradp, hessp, intsimp, linsolve, eqsolve, sqpsolve GAUSS Programming Controlling Execution Flow ? If Statement if。 ) /( 39。 ) /( 39。 )V a r? ? ? ? ???δ Z Ω ZZ Ω y δ Z Ω Z11222 2 1 2 2? ?? ? ? ? ?( 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。 )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