---------------------------------------------------------------------- name: log: analysis.log log type: text opened on: 01 Dec 2023, 10:00:00 . **************************************************** . * Cross-Sectional Spatial Regression in Stata: . * Crime in Columbus Neighborhoods . **************************************************** . clear all . macro drop _all . set more off . . *--------------------------------------------------- . * Section 3: Setup and data loading . *--------------------------------------------------- . * 3.1 Spatial weight matrix . use "https://github.com/quarcs-lab/data-open/raw/master/Columbus/columbus/Wqueen_fromStata_spmat.dta", clear . gen id = _n . order id, first . spset id Spaceset variable: id Data has been set. . spmatrix fromdata WqueenS_fromStata15 = v*, normalize(row) replace . spmatrix summarize WqueenS_fromStata15 Summary of spatial-weighting matrix WqueenS_fromStata15 Dimension: 49 x 49 Stored as type: 49 x 49 Number of stored links: 230 Percent of stored links > 0: 100.0% Total stored links: 230 . . * 3.2 Dataset . use "https://github.com/quarcs-lab/data-open/raw/master/Columbus/columbus/columbusDbase.dta", clear . spset id Sp aceset variable: id Data has been set. . label var CRIME "Crime" . label var INC "Income" . label var HOVAL "House value" . . *--------------------------------------------------- . * Section 4: OLS baseline and spatial diagnostics . *--------------------------------------------------- . * 4.1 OLS regression . regress CRIME INC HOVAL Source | SS df MS Number of obs = 49 -------------+---------------------------------- F(2, 46) = 28.39 Model | 5765.1758 2 2882.58790 Prob > F = 0.0000 Residual | 4670.9736 46 101.54290 R-squared = 0.5524 -------------+---------------------------------- Adj R-squared = 0.5330 Total | 10436.1494 48 217.41978 Root MSE = 10.077 ------------------------------------------------------------------------------ CRIME | Coefficient Std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- INC | -1.597311 .3341308 -4.78 0.000 -2.269967 -.9246553 HOVAL | -.2739315 .1031999 -2.65 0.011 -.4816777 -.0661854 _cons | 68.61896 4.735292 14.49 0.000 59.08750 78.15042 ------------------------------------------------------------------------------ . eststo OLS . . estat ic Akaike's information criterion and Bayesian information criterion -------------+---------------------------------------------------------------- Model | N ll(null) ll(model) df AIC BIC -------------+---------------------------------------------------------------- . | 49 -229.3837 -210.0794 3 426.1588 431.7981 | 3 378.1588* 383.7981 -------------+---------------------------------------------------------------- Note: BIC uses N = number of observations. See [R] IC note. * indicates the best model using the respective criterion. . mat s = r(S) . quietly estadd scalar AIC = s[1,5] . . * 4.2 Moran's I test on OLS residuals . regress CRIME INC HOVAL Source | SS df MS Number of obs = 49 -------------+---------------------------------- F(2, 46) = 28.39 Model | 5765.1758 2 2882.58790 Prob > F = 0.0000 Residual | 4670.9736 46 101.54290 R-squared = 0.5524 -------------+---------------------------------- Adj R-squared = 0.5330 Total | 10436.1494 48 217.41978 Root MSE = 10.077 ------------------------------------------------------------------------------ CRIME | Coefficient Std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- INC | -1.597311 .3341308 -4.78 0.000 -2.269967 -.9246553 HOVAL | -.2739315 .1031999 -2.65 0.011 -.4816777 -.0661854 _cons | 68.61896 4.735292 14.49 0.000 59.08750 78.15042 ------------------------------------------------------------------------------ . estat moran, errorlag(WqueenS_fromStata15) Moran's I test for spatial dependence in the error distribution H0: error is i.i.d. I = 0.2222 E(I) = -0.0208 sd(I) = 0.0855 z = 2.84 Prob > z = 0.0045 . . * 4.3 LM tests (requires spatwmat/spatdiag) . spatwmat using "https://github.com/quarcs-lab/data-open/raw/master/Columbus/columbus/Wqueen_fromStata_spmat.dta", name(WqueenS_spatwmat) eigenval(eWqueenS_spatwmat) standardize . . quietly reg CRIME INC HOVAL . spatdiag, weights(WqueenS_spatwmat) Spatial Diagnostics ------------------- Diagnostic Statistic df p-value --------------------------------------------------------------- Spatial error: Moran's I 2.838 1 0.005 Lagrange multiplier 5.327 1 0.021 Robust Lagrange multiplier 2.196 1 0.138 Spatial lag: Lagrange multiplier 3.412 1 0.065 Robust Lagrange multiplier 0.281 1 0.596 --------------------------------------------------------------- . . *--------------------------------------------------- . * Section 5: First-generation spatial models . *--------------------------------------------------- . * 5.1 SAR (Spatial Autoregressive / Spatial Lag) . spregress CRIME INC HOVAL, ml dvarlag(WqueenS_fromStata15) Spatial autoregressive model Number of obs = 49 Maximum likelihood estimates Wald chi2(2) = 54.83 Prob > chi2 = 0.0000 Log-likelihood = -184.9262 Pseudo R2 = 0.5830 ------------------------------------------------------------------------------ CRIME | Coefficient Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- CRIME | INC | -1.031218 .3359487 -3.07 0.002 -1.689665 -.3727706 HOVAL | -.2654187 .0921835 -2.88 0.004 -.4460950 -.0847425 _cons | 45.07187 7.840615 5.75 0.000 29.70452 60.43923 -------------+---------------------------------------------------------------- WqueenS_fr~5 | CRIME | .4282614 .1228467 3.49 0.000 .1874863 .6690364 -------------+---------------------------------------------------------------- /sigma2_e | 80.93012 16.36204 54.90534 119.31143 ------------------------------------------------------------------------------ . eststo SAR . . estat ic Akaike's information criterion and Bayesian information criterion -------------+---------------------------------------------------------------- Model | N ll(null) ll(model) df AIC BIC -------------+---------------------------------------------------------------- . | 49 -184.9262 5 379.8524 389.2615 | 5 373.8524* 383.2615 -------------+---------------------------------------------------------------- Note: BIC uses N = number of observations. See [R] IC note. . mat s = r(S) . quietly estadd scalar AIC = s[1,5] . . estat impact Average impacts Delta-method dy/dx std. err. z P>|z| [95% conf. interval] ---------------------------------------------------------------------------------- Direct | INC | -1.1024 .3486 -3.16 0.002 -1.7857 -.4191 HOVAL | -.2838 .0983 -2.89 0.004 -.4765 -.0911 ---------------------------------------------------------------------------------- Indirect | INC | -.7594 .3712 -2.05 0.041 -1.4870 -.0319 HOVAL | -.1954 .1123 -1.74 0.082 -.4156 .0247 ---------------------------------------------------------------------------------- Total | INC | -1.8618 .5803 -3.21 0.001 -2.9991 -.7245 HOVAL | -.4792 .1722 -2.78 0.005 -.8167 -.1417 ---------------------------------------------------------------------------------- . . * 5.2 SEM (Spatial Error Model) . spregress CRIME INC HOVAL, ml errorlag(WqueenS_fromStata15) Spatial error model Number of obs = 49 Maximum likelihood estimates Wald chi2(2) = 50.51 Prob > chi2 = 0.0000 Log-likelihood = -184.3792 Pseudo R2 = 0.5877 ------------------------------------------------------------------------------ CRIME | Coefficient Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- CRIME | INC | -.9376126 .3393120 -2.76 0.006 -1.6026520 -.2725732 HOVAL | -.3023101 .0909284 -3.32 0.001 -.4805264 -.1240937 _cons | 59.62283 5.472159 10.90 0.000 48.89758 70.34808 -------------+---------------------------------------------------------------- WqueenS_fr~5 | lambda | .5623481 .1329743 4.23 0.000 .3017234 .8229728 -------------+---------------------------------------------------------------- /sigma2_e | 74.68211 15.10203 50.71684 109.96143 ------------------------------------------------------------------------------ . eststo SEM . . estat ic Akaike's information criterion and Bayesian information criterion -------------+---------------------------------------------------------------- Model | N ll(null) ll(model) df AIC BIC -------------+---------------------------------------------------------------- . | 49 -184.3792 5 378.7584 388.1675 | 5 372.7584* 382.1675 -------------+---------------------------------------------------------------- . mat s = r(S) . quietly estadd scalar AIC = s[1,5] . . estat impact Average impacts Delta-method dy/dx std. err. z P>|z| [95% conf. interval] ---------------------------------------------------------------------------------- Direct | INC | -.9376 .3393 -2.76 0.006 -1.6027 -.2726 HOVAL | -.3023 .0909 -3.32 0.001 -.4805 -.1241 ---------------------------------------------------------------------------------- Indirect | INC | .0000 .0000 . . . . HOVAL | .0000 .0000 . . . . ---------------------------------------------------------------------------------- Total | INC | -.9376 .3393 -2.76 0.006 -1.6027 -.2726 HOVAL | -.3023 .0909 -3.32 0.001 -.4805 -.1241 ---------------------------------------------------------------------------------- . . *--------------------------------------------------- . * Section 6: Models with spatial lags of X . *--------------------------------------------------- . * 6.1 SLX (Spatial Lag of X) . spregress CRIME INC HOVAL, ml ivarlag(WqueenS_fromStata15: INC HOVAL) Spatial lag of X model Number of obs = 49 Maximum likelihood estimates Wald chi2(4) = 59.12 Prob > chi2 = 0.0000 Log-likelihood = -187.5218 Pseudo R2 = 0.5607 ------------------------------------------------------------------------------ CRIME | Coefficient Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- CRIME | INC | -1.472514 .3564127 -4.13 0.000 -2.171071 -.7739571 HOVAL | -.3087321 .0942617 -3.27 0.001 -.4934816 -.1239826 _cons | 74.27642 7.564388 9.82 0.000 59.45050 89.10233 -------------+---------------------------------------------------------------- WqueenS_fr~5 | INC | .4965121 .5309834 0.94 0.350 -.5442062 1.537231 HOVAL | -.1487230 .2529164 -0.59 0.557 -.6444300 .3469841 -------------+---------------------------------------------------------------- /sigma2_e | 87.47219 17.68413 59.34671 128.84631 ------------------------------------------------------------------------------ . eststo SLX . . estat ic Akaike's information criterion and Bayesian information criterion -------------+---------------------------------------------------------------- Model | N ll(null) ll(model) df AIC BIC -------------+---------------------------------------------------------------- . | 49 -187.5218 6 387.0436 398.3325 | 6 379.0436* 390.3325 -------------+---------------------------------------------------------------- . mat s = r(S) . quietly estadd scalar AIC = s[1,5] . . estat impact Average impacts Delta-method dy/dx std. err. z P>|z| [95% conf. interval] ---------------------------------------------------------------------------------- Direct | INC | -1.4725 .3564 -4.13 0.000 -2.1711 -.7740 HOVAL | -.3087 .0943 -3.27 0.001 -.4935 -.1240 ---------------------------------------------------------------------------------- Indirect | INC | .4965 .5310 0.94 0.350 -.5442 1.5372 HOVAL | -.1487 .2529 -0.59 0.557 -.6444 .3470 ---------------------------------------------------------------------------------- Total | INC | -.9760 .5825 -1.68 0.094 -2.1177 .1657 HOVAL | -.4575 .2639 -1.73 0.083 -.9747 .0598 ---------------------------------------------------------------------------------- . . * 6.2 SDM (Spatial Durbin Model) . spregress CRIME INC HOVAL, ml dvarlag(WqueenS_fromStata15) ivarlag(WqueenS_fromStata15: INC HOVAL) Spatial Durbin model Number of obs = 49 Maximum likelihood estimates Wald chi2(4) = 55.48 Prob > chi2 = 0.0000 Log-likelihood = -183.1451 Pseudo R2 = 0.5983 ------------------------------------------------------------------------------ CRIME | Coefficient Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- CRIME | INC | -1.012437 .3620147 -2.80 0.005 -1.721972 -.3029020 HOVAL | -.3005824 .0887134 -3.39 0.001 -.4744575 -.1267072 _cons | 50.87235 9.843765 5.17 0.000 31.57882 70.16589 -------------+---------------------------------------------------------------- WqueenS_fr~5 | CRIME | .4362571 .1398223 3.12 0.002 .1622104 .7103038 INC | .5344928 .6075781 0.88 0.379 -.6563393 1.725325 HOVAL | -.1756174 .3017351 -0.58 0.561 -.7670074 .4157726 -------------+---------------------------------------------------------------- /sigma2_e | 72.93415 14.75478 49.53071 107.38752 ------------------------------------------------------------------------------ . eststo SDM . . estat ic Akaike's information criterion and Bayesian information criterion -------------+---------------------------------------------------------------- Model | N ll(null) ll(model) df AIC BIC -------------+---------------------------------------------------------------- . | 49 -183.1451 7 380.2902 393.4590 | 7 374.2902* 387.4590 -------------+---------------------------------------------------------------- . mat s = r(S) . quietly estadd scalar AIC = s[1,5] . . estat impact Average impacts Delta-method dy/dx std. err. z P>|z| [95% conf. interval] ---------------------------------------------------------------------------------- Direct | INC | -1.0793 .3757 -2.87 0.004 -1.8157 -.3429 HOVAL | -.3208 .0922 -3.48 0.001 -.5015 -.1401 ---------------------------------------------------------------------------------- Indirect | INC | -.2954 .8203 -0.36 0.719 -1.9031 1.3123 HOVAL | -.4531 .4816 -0.94 0.347 -1.3971 .4909 ---------------------------------------------------------------------------------- Total | INC | -1.3747 .8952 -1.54 0.125 -3.1293 .3799 HOVAL | -.7740 .5102 -1.52 0.129 -1.7740 .2260 ---------------------------------------------------------------------------------- . . *--------------------------------------------------- . * Section 7: Wald specification tests from SDM . *--------------------------------------------------- . quietly spregress CRIME INC HOVAL, ml dvarlag(WqueenS_fromStata15) ivarlag(WqueenS_fromStata15: INC HOVAL) . . * Wald test: Reduce to SLX? (rho = 0; NO if p < 0.05) . test ([WqueenS_fromStata15]CRIME = 0) ( 1) [WqueenS_fromStata15]CRIME = 0 chi2( 1) = 7.76 Prob > chi2 = 0.0053 . . * Wald test: Reduce to SAR? (theta = 0; NO if p < 0.05) . test ([WqueenS_fromStata15]INC = 0) ([WqueenS_fromStata15]HOVAL = 0) ( 1) [WqueenS_fromStata15]INC = 0 ( 2) [WqueenS_fromStata15]HOVAL = 0 chi2( 2) = 3.51 Prob > chi2 = 0.1726 . . * Wald test: Reduce to SEM? (common factor; NO if p < 0.05) . testnl ([WqueenS_fromStata15]INC = -[WqueenS_fromStata15]CRIME*[CRIME]INC) ([WqueenS_fromStata15]HOVAL = -[WqueenS_fromStata15]CRIME*[CRIME]HOVAL) (1) [WqueenS_fromStata15]INC = -[WqueenS_fromStata15]CRIME*[CRIME]INC (2) [WqueenS_fromStata15]HOVAL = -[WqueenS_fromStata15]CRIME*[CRIME]HOVAL chi2( 2) = 2.08 Prob > chi2 = 0.3535 . . *--------------------------------------------------- . * Section 8: Extended spatial models . *--------------------------------------------------- . * 8.1 SDEM (Spatial Durbin Error Model) . spregress CRIME INC HOVAL, ml ivarlag(WqueenS_fromStata15: INC HOVAL) errorlag(WqueenS_fromStata15) Spatial Durbin error model Number of obs = 49 Maximum likelihood estimates Wald chi2(4) = 49.60 Prob > chi2 = 0.0000 Log-likelihood = -183.3561 Pseudo R2 = 0.5965 ------------------------------------------------------------------------------ CRIME | Coefficient Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- CRIME | INC | -.8725810 .3634289 -2.40 0.016 -1.5848886 -.1602734 HOVAL | -.3287133 .0895607 -3.67 0.000 -.5042491 -.1531775 _cons | 62.40312 6.296218 9.91 0.000 50.06277 74.74347 -------------+---------------------------------------------------------------- WqueenS_fr~5 | INC | .4575841 .5463231 0.84 0.402 -.6131896 1.528358 HOVAL | -.1082341 .2814122 -0.38 0.700 -.6597918 .4433236 lambda | .5424611 .1504228 3.61 0.000 .2476378 .8372843 -------------+---------------------------------------------------------------- /sigma2_e | 72.32144 14.62892 49.11537 106.52419 ------------------------------------------------------------------------------ . eststo SDEM . . estat ic Akaike's information criterion and Bayesian information criterion -------------+---------------------------------------------------------------- Model | N ll(null) ll(model) df AIC BIC -------------+---------------------------------------------------------------- . | 49 -183.3561 7 380.7122 393.8810 | 7 374.7122* 387.8810 -------------+---------------------------------------------------------------- . mat s = r(S) . quietly estadd scalar AIC = s[1,5] . . estat impact Average impacts Delta-method dy/dx std. err. z P>|z| [95% conf. interval] ---------------------------------------------------------------------------------- Direct | INC | -.8726 .3634 -2.40 0.016 -1.5849 -.1603 HOVAL | -.3287 .0896 -3.67 0.000 -.5042 -.1532 ---------------------------------------------------------------------------------- Indirect | INC | .4576 .5463 0.84 0.402 -.6132 1.5284 HOVAL | -.1082 .2814 -0.38 0.700 -.6598 .4433 ---------------------------------------------------------------------------------- Total | INC | -.4150 .6178 -0.67 0.502 -1.6259 .7960 HOVAL | -.4369 .2882 -1.52 0.130 -1.0018 .1279 ---------------------------------------------------------------------------------- . . * 8.2 SAC (Spatial Autoregressive Combined) . spregress CRIME INC HOVAL, ml dvarlag(WqueenS_fromStata15) errorlag(WqueenS_fromStata15) Spatial autoregressive combined model Number of obs = 49 Maximum likelihood estimates Wald chi2(2) = 51.17 Prob > chi2 = 0.0000 Log-likelihood = -184.0543 Pseudo R2 = 0.5905 ------------------------------------------------------------------------------ CRIME | Coefficient Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- CRIME | INC | -1.037145 .3295422 -3.15 0.002 -1.683036 -.3912536 HOVAL | -.2819738 .0937163 -3.01 0.003 -.4656545 -.0982931 _cons | 50.36122 9.268145 5.43 0.000 32.19601 68.52644 -------------+---------------------------------------------------------------- WqueenS_fr~5 | CRIME | .3172541 .1870263 1.70 0.090 -.0493107 .6838189 lambda | .3818241 .2414286 1.58 0.114 -.0913672 .8550154 -------------+---------------------------------------------------------------- /sigma2_e | 74.28175 15.02908 50.44438 109.37122 ------------------------------------------------------------------------------ . eststo SAC . . estat ic Akaike's information criterion and Bayesian information criterion -------------+---------------------------------------------------------------- Model | N ll(null) ll(model) df AIC BIC -------------+---------------------------------------------------------------- . | 49 -184.0543 6 380.1086 391.3975 | 6 374.1086* 385.3975 -------------+---------------------------------------------------------------- . mat s = r(S) . quietly estadd scalar AIC = s[1,5] . . estat impact Average impacts Delta-method dy/dx std. err. z P>|z| [95% conf. interval] ---------------------------------------------------------------------------------- Direct | INC | -1.0806 .3444 -3.14 0.002 -1.7557 -.4056 HOVAL | -.2937 .0976 -3.01 0.003 -.4850 -.1024 ---------------------------------------------------------------------------------- Indirect | INC | -.4829 .3481 -1.39 0.165 -1.1651 .1994 HOVAL | -.1311 .0890 -1.47 0.141 -.3056 .0434 ---------------------------------------------------------------------------------- Total | INC | -1.5635 .5267 -2.97 0.003 -2.5957 -.5312 HOVAL | -.4248 .1505 -2.82 0.005 -.7197 -.1299 ---------------------------------------------------------------------------------- . . * 8.3 GNS (General Nesting Spatial) . spregress CRIME INC HOVAL, ml dvarlag(WqueenS_fromStata15) ivarlag(WqueenS_fromStata15: INC HOVAL) errorlag(WqueenS_fromStata15) General nesting spatial model Number of obs = 49 Maximum likelihood estimates Wald chi2(4) = 50.24 Prob > chi2 = 0.0000 Log-likelihood = -182.8123 Pseudo R2 = 0.6012 ------------------------------------------------------------------------------ CRIME | Coefficient Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- CRIME | INC | -.9525834 .3712653 -2.57 0.010 -1.6802509 -.2249160 HOVAL | -.3061482 .0908217 -3.37 0.001 -.4841555 -.1281409 _cons | 52.46351 10.46128 5.02 0.000 31.95977 72.96724 -------------+---------------------------------------------------------------- WqueenS_fr~5 | CRIME | .3497814 .1812371 1.93 0.054 -.0054368 .7049996 INC | .4021345 .6234862 0.64 0.519 -.8198751 1.624144 HOVAL | -.1178234 .3092185 -0.38 0.703 -.7238805 .4882338 lambda | .2487314 .2812653 0.88 0.377 -.3025383 .8000012 -------------+---------------------------------------------------------------- /sigma2_e | 70.81472 14.33318 48.09246 104.27931 ------------------------------------------------------------------------------ . eststo GNS . . estat ic Akaike's information criterion and Bayesian information criterion -------------+---------------------------------------------------------------- Model | N ll(null) ll(model) df AIC BIC -------------+---------------------------------------------------------------- . | 49 -182.8123 8 381.6246 396.6733 | 8 375.6246* 390.6733 -------------+---------------------------------------------------------------- . mat s = r(S) . quietly estadd scalar AIC = s[1,5] . . estat impact Average impacts Delta-method dy/dx std. err. z P>|z| [95% conf. interval] ---------------------------------------------------------------------------------- Direct | INC | -1.0104 .3883 -2.60 0.009 -1.7715 -.2494 HOVAL | -.3264 .0950 -3.44 0.001 -.5126 -.1403 ---------------------------------------------------------------------------------- Indirect | INC | -.1344 .8564 -0.16 0.875 -1.8130 1.5441 HOVAL | -.3897 .5012 -0.78 0.437 -1.3721 .5927 ---------------------------------------------------------------------------------- Total | INC | -1.1449 .9527 -1.20 0.229 -3.0122 .7224 HOVAL | -.7161 .5284 -1.35 0.176 -1.7517 .3195 ---------------------------------------------------------------------------------- . . *--------------------------------------------------- . * Section 9: Model comparison . *--------------------------------------------------- . esttab OLS SAR SEM SLX SDM SDEM SAC GNS, label stats(AIC) mtitle("OLS" "SAR" "SEM" "SLX" "SDM" "SDEM" "SAC" "GNS") ------------------------------------------------------------------------------------------------------------ (1) (2) (3) (4) (5) (6) (7) (8) OLS SAR SEM SLX SDM SDEM SAC GNS ------------------------------------------------------------------------------------------------------------ Crime Income -1.597*** -1.031*** -0.938*** -1.473*** -1.012*** -0.873** -1.037*** -0.953** (-4.78) (-3.07) (-2.76) (-4.13) (-2.80) (-2.40) (-3.15) (-2.57) House value -0.274*** -0.265*** -0.302*** -0.309*** -0.301*** -0.329*** -0.282*** -0.306*** (-2.65) (-2.88) (-3.32) (-3.27) (-3.39) (-3.67) (-3.01) (-3.37) Constant 68.619*** 45.072*** 59.623*** 74.276*** 50.872*** 62.403*** 50.361*** 52.464*** (14.49) (5.75) (10.90) (9.82) (5.17) (9.91) (5.43) (5.02) ------------------------------------------------------------------------------------------------------------ AIC 378.159 373.852 372.758 379.044 374.290 374.712 374.109 375.625 ------------------------------------------------------------------------------------------------------------ t statistics in parentheses * p<0.1, ** p<0.05, *** p<0.01 . . log close name: log: analysis.log log type: text closed on: 01 Dec 2023, 10:15:00 ----------------------------------------------------------------------