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 4.5.1 (2025-06-13 ucrt)
Platform: x86_64-w64-mingw32/x64
Running under: Windows 11 x64 (build 22631)
Matrix products: default
LAPACK version 3.12.1
locale:
[1] LC_COLLATE=English_United Kingdom.utf8
[2] LC_CTYPE=English_United Kingdom.utf8
[3] LC_MONETARY=English_United Kingdom.utf8
[4] LC_NUMERIC=C
[5] LC_TIME=English_United Kingdom.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_0.0.6 splm_1.6-5 lfe_3.1.1
[4] plm_2.6-6 viridis_0.6.5 viridisLite_0.4.2
[7] tmap_4.1 ggplot2_3.5.2 spatialreg_1.3-6
[10] Matrix_1.7-3 spdep_1.3-13 spData_2.3.4
[13] mapview_2.11.2 sf_1.0-21
loaded via a namespace (and not attached):
[1] Rdpack_2.6.4 DBI_1.2.3
[3] deldir_2.0-4 gridExtra_2.3
[5] tmaptools_3.2 s2_1.1.9
[7] logger_0.4.0 sandwich_3.1-1
[9] rlang_1.1.6 magrittr_2.0.3
[11] dreamerr_1.5.0 multcomp_1.4-28
[13] e1071_1.7-16 compiler_4.5.1
[15] png_0.1-8 vctrs_0.6.5
[17] stringr_1.5.1 pkgconfig_2.0.3
[19] wk_0.9.4 fastmap_1.2.0
[21] lwgeom_0.2-14 leafem_0.2.4
[23] rmarkdown_2.29 spacesXYZ_1.6-0
[25] miscTools_0.6-28 xfun_0.52
[27] satellite_1.0.5 jsonlite_2.0.0
[29] stringmagic_1.2.0 collapse_2.1.2
[31] terra_1.8-54 parallel_4.5.1
[33] LearnBayes_2.15.1 R6_2.6.1
[35] stringi_1.8.7 RColorBrewer_1.1-3
[37] boot_1.3-31 numDeriv_2016.8-1.1
[39] lmtest_0.9-40 stars_0.6-8
[41] Rcpp_1.0.14 knitr_1.50
[43] zoo_1.8-14 base64enc_0.1-3
[45] leaflet.providers_2.0.0 splines_4.5.1
[47] tidyselect_1.2.1 rstudioapi_0.17.1
[49] dichromat_2.0-0.1 abind_1.4-8
[51] maptiles_0.10.0 maxLik_1.5-2.1
[53] codetools_0.2-20 lattice_0.22-7
[55] tibble_3.3.0 leafsync_0.1.0
[57] withr_3.0.2 coda_0.19-4.1
[59] evaluate_1.0.4 survival_3.8-3
[61] fixest_0.12.1 units_0.8-7
[63] proxy_0.4-27 pillar_1.10.2
[65] KernSmooth_2.23-26 stats4_4.5.1
[67] generics_0.1.4 sp_2.2-0
[69] scales_1.4.0 xtable_1.8-4
[71] class_7.3-23 glue_1.8.0
[73] tools_4.5.1 leaflegend_1.2.1
[75] data.table_1.17.6 RSpectra_0.16-2
[77] dotCall64_1.2 mvtnorm_1.3-3
[79] XML_3.99-0.18 grid_4.5.1
[81] rbibutils_2.3 crosstalk_1.2.1
[83] bdsmatrix_1.3-7 colorspace_2.1-1
[85] nlme_3.1-168 cols4all_0.8
[87] raster_3.6-32 Formula_1.2-5
[89] cli_3.6.5 spam_2.11-1
[91] dplyr_1.1.4 gtable_0.3.6
[93] digest_0.6.37 classInt_0.4-11
[95] TH.data_1.1-3 htmlwidgets_1.6.4
[97] farver_2.1.2 htmltools_0.5.8.1
[99] lifecycle_1.0.4 leaflet_2.2.2
[101] 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 links# There are some empty neighbour sets. Lets impute those with the nearest neighbour.
k2.nb <- knearneigh(coords, k = 1)
# Replace zero
nolink_ids <- which(card(dist_15.nb) == 0)
dist_15.nb[card(dist_15.nb) == 0] <- k2.nb$nn[nolink_ids, ]
summary(dist_15.nb)Neighbour 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:
mod_1.sar <- lagsarlm(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) # we could here extend to SDM
summary(mod_1.sar)
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 \(({\bm I_N}-\rho {\bm W})^{-1}\).
W <- listw2mat(dist_15.lw)
I <- diag(dim(W)[1])
rho <- unname(mod_1.sar$rho)
M <- solve(I - rho*W)
M[1:10, 1:10] 1 2 3 4 5
1 1.164650997 0.002433319 0.004089559 0.004034508 0.006545994
2 0.010706605 1.407336301 0.643881932 0.370049927 0.464794934
3 0.011246286 0.402426207 1.474021599 0.429011868 0.641526285
4 0.008875918 0.185024963 0.343209495 1.684533322 0.614086824
5 0.012000989 0.193664556 0.427684190 0.511739020 1.560840834
6 0.010741524 0.192552594 0.452940016 0.631452476 0.672787841
7 0.012779708 0.141953871 0.299247377 0.418234186 0.616895800
8 0.014769006 0.125781189 0.253122442 0.295553039 0.500919513
9 0.011708131 0.147549264 0.309080773 0.568442619 0.629156269
10 0.009937859 0.152900148 0.306652041 0.727001926 0.553973310
6 7 8 9 10
1 0.004882511 0.005808958 0.00872714 0.005854065 0.003613767
2 0.385105188 0.283907742 0.32703109 0.324608380 0.244640237
3 0.566175019 0.374059222 0.41132397 0.424986063 0.306652041
4 0.631452476 0.418234186 0.38421895 0.625286881 0.581601541
5 0.560656534 0.514079833 0.54266281 0.576726580 0.369315540
6 1.571175245 0.558170218 0.46513922 0.661184961 0.543820047
7 0.558170218 1.475511568 0.58520461 0.614170880 0.463886540
8 0.357799398 0.450157392 1.46638195 0.474994894 0.272339890
9 0.601077237 0.558337164 0.56135760 1.581077095 0.517983092
10 0.679775059 0.579858175 0.44255232 0.712226751 1.5600831382) Create an N x N effects matrix for the effect of the non-EU citizens. What is the effect of unit 6 on unit 10? Why is this larger than the effect of unit 5 on unit 8?
# For beta 1
beta <- mod_1.sar$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 of 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 0.0004939013 0.010385844 0.010879745
per_asian -0.0001224192 -0.002574253 -0.002696672
per_black -0.0006142789 -0.012917166 -0.013531445
per_other 0.0028294759 0.059498722 0.062328198
per_nonUK_EU 0.0012791011 0.026897166 0.028176267
per_nonEU 0.0026523198 0.055773451 0.058425770
log(POPDEN) 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 0.0004516 0.0021828 1.260e-04 8.945e-05
per_asian -0.0001250 0.0001663 9.601e-06 9.601e-06
per_black -0.0005971 0.0003198 1.847e-05 1.847e-05
per_other 0.0027945 0.0010346 5.973e-05 5.973e-05
per_nonUK_EU 0.0013252 0.0005347 3.087e-05 3.087e-05
per_nonEU 0.0026841 0.0005053 2.918e-05 2.918e-05
log(POPDEN) 0.0267885 0.0040421 2.334e-04 2.334e-04
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
per_mixed -0.0037399 -0.0009906 0.0005289 1.927e-03 0.0047141
per_asian -0.0004485 -0.0002400 -0.0001155 6.451e-06 0.0001544
per_black -0.0011900 -0.0008267 -0.0006154 -3.761e-04 0.0000222
per_other 0.0005838 0.0021063 0.0028424 3.425e-03 0.0048597
per_nonUK_EU 0.0002841 0.0009735 0.0013422 1.684e-03 0.0023795
per_nonEU 0.0017103 0.0023158 0.0026863 3.051e-03 0.0037285
log(POPDEN) 0.0193365 0.0243128 0.0264296 2.976e-02 0.0353543
========================================================
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 0.009806 0.047990 0.0027707 0.0027707
per_asian -0.002794 0.003794 0.0002190 0.0002190
per_black -0.013020 0.007720 0.0004457 0.0004457
per_other 0.060285 0.024954 0.0014407 0.0014407
per_nonUK_EU 0.028365 0.012363 0.0007138 0.0007138
per_nonEU 0.058650 0.017522 0.0010116 0.0010116
log(POPDEN) 0.579295 0.133237 0.0076924 0.0076924
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
per_mixed -0.082940 -0.019801 0.011368 0.0409128 0.1023618
per_asian -0.009843 -0.005393 -0.002477 0.0001489 0.0034666
per_black -0.029249 -0.017409 -0.012866 -0.0079063 0.0004804
per_other 0.015476 0.044411 0.057650 0.0748676 0.1160783
per_nonUK_EU 0.006968 0.020562 0.027676 0.0346838 0.0590104
per_nonEU 0.032639 0.047138 0.056561 0.0668895 0.1005717
log(POPDEN) 0.382252 0.487462 0.562210 0.6450783 0.8946578
========================================================
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 0.010257 0.050129 0.0028942 0.0028942
per_asian -0.002918 0.003956 0.0002284 0.0002284
per_black -0.013618 0.008010 0.0004624 0.0004624
per_other 0.063079 0.025854 0.0014927 0.0014927
per_nonUK_EU 0.029691 0.012835 0.0007410 0.0007410
per_nonEU 0.061334 0.017879 0.0010323 0.0010323
log(POPDEN) 0.606083 0.135189 0.0078052 0.0078052
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
per_mixed -0.086392 -0.020740 0.011934 0.0430979 0.1076969
per_asian -0.010270 -0.005633 -0.002607 0.0001556 0.0036279
per_black -0.030105 -0.018150 -0.013530 -0.0082778 0.0005026
per_other 0.016217 0.046569 0.060990 0.0782495 0.1205891
per_nonUK_EU 0.007314 0.021705 0.028962 0.0363637 0.0614172
per_nonEU 0.034601 0.049757 0.059313 0.0698470 0.1033319
log(POPDEN) 0.402366 0.512824 0.588171 0.6755833 0.92406944) 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
# Spatial Dubrbin model
mod_1.durb <- lagsarlm(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)
summary(mod_1.durb)
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 0.0040597904 -0.027843988 -0.023784197
per_asian -0.0003101210 -0.002332322 -0.002642443
per_black -0.0007447486 -0.017299184 -0.018043932
per_other 0.0030823781 0.083398408 0.086480787
per_nonUK_EU 0.0019115634 0.103104372 0.105015935
per_nonEU 0.0022824096 0.129706442 0.131988851
log(POPDEN) 0.0269491699 0.044653927 0.071603096
========================================================
Simulation results ( variance matrix):
========================================================
Simulated standard errors
Direct Indirect Total
per_mixed 0.0023787890 0.159440020 0.160531267
per_asian 0.0002376030 0.009744063 0.009787986
per_black 0.0004261366 0.027466212 0.027684055
per_other 0.0013137361 0.090284099 0.090914927
per_nonUK_EU 0.0006666248 0.060470345 0.060806237
per_nonEU 0.0006382028 0.062911637 0.063196993
log(POPDEN) 0.0044066912 0.339875074 0.342909103
Simulated z-values:
Direct Indirect Total
per_mixed 1.655641 -0.26404652 -0.23771794
per_asian -1.289660 -0.33533651 -0.36513816
per_black -1.722271 -0.61269105 -0.63438053
per_other 2.332877 1.00042307 1.02719194
per_nonUK_EU 2.999279 1.95740426 1.97947300
per_nonEU 3.637737 2.29844074 2.32479863
log(POPDEN) 6.039395 -0.02530795 0.05252765
Simulated p-values:
Direct Indirect Total
per_mixed 0.09779462 0.791744 0.812100
per_asian 0.19716879 0.737371 0.715008
per_black 0.08502038 0.540081 0.525833
per_other 0.01965459 0.317106 0.304330
per_nonUK_EU 0.00270619 0.050300 0.047763
per_nonEU 0.00027504 0.021537 0.020083
log(POPDEN) 1.5469e-09 0.979809 0.9581085) Please repeat with a Durbin Error model. Why are the impacts here identical to the coefficients?
# Spatial Dubrbin model
mod_1.durbe <- errorsarlm(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)
summary(mod_1.durbe)
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 0.0055333298 0.0010713981 0.0066047279
per_asian -0.0001715584 -0.0006061567 -0.0007777151
per_black -0.0005794704 -0.0019173261 -0.0024967965
per_other 0.0020339226 0.0101412504 0.0121751729
per_nonUK_EU 0.0008625423 0.0092561971 0.0101187394
per_nonEU 0.0013582217 0.0056356426 0.0069938643
log(POPDEN) 0.0271682392 -0.0137089088 0.0134593305
========================================================
Standard errors:
Direct Indirect Total
per_mixed 0.0022368774 0.0081932151 0.0089412446
per_asian 0.0002418282 0.0007008041 0.0007540247
per_black 0.0003442649 0.0013099673 0.0013639632
per_other 0.0011253352 0.0049697880 0.0051074709
per_nonUK_EU 0.0005890159 0.0021762395 0.0022231333
per_nonEU 0.0005348024 0.0018554128 0.0020196122
log(POPDEN) 0.0035423870 0.0112895670 0.0132006518
========================================================
Z-values:
Direct Indirect Total
per_mixed 2.4736849 0.1307665 0.738681
per_asian -0.7094226 -0.8649446 -1.031419
per_black -1.6832108 -1.4636442 -1.830545
per_other 1.8073926 2.0405801 2.383797
per_nonUK_EU 1.4643785 4.2532989 4.551567
per_nonEU 2.5396703 3.0374063 3.462974
log(POPDEN) 7.6694724 -1.2142989 1.019596
p-values:
Direct Indirect Total
per_mixed 0.013373 0.8959600 0.46010070
per_asian 0.478062 0.3870692 0.30234457
per_black 0.092334 0.1432912 0.06716843
per_other 0.070701 0.0412926 0.01713506
per_nonUK_EU 0.143091 2.1064e-05 5.3248e-06
per_nonEU 0.011096 0.0023862 0.00053424
log(POPDEN) 1.7319e-14 0.2246336 0.30792015