11 Exercises III
\[ \newcommand{\tr}{\mathrm{tr}} \newcommand{\rank}{\mathrm{rank}} \newcommand{\plim}{\operatornamewithlimits{plim}} \newcommand{\diag}{\mathrm{diag}} \newcommand{\bm}[1]{\boldsymbol{\mathbf{#1}}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Exp}{\mathrm{E}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand\given[1][]{\:#1\vert\:} \newcommand{\irow}[1]{% \begin{pmatrix}#1\end{pmatrix} } \]
Required packages
Session info
R version 4.6.0 (2026-04-24 ucrt)
Platform: x86_64-w64-mingw32/x64
Running under: Windows 11 x64 (build 26200)
Matrix products: default
LAPACK version 3.12.1
locale:
[1] LC_COLLATE=German_Germany.utf8 LC_CTYPE=German_Germany.utf8
[3] LC_MONETARY=German_Germany.utf8 LC_NUMERIC=C
[5] LC_TIME=German_Germany.utf8
time zone: Europe/Berlin
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods
[7] base
other attached packages:
[1] SDPDmod_0.0.7 splm_1.6-5 lfe_3.1.1
[4] plm_2.6-7 viridis_0.6.5 viridisLite_0.4.3
[7] tmap_4.4 ggplot2_4.0.3 spatialreg_1.4-3
[10] Matrix_1.7-5 spdep_1.4-2 spData_2.3.5
[13] mapview_2.11.4 sf_1.1-1
loaded via a namespace (and not attached):
[1] Rdpack_2.6.6 DBI_1.3.0
[3] deldir_2.0-4 gridExtra_2.3
[5] tmaptools_3.3 s2_1.1.11
[7] logger_0.4.2 sandwich_3.1-1
[9] rlang_1.2.0 magrittr_2.0.5
[11] dreamerr_1.5.0 multcomp_1.4-30
[13] otel_0.2.0 e1071_1.7-17
[15] compiler_4.6.0 png_0.1-9
[17] vctrs_0.7.3 stringr_1.6.0
[19] pkgconfig_2.0.3 wk_0.9.5
[21] fastmap_1.2.0 backports_1.5.1
[23] lwgeom_0.2-16 leafem_0.2.5
[25] rmarkdown_2.31 spacesXYZ_1.6-0
[27] miscTools_0.6-30 xfun_0.57
[29] satellite_1.0.6 jsonlite_2.0.0
[31] stringmagic_1.2.0 collapse_2.1.7
[33] terra_1.9-27 parallel_4.6.0
[35] LearnBayes_2.15.2 R6_2.6.1
[37] stringi_1.8.7 RColorBrewer_1.1-3
[39] boot_1.3-32 numDeriv_2016.8-1.1
[41] lmtest_0.9-40 stars_0.7-2
[43] Rcpp_1.1.1-1.1 knitr_1.51
[45] zoo_1.8-15 base64enc_0.1-6
[47] splines_4.6.0 tidyselect_1.2.1
[49] rstudioapi_0.18.0 abind_1.4-8
[51] maptiles_0.11.0 maxLik_1.5-2.2
[53] codetools_0.2-20 lattice_0.22-9
[55] tibble_3.3.1 leafsync_0.1.0
[57] withr_3.0.2 S7_0.2.2
[59] coda_0.19-4.1 evaluate_1.0.5
[61] marginaleffects_0.32.0 survival_3.8-6
[63] fixest_0.14.1 units_1.0-1
[65] proxy_0.4-29 pillar_1.11.1
[67] KernSmooth_2.23-26 stats4_4.6.0
[69] generics_0.1.4 sp_2.2-1
[71] scales_1.4.0 xtable_1.8-8
[73] class_7.3-23 glue_1.8.1
[75] tools_4.6.0 leaflegend_1.2.8
[77] data.table_1.18.4 RSpectra_0.16-2
[79] dotCall64_1.2 mvtnorm_1.3-7
[81] XML_3.99-0.23 grid_4.6.0
[83] rbibutils_2.4.1 crosstalk_1.2.2
[85] bdsmatrix_1.3-7 colorspace_2.1-2
[87] nlme_3.1-169 cols4all_0.10
[89] raster_3.6-32 Formula_1.2-5
[91] cli_3.6.6 spam_2.11-4
[93] dplyr_1.2.1 gtable_0.3.6
[95] digest_0.6.39 classInt_0.4-11
[97] TH.data_1.1-5 htmlwidgets_1.6.4
[99] farver_2.1.2 htmltools_0.5.9
[101] lifecycle_1.0.5 leaflet_2.2.3
[103] microbenchmark_1.5.0 MASS_7.3-65 Reload data from pervious session
load("_data/msoa2_spatial.RData")11.1 Environmental inequality (continued)
Let’s use the same neighbours weights definition as before:
coords <- st_centroid(msoa.spdf)Warning: st_centroid assumes attributes are constant over
geometries# Neighbours within 3km distance
dist_15.nb <- dnearneigh(coords, d1 = 0, d2 = 2500)Warning in dnearneigh(coords, d1 = 0, d2 = 2500): neighbour object
has 6 sub-graphssummary(dist_15.nb)Neighbour list object:
Number of regions: 983
Number of nonzero links: 15266
Percentage nonzero weights: 1.579859
Average number of links: 15.53001
4 regions with no links:
158, 463, 478, 505
6 disjoint connected subgraphs
Link number distribution:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
4 5 9 23 19 26 36 31 53 39 61 63 59 48 42 35 24 31 28 30 27 26
22 23 24 25 26 27 28 29 30 31 32 33 34
25 19 38 29 32 38 26 16 20 10 8 1 2
5 least connected regions:
160 469 474 597 959 with 1 link
2 most connected regions:
565 567 with 34 linksNeighbour list object:
Number of regions: 983
Number of nonzero links: 15270
Percentage nonzero weights: 1.580273
Average number of links: 15.53408
6 disjoint connected subgraphs
Link number distribution:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
9 9 23 19 26 36 31 53 39 61 63 59 48 42 35 24 31 28 30 27 26 25
23 24 25 26 27 28 29 30 31 32 33 34
19 38 29 32 38 26 16 20 10 8 1 2
9 least connected regions:
158 160 463 469 474 478 505 597 959 with 1 link
2 most connected regions:
565 567 with 34 links# listw object with row-normalization
dist_15.lw <- nb2listw(dist_15.nb, style = "W")and estiamte the spatial SAR model:
Call:
lagsarlm(formula = log(no2) ~ per_mixed + per_asian + per_black +
per_other + per_nonUK_EU + per_nonEU + log(POPDEN), data = msoa.spdf,
listw = dist_15.lw, Durbin = FALSE)
Residuals:
Min 1Q Median 3Q Max
-0.2140485 -0.0267085 -0.0021421 0.0238337 0.3505513
Type: lag
Coefficients: (asymptotic standard errors)
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.7004e-02 1.8122e-02 -0.9383 0.348110
per_mixed 3.4376e-04 1.4758e-03 0.2329 0.815810
per_asian -8.5205e-05 1.1494e-04 -0.7413 0.458507
per_black -4.2754e-04 2.3468e-04 -1.8218 0.068484
per_other 1.9693e-03 7.4939e-04 2.6279 0.008591
per_nonUK_EU 8.9027e-04 3.9638e-04 2.2460 0.024703
per_nonEU 1.8460e-03 3.5159e-04 5.2506 1.516e-07
log(POPDEN) 1.8650e-02 2.7852e-03 6.6963 2.138e-11
Rho: 0.9684, LR test value: 2002.5, p-value: < 2.22e-16
Asymptotic standard error: 0.0063124
z-value: 153.41, p-value: < 2.22e-16
Wald statistic: 23535, p-value: < 2.22e-16
Log likelihood: 1562.401 for lag model
ML residual variance (sigma squared): 0.0020568, (sigma: 0.045352)
Number of observations: 983
Number of parameters estimated: 10
AIC: -3104.8, (AIC for lm: -1104.3)
LM test for residual autocorrelation
test value: 108.97, p-value: < 2.22e-161) Please calculate the true multiplier matrix of this SAR model.
The multiplier matrix is given by \(({\bm I_N}-\rho {\bm W})^{-1}\). \({\bm W}\) you get via listw2mat(). The diagonal matrix \({\bm I_N}\), you can construct by using diag() (using the right dimension of W). \(\rho\), you get from the model object, and you can use solve() to calculate an inverse.
1 2 3 4 5
1 1.164650997 0.002433319 0.004089559 0.004034508 0.006545994
2 0.010706605 1.407336301 0.643881932 0.370049927 0.464794934
3 0.011246286 0.402426207 1.474021599 0.429011868 0.641526285
4 0.008875918 0.185024963 0.343209495 1.684533322 0.614086824
5 0.012000989 0.193664556 0.427684190 0.511739020 1.560840834
6 0.010741524 0.192552594 0.452940016 0.631452476 0.672787841
7 0.012779708 0.141953871 0.299247377 0.418234186 0.616895800
8 0.014769006 0.125781189 0.253122442 0.295553039 0.500919513
9 0.011708131 0.147549264 0.309080773 0.568442619 0.629156269
10 0.009937859 0.152900148 0.306652041 0.727001926 0.553973310
6 7 8 9 10
1 0.004882511 0.005808958 0.00872714 0.005854065 0.003613767
2 0.385105188 0.283907742 0.32703109 0.324608380 0.244640236
3 0.566175019 0.374059222 0.41132397 0.424986063 0.306652041
4 0.631452476 0.418234186 0.38421895 0.625286881 0.581601541
5 0.560656534 0.514079833 0.54266281 0.576726579 0.369315540
6 1.571175245 0.558170218 0.46513922 0.661184961 0.543820047
7 0.558170218 1.475511568 0.58520461 0.614170880 0.463886540
8 0.357799398 0.450157392 1.46638195 0.474994894 0.272339890
9 0.601077237 0.558337164 0.56135760 1.581077095 0.517983092
10 0.679775059 0.579858174 0.44255232 0.712226751 1.5600831382) Assume we are interested in the effect for non-EU citizens (beta for “per_nonEU”). Please create an N x N effects matrix (using the multiplier above) for the effect of the shar of non-EU citizens on the outcome of each other unit. What is the effect of unit 6 on unit 10? Why is this larger than the effect of unit 5 on unit 8?
# For beta 1
beta <- mod_1.sar$coefficients
effM <- beta["per_nonEU"] * M
effM[1:10, 1:10] 1 2 3 4 5
1 2.149995e-03 4.492010e-06 7.549498e-06 7.447872e-06 1.208418e-05
2 1.976484e-05 2.598002e-03 1.188633e-03 6.831278e-04 8.580311e-04
3 2.076112e-05 7.428958e-04 2.721106e-03 7.919740e-04 1.184285e-03
4 1.638532e-05 3.415639e-04 6.335792e-04 3.109720e-03 1.133630e-03
5 2.215433e-05 3.575129e-04 7.895231e-04 9.446918e-04 2.881378e-03
6 1.982931e-05 3.554602e-04 8.361464e-04 1.165688e-03 1.241995e-03
7 2.359188e-05 2.620528e-04 5.524233e-04 7.720780e-04 1.138816e-03
8 2.726421e-05 2.321974e-04 4.672747e-04 5.456034e-04 9.247186e-04
9 2.161370e-05 2.723822e-04 5.705762e-04 1.049369e-03 1.161449e-03
10 1.834571e-05 2.822601e-04 5.660926e-04 1.342076e-03 1.022658e-03
6 7 8 9 10
1 9.013321e-06 1.072358e-05 1.611067e-05 1.080685e-05 6.671166e-06
2 7.109204e-04 5.241057e-04 6.037132e-04 5.992408e-04 4.516162e-04
3 1.045183e-03 6.905291e-04 7.593214e-04 7.845422e-04 5.660926e-04
4 1.165688e-03 7.720780e-04 7.092844e-04 1.154306e-03 1.073661e-03
5 1.034996e-03 9.490131e-04 1.001778e-03 1.064662e-03 6.817721e-04
6 2.900456e-03 1.030406e-03 8.586666e-04 1.220575e-03 1.003915e-03
7 1.030406e-03 2.723857e-03 1.080312e-03 1.133785e-03 8.563541e-04
8 6.605128e-04 8.310095e-04 2.707003e-03 8.768606e-04 5.027509e-04
9 1.109614e-03 1.030714e-03 1.036290e-03 2.918735e-03 9.562187e-04
10 1.254893e-03 1.070443e-03 8.169703e-04 1.314801e-03 2.879979e-03# "Effect" of unit 6 on unit 10
effM[10, 6][1] 0.001254893# "Effect" of unit 5 on unit 8
effM[8, 5][1] 0.00092471863) Calculate and interpret the summary impact measures for the SAR model.
mod_1.sar.imp <- impacts(mod_1.sar, listw = dist_15.lw, R = 300)
summary(mod_1.sar.imp)Impact measures (lag, exact):
Direct Indirect Total
per_mixed dy/dx 0.0004939013 0.010385844 0.010879745
per_asian dy/dx -0.0001224192 -0.002574253 -0.002696672
per_black dy/dx -0.0006142789 -0.012917166 -0.013531445
per_other dy/dx 0.0028294759 0.059498722 0.062328198
per_nonUK_EU dy/dx 0.0012791011 0.026897166 0.028176267
per_nonEU dy/dx 0.0026523198 0.055773451 0.058425770
log(POPDEN) dy/dx 0.0267960076 0.563471199 0.590267206
========================================================
Simulation results ( variance matrix):
Direct:
Iterations = 1:300
Thinning interval = 1
Number of chains = 1
Sample size per chain = 300
1. Empirical mean and standard deviation for each variable,
plus standard error of the mean:
Mean SD Naive SE Time-series SE
per_mixed dy/dx 0.0004517 0.0020542 1.186e-04 1.186e-04
per_asian dy/dx -0.0001178 0.0001607 9.280e-06 9.280e-06
per_black dy/dx -0.0006048 0.0003269 1.888e-05 2.056e-05
per_other dy/dx 0.0028529 0.0010074 5.816e-05 5.816e-05
per_nonUK_EU dy/dx 0.0012637 0.0005787 3.341e-05 3.341e-05
per_nonEU dy/dx 0.0026625 0.0004423 2.553e-05 2.553e-05
log(POPDEN) dy/dx 0.0267119 0.0038793 2.240e-04 2.240e-04
2. Quantiles for each variable:
2.5% 25% 50% 75%
per_mixed dy/dx -0.0037476 -0.0007716 0.0003728 1.826e-03
per_asian dy/dx -0.0004024 -0.0002322 -0.0001275 -1.117e-05
per_black dy/dx -0.0011889 -0.0008643 -0.0006187 -3.758e-04
per_other dy/dx 0.0011247 0.0021310 0.0027917 3.503e-03
per_nonUK_EU dy/dx 0.0001476 0.0008990 0.0012477 1.616e-03
per_nonEU dy/dx 0.0017721 0.0023876 0.0026598 2.960e-03
log(POPDEN) dy/dx 0.0198258 0.0236060 0.0266380 2.938e-02
97.5%
per_mixed dy/dx 4.325e-03
per_asian dy/dx 1.967e-04
per_black dy/dx 1.955e-05
per_other dy/dx 4.790e-03
per_nonUK_EU dy/dx 2.531e-03
per_nonEU dy/dx 3.459e-03
log(POPDEN) dy/dx 3.459e-02
========================================================
Indirect:
Iterations = 1:300
Thinning interval = 1
Number of chains = 1
Sample size per chain = 300
1. Empirical mean and standard deviation for each variable,
plus standard error of the mean:
Mean SD Naive SE Time-series SE
per_mixed dy/dx 0.008810 0.048179 0.0027816 0.0027816
per_asian dy/dx -0.002617 0.003819 0.0002205 0.0002205
per_black dy/dx -0.013267 0.008121 0.0004689 0.0005122
per_other dy/dx 0.062893 0.026819 0.0015484 0.0015484
per_nonUK_EU dy/dx 0.027065 0.012211 0.0007050 0.0007050
per_nonEU dy/dx 0.058749 0.016583 0.0009574 0.0009574
log(POPDEN) dy/dx 0.586973 0.149351 0.0086228 0.0086228
2. Quantiles for each variable:
2.5% 25% 50% 75%
per_mixed dy/dx -0.097259 -0.017613 0.008571 0.0388572
per_asian dy/dx -0.009696 -0.005022 -0.002475 -0.0002205
per_black dy/dx -0.029321 -0.018123 -0.013063 -0.0083645
per_other dy/dx 0.021774 0.044679 0.058343 0.0789462
per_nonUK_EU dy/dx 0.003496 0.019280 0.026767 0.0347125
per_nonEU dy/dx 0.033520 0.048078 0.057930 0.0669214
log(POPDEN) dy/dx 0.386684 0.488729 0.566325 0.6498024
97.5%
per_mixed dy/dx 0.1044319
per_asian dy/dx 0.0049233
per_black dy/dx 0.0004071
per_other dy/dx 0.1147984
per_nonUK_EU dy/dx 0.0534923
per_nonEU dy/dx 0.0942100
log(POPDEN) dy/dx 0.9583151
========================================================
Total:
Iterations = 1:300
Thinning interval = 1
Number of chains = 1
Sample size per chain = 300
1. Empirical mean and standard deviation for each variable,
plus standard error of the mean:
Mean SD Naive SE Time-series SE
per_mixed dy/dx 0.009262 0.050183 0.0028973 0.0028973
per_asian dy/dx -0.002735 0.003973 0.0002294 0.0002294
per_black dy/dx -0.013872 0.008414 0.0004858 0.0005311
per_other dy/dx 0.065745 0.027659 0.0015969 0.0015969
per_nonUK_EU dy/dx 0.028328 0.012736 0.0007353 0.0007353
per_nonEU dy/dx 0.061412 0.016868 0.0009739 0.0009739
log(POPDEN) dy/dx 0.613685 0.151457 0.0087444 0.0087444
2. Quantiles for each variable:
2.5% 25% 50% 75%
per_mixed dy/dx -0.101074 -0.018369 0.009006 0.0407207
per_asian dy/dx -0.010075 -0.005251 -0.002587 -0.0002339
per_black dy/dx -0.030385 -0.019110 -0.013707 -0.0087500
per_other dy/dx 0.022912 0.046869 0.061246 0.0824820
per_nonUK_EU dy/dx 0.003644 0.020072 0.027897 0.0364017
per_nonEU dy/dx 0.035662 0.050664 0.060696 0.0695733
log(POPDEN) dy/dx 0.409671 0.515044 0.592427 0.6773606
97.5%
per_mixed dy/dx 0.1082400
per_asian dy/dx 0.0051375
per_black dy/dx 0.0004262
per_other dy/dx 0.1184796
per_nonUK_EU dy/dx 0.0555344
per_nonEU dy/dx 0.0971066
log(POPDEN) dy/dx 0.98825604) Is SAR the right model choice or would you rather estimate a different model? Please run a Durbin model and caculate its impact summary measures
Call:
lagsarlm(formula = log(no2) ~ per_mixed + per_asian + per_black +
per_other + per_nonUK_EU + per_nonEU + log(POPDEN), data = msoa.spdf,
listw = dist_15.lw, Durbin = TRUE)
Residuals:
Min 1Q Median 3Q Max
-0.1854009 -0.0263818 -0.0020816 0.0229647 0.3321974
Type: mixed
Coefficients: (asymptotic standard errors)
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.00409824 0.01983728 -0.2066 0.83633
per_mixed 0.00434535 0.00218712 1.9868 0.04695
per_asian -0.00028620 0.00023959 -1.1945 0.23227
per_black -0.00056734 0.00034455 -1.6466 0.09964
per_other 0.00222708 0.00112918 1.9723 0.04857
per_nonUK_EU 0.00085417 0.00059478 1.4361 0.15097
per_nonEU 0.00095220 0.00052681 1.8075 0.07069
log(POPDEN) 0.02649122 0.00320358 8.2693 2.220e-16
lag.per_mixed -0.00475294 0.00315799 -1.5051 0.13231
lag.per_asian 0.00024092 0.00028983 0.8312 0.40584
lag.per_black 0.00025812 0.00054125 0.4769 0.63344
lag.per_other -0.00074506 0.00176141 -0.4230 0.67230
lag.per_nonUK_EU 0.00094549 0.00100320 0.9425 0.34595
lag.per_nonEU 0.00130970 0.00078547 1.6674 0.09544
lag.log(POPDEN) -0.02526415 0.00588517 -4.2928 1.764e-05
Rho: 0.98286, LR test value: 1536.9, p-value: < 2.22e-16
Asymptotic standard error: 0.0051804
z-value: 189.73, p-value: < 2.22e-16
Wald statistic: 35997, p-value: < 2.22e-16
Log likelihood: 1576.566 for mixed model
ML residual variance (sigma squared): 0.001969, (sigma: 0.044374)
Number of observations: 983
Number of parameters estimated: 17
AIC: -3119.1, (AIC for lm: -1584.3)
LM test for residual autocorrelation
test value: 103.97, p-value: < 2.22e-16# Impact measures of the Durbin Error model
mod_1.durb.imp <- impacts(mod_1.durb, listw = dist_15.lw, R = 300)
summary(mod_1.durb.imp, zstats = TRUE, short = TRUE)Impact measures (mixed, exact):
Direct Indirect Total
per_mixed dy/dx 0.0040597904 -0.027843988 -0.023784197
per_asian dy/dx -0.0003101210 -0.002332322 -0.002642443
per_black dy/dx -0.0007447486 -0.017299184 -0.018043932
per_other dy/dx 0.0030823781 0.083398408 0.086480787
per_nonUK_EU dy/dx 0.0019115634 0.103104372 0.105015935
per_nonEU dy/dx 0.0022824096 0.129706442 0.131988851
log(POPDEN) dy/dx 0.0269491699 0.044653927 0.071603096
========================================================
Simulation results ( variance matrix):
========================================================
Simulated standard errors
Direct Indirect Total
per_mixed dy/dx 0.0025286709 0.16959882 0.17081739
per_asian dy/dx 0.0002471124 0.01324621 0.01329233
per_black dy/dx 0.0003792924 0.02724657 0.02737951
per_other dy/dx 0.0012852899 0.09795036 0.09852127
per_nonUK_EU dy/dx 0.0006740466 0.08119923 0.08150104
per_nonEU dy/dx 0.0006368414 0.09916611 0.09945587
log(POPDEN) dy/dx 0.0044540782 0.41173991 0.41468431
Simulated z-values:
Direct Indirect Total
per_mixed dy/dx 1.566054 -0.13766305 -0.11349814
per_asian dy/dx -1.342328 -0.28025883 -0.30424129
per_black dy/dx -2.000899 -0.72203648 -0.74624933
per_other dy/dx 2.373590 0.88701658 0.91284191
per_nonUK_EU dy/dx 2.953143 1.47548381 1.49444356
per_nonEU dy/dx 3.632177 1.52140146 1.54022678
log(POPDEN) dy/dx 6.044093 0.02279423 0.08755131
Simulated p-values:
Direct Indirect Total
per_mixed dy/dx 0.11733602 0.89051 0.90964
per_asian dy/dx 0.17948959 0.77928 0.76094
per_black dy/dx 0.04540330 0.47027 0.45552
per_other dy/dx 0.01761612 0.37507 0.36133
per_nonUK_EU dy/dx 0.00314556 0.14008 0.13506
per_nonEU dy/dx 0.00028104 0.12816 0.12351
log(POPDEN) dy/dx 1.5025e-09 0.98181 0.930235) Please repeat with a Durbin Error model. Why are the impacts here identical to the coefficients?
Call:
errorsarlm(formula = log(no2) ~ per_mixed + per_asian + per_black +
per_other + per_nonUK_EU + per_nonEU + log(POPDEN), data = msoa.spdf,
listw = dist_15.lw, Durbin = TRUE)
Residuals:
Min 1Q Median 3Q Max
-0.1839285 -0.0254426 -0.0027042 0.0216084 0.2944840
Type: error
Coefficients: (asymptotic standard errors)
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.64939215 0.24748370 10.7053 < 2.2e-16
per_mixed 0.00553333 0.00223688 2.4737 0.013373
per_asian -0.00017156 0.00024183 -0.7094 0.478062
per_black -0.00057947 0.00034426 -1.6832 0.092334
per_other 0.00203392 0.00112534 1.8074 0.070701
per_nonUK_EU 0.00086254 0.00058902 1.4644 0.143091
per_nonEU 0.00135822 0.00053480 2.5397 0.011096
log(POPDEN) 0.02716824 0.00354239 7.6695 1.732e-14
lag.per_mixed 0.00107140 0.00819322 0.1308 0.895960
lag.per_asian -0.00060616 0.00070080 -0.8649 0.387069
lag.per_black -0.00191733 0.00130997 -1.4636 0.143291
lag.per_other 0.01014125 0.00496979 2.0406 0.041293
lag.per_nonUK_EU 0.00925620 0.00217624 4.2533 2.106e-05
lag.per_nonEU 0.00563564 0.00185541 3.0374 0.002386
lag.log(POPDEN) -0.01370891 0.01128957 -1.2143 0.224634
Lambda: 0.99424, LR test value: 1527.8, p-value: < 2.22e-16
Asymptotic standard error: 0.0024921
z-value: 398.96, p-value: < 2.22e-16
Wald statistic: 159170, p-value: < 2.22e-16
Log likelihood: 1572.051 for error model
ML residual variance (sigma squared): 0.0019546, (sigma: 0.044211)
Number of observations: 983
Number of parameters estimated: 17
AIC: -3110.1, (AIC for lm: -1584.3)# Impact measures of the Durbin model
mod_1.durbe.imp <- impacts(mod_1.durbe, listw = dist_15.lw, R = 300)
summary(mod_1.durbe.imp, zstats = TRUE, short = TRUE)Impact measures (SDEM, glht, n):
Direct Indirect Total
per_mixed dy/dx 0.0055333298 0.0010713981 0.0066047279
per_asian dy/dx -0.0001715584 -0.0006061567 -0.0007777151
per_black dy/dx -0.0005794704 -0.0019173261 -0.0024967965
per_other dy/dx 0.0020339226 0.0101412504 0.0121751729
per_nonUK_EU dy/dx 0.0008625423 0.0092561971 0.0101187394
per_nonEU dy/dx 0.0013582217 0.0056356426 0.0069938643
log(POPDEN) dy/dx 0.0271682392 -0.0137089088 0.0134593305
========================================================
Standard errors:
Direct Indirect Total
per_mixed dy/dx 0.0022368774 0.0081932151 0.0089412446
per_asian dy/dx 0.0002418282 0.0007008041 0.0007540247
per_black dy/dx 0.0003442649 0.0013099673 0.0013639632
per_other dy/dx 0.0011253352 0.0049697880 0.0051074709
per_nonUK_EU dy/dx 0.0005890159 0.0021762395 0.0022231333
per_nonEU dy/dx 0.0005348024 0.0018554128 0.0020196122
log(POPDEN) dy/dx 0.0035423870 0.0112895670 0.0132006518
========================================================
Z-values:
Direct Indirect Total
per_mixed dy/dx 2.4736849 0.1307665 0.738681
per_asian dy/dx -0.7094226 -0.8649446 -1.031419
per_black dy/dx -1.6832108 -1.4636442 -1.830545
per_other dy/dx 1.8073926 2.0405801 2.383797
per_nonUK_EU dy/dx 1.4643785 4.2532989 4.551567
per_nonEU dy/dx 2.5396703 3.0374063 3.462974
log(POPDEN) dy/dx 7.6694724 -1.2142989 1.019596
p-values:
Direct Indirect Total
per_mixed dy/dx 0.013373 0.8959600 0.46010070
per_asian dy/dx 0.478062 0.3870692 0.30234457
per_black dy/dx 0.092334 0.1432912 0.06716843
per_other dy/dx 0.070701 0.0412926 0.01713506
per_nonUK_EU dy/dx 0.143091 2.1064e-05 5.3248e-06
per_nonEU dy/dx 0.011096 0.0023862 0.00053424
log(POPDEN) dy/dx 1.7319e-14 0.2246336 0.30792015