Intro

The study examines whether household income levels (measured by the income decile variable hinctnta) explain regional variation in institutional trust across four Western European countries: Belgium (BE), France (FR), the Netherlands (NL), and Luxembourg (LU). Four dimensions of trust are analysed:

Var Desc Sca;e
trstplt Trust in polititians 0–10
trstprt Trust in parliament 0–10
trstlgl Trust in legal system 0–10
trstplc Trust in police 0–10

Why spatial?

Standard OLS regression assumes independence of observations. In the case of regional NUTS-2 data, this assumption is violated—neighboring regions tend to exhibit similar levels of trust due to shared social networks, common institutional history, migration, and diffusion of norms.

Country dummies?

Preliminary diagnostics reveal a fundamental issue: hinctnta is highly significant in a simple model y ~ x (p < 0.001), but the effect disappears once country dummies (BE/FR/NL/LU) are included. Moran’s I for the dependent variable is very high (e.g., 0.81 for trstplt), yet OLS residuals with country dummies no longer exhibit spatial autocorrelation.

Interpretation/Assumption: country dummies simultaneously absorb both the income effect (richer countries tend to have higher trust) and the entire spatial structure (regions within the same country are geographically closer). This results in multicollinearity between country fixed effects and the income variable.

Solution: estimate models both without country fixed effects (where the income effect operates across and within countries) and with country fixed effects (SDEM_FE — capturing the pure within-country effect). Comparing these specifications allows to identify the level at which the income mechanism operates.

##Model Hierarchy

8 models are estimatedi:

Model Formula Spatial Mechanism Country FE
OLS_1 y ~ x lack no
OLS_2 y ~ x + controls lack no
OLS_3 y ~ x + lag_x + controls neighbours context in x no
SAR OLS_2 + ρWy spillover y no
SEM OLS_2 + λu error correlation no
SDM OLS_3 + ρWy + lag_gdp spillover y + effects WX no
SDEM OLS_3 + λu + lag_gdp errors correlation + WX effects no
SDEM_FE SDEM + dummies errors correlation + WX effects yes

Why SDEM? The SDEM (Spatial Durbin Error Model) combines three mechanisms: (1) the direct effect of a region’s own income (β on x), (2) the indirect effect of neighboring income context (θ on lag_x), (3) spatial correlation in unobserved factors (λ).

Unlike the SDM, the SDEM does not assume that neighboring regions’ trust directly affects local trust (no ρ). Instead, indirect effects operate through the income channel (WX). This makes it a more conservative and interpretable specification when a direct spillover of the dependent variable is theoretically difficult to justify.

Data

Distribution

(LU has 1 region NUTS-2)
Cntry NUTS-2 count Obs. ESS hinctnta M hinctnta SD trstplt M trstprt M trstlgl M trstplc M
Belgia 11 7492 5.85 0.34 3.57 3.52 4.02 4.81
Francja 22 8748 5.28 0.39 2.97 2.79 3.94 4.99
Holandia 12 7376 6.13 0.35 4.34 4.43 4.73 5.09

Spatial maps — distribution of trust

Scatter hinctnta vs trust

Interpretation: For trstplt and trstprt, a strong positive relationship is observed (R² ≈ 0.75, β > 0.9, p < 0.001) — regions with higher income exhibit higher levels of political trust. This effect operates both between countries (with the Netherlands located in the upper-right corner and France in the lower-left) and within countries (with clear variation visible across regions within France). For trstplc, the relationship is weaker and more dispersed.

Spatial autocorrelation

Morans test

How estimated: Moran I = \(\frac{n}{\sum_{i,j}w_{ij}} \cdot \frac{\sum_{i,j}w_{ij}(y_i-\bar{y})(y_j-\bar{y})}{\sum_i(y_i-\bar{y})^2}\), where \(w_{ij}\) denotes the elements of the spatial weights matrix. Statistical test: H₀ = values are randomly distributed in space. A Moran’s I value close to 1 indicates strong positive spatial autocorrelation (similar values cluster together).

compute Moran’s I for: (1) the observed y — to assess whether trust is spatially clustered, (2) the residuals from OLS_2 — to determine whether spatial structure remains after controlling for explanatory variables.

Var Test Moran I p-value
Trust in politicians Observed y 0.8129 0.0000 ***
Trust in politicians Residuals OLS_2 0.1537 0.0324 **
Trust in politicians Residuals SAR -0.0415 0.5808
Trust in politicians Residuals SEM -0.0201 0.4891
Trust in politicians Residuals SDM -0.0645 0.6694
Trust in politicians Residuals SDEM 0.0242 0.3113
Trust in politicians Residuals SDEM_FE -0.0183 0.4818
Trust in parliament Observed y 0.8715 0.0000 ***
Trust in parliament Residuals OLS_2 0.1406 0.0460 **
Trust in parliament Residuals SAR -0.1050 0.8042
Trust in parliament Residuals SEM -0.0483 0.6047
Trust in parliament Residuals SDM -0.1107 0.8175
Trust in parliament Residuals SDEM -0.0594 0.6487
Trust in parliament Residuals SDEM_FE -0.0309 0.5330
Trust in legal system Observed y 0.7058 0.0000 ***
Trust in legal system Residuals OLS_2 0.1904 0.0137 **
Trust in legal system Residuals SAR -0.0559 0.6339
Trust in legal system Residuals SEM -0.0381 0.5630
Trust in legal system Residuals SDM -0.0077 0.4382
Trust in legal system Residuals SDEM -0.0055 0.4296
Trust in legal system Residuals SDEM_FE -0.0094 0.4452
Trust in police Observed y 0.1538 0.0353 **
Trust in police Residuals OLS_2 0.1191 0.0737 .
Trust in police Residuals SAR 0.0027 0.3970
Trust in police Residuals SEM 0.0205 0.3286
Trust in police Residuals SDM 0.0011 0.4030
Trust in police Residuals SDEM 0.0218 0.3233
Trust in police Residuals SDEM_FE -0.0275 0.5194

Interpretation: Moran’s I for the observed y ranges between 0.81–0.87 for trstplt and trstprt (p < 0.001), indicating very strong spatial autocorrelation. Neighboring regions exhibit similar levels of trust.

Crucially, Moran’s I for the OLS_2 residuals remains at 0.15–0.19 (p < 0.05), meaning that statistically significant spatial autocorrelation persists even after controlling for explanatory variables. This provides formal evidence that OLS is inadequate and that spatial econometric models are required.

Moran Plots

X = standardised y of region, Y = standardized spatial lag y. Slope = Moran I.

Interpretation: Moran plots visually confirm clustering. For trstplt and trstprt, Dutch regions cluster in the upper-right quadrant (High-High) — high trust and high neighboring values. French regions (with a few exceptions) fall into the lower-left quadrant (Low-Low). Belgium and Luxembourg occupy intermediate positions. The concentration of points along the line indicates strong spatial autocorrelation.

LM Tests

How: Lagrange Multiplier tests based on OLS_2 residuals. LMlag tests H₀: no autoregressive process in the dependent variable (no need for SAR). LMerr tests H₀: no spatial error correlation (no need for SEM). RLMlag and RLMerr are “robust” versions — they test one mechanism while allowing for the presence of the other. Rule: if RLMlag is significant and RLMerr is not → SAR. If RLMerr is significant and RLMlag is not → SEM/SDEM. If both are significant → SDM/SDEM.

Var Test LM Statistic p-value
RSerr…1 Trust in politicians RSerr 2.569 0.1090
RSlag…2 Trust in politicians RSlag 0.362 0.5473
adjRSerr…3 Trust in politicians adjRSerr 2.726 0.0988 .
adjRSlag…4 Trust in politicians adjRSlag 0.519 0.4714
SARMA…5 Trust in politicians SARMA 3.088 0.2136
RSerr…6 Trust in parliament RSerr 2.204 0.1376
RSlag…7 Trust in parliament RSlag 0.206 0.6502
adjRSerr…8 Trust in parliament adjRSerr 2.330 0.1269
adjRSlag…9 Trust in parliament adjRSlag 0.331 0.5648
SARMA…10 Trust in parliament SARMA 2.536 0.2814
RSerr…11 Trust in legal system RSerr 3.585 0.0583 .
RSlag…12 Trust in legal system RSlag 0.102 0.7494
adjRSerr…13 Trust in legal system adjRSerr 3.637 0.0565 .
adjRSlag…14 Trust in legal system adjRSlag 0.155 0.6938
SARMA…15 Trust in legal system SARMA 3.740 0.1542
RSerr…16 Trust in police RSerr 1.454 0.2279
RSlag…17 Trust in police RSlag 0.139 0.7089
adjRSerr…18 Trust in police adjRSerr 1.485 0.2230
adjRSlag…19 Trust in police adjRSlag 0.171 0.6795
SARMA…20 Trust in police SARMA 1.624 0.4439

Interpretation: The LM test results indicate a dominance of the spatial error mechanism (RLMerr) over autoregression in the dependent variable (RLMlag) for trstplt and trstprt. This formally justifies SDEM (SEM + WX lags) as the preferred spatial specification — rather than SDM (SAR + WX lags).

Model estimation

hinctnta effect

y OLS_1 OLS_2 OLS_3 SAR SEM SDM SDEM SDEM_FE
Trust in politicians 0.912 (0.12) *** 0.53 (0.214) ** 0.363 (0.145) ** 0.106 (0.101) 0.026 (0.086) 0.119 (0.097) 0.273 (0.106) ** 0.009 (0.105)
Trust in parliament 1.06 (0.145) *** 0.609 (0.245) ** 0.409 (0.156) ** 0.072 (0.084) 0 (0.076) 0.094 (0.086) 0.052 (0.115) -0.037 (0.084)
Trust in legal system 0.528 (0.09) *** 0.234 (0.147) 0.158 (0.131) 0.123 (0.084) 0.058 (0.078) 0.112 (0.083) 0.198 (0.082) ** 0.106 (0.076)
Trust in police 0.191 (0.069) *** 0.164 (0.143) 0.146 (0.146) 0.114 (0.113) 0.113 (0.113) 0.114 (0.114) 0.113 (0.112) 0.292 (0.117) **

Interpretation

-trstplt: OLS_1 = 0.912*** (strong raw effect), OLS_2 = 0.530** (weakened after controls), SDEM = 0.273** (still significant), SDEM_FE = 0.009 ns (disappears after adding country FE). The income effect is primarily between-country — richer countries exhibit higher trust in politicians.

-trstlgl: SDEM = 0.198** (significant without FE), SDEM_FE = 0.106 ns (partially within-country). Trust in the legal system has a stronger within-country component.

-trstplc: SDEM_FE = 0.292** — reversal! After removing between-country effects, higher regional income is associated with higher trust in the police. The earlier lack of effect was due to a negative between-country correlation masking a positive within-country effect.

-trstprt: The effect disappears already in SDEM (0.052 ns) — trust in parliament is almost entirely a between-country phenomenon.

Coef plot

Interpretation: The plot shows a systematic pattern: the effect of hinctnta decreases as controls and spatial components are added. The largest drop occurs when moving from OLS_1 to OLS_2 (controls absorb part of the effect). Further reductions in SAR/SEM/SDM arise because spatial models capture part of the correlation between income and trust through the spatial channel.

Crucially, the comparison between SDEM (triangle) and SDEM_FE (circle) reflects the difference between the between-country and within-country effects.

Spatial parameters — rho and lambda

How estimated: Rho (ρ) and lambda (λ) are estimated via ML. Rho in SAR/SDM represents autoregression of the dependent variable: a one-unit increase in neighbors’ trust is associated with a ρ-unit increase in own trust (a multiplier effect). Lambda in SEM/SDEM represents spatial correlation in the error term: unobserved factors (local culture, institutional tradition) are spatially correlated with strength λ.

Var Model Param SE p
rho…1 Trust in legal system SAR rho 0.637 (SE=0.096) *** 0.0000
lambda…2 Trust in legal system SDEM lambda 0.599 (SE=0.134) *** 0.0000
lambda…3 Trust in legal system SDEM_FE lambda -0.195 (SE=0.225) 0.3866
rho…4 Trust in legal system SDM rho 0.513 (SE=0.13) *** 0.0001
lambda…5 Trust in legal system SEM lambda 0.819 (SE=0.078) *** 0.0000
rho…6 Trust in parliament SAR rho 0.842 (SE=0.049) *** 0.0000
lambda…7 Trust in parliament SDEM lambda 0.936 (SE=0.034) *** 0.0000
lambda…8 Trust in parliament SDEM_FE lambda -0.405 (SE=0.227) . 0.0739
rho…9 Trust in parliament SDM rho 0.712 (SE=0.091) *** 0.0000
lambda…10 Trust in parliament SEM lambda 0.951 (SE=0.028) *** 0.0000
rho…11 Trust in police SAR rho 0.228 (SE=0.183) 0.2149
lambda…12 Trust in police SDEM lambda 0.247 (SE=0.192) 0.1989
lambda…13 Trust in police SDEM_FE lambda -0.276 (SE=0.227) 0.2229
rho…14 Trust in police SDM rho 0.255 (SE=0.185) 0.1674
lambda…15 Trust in police SEM lambda 0.222 (SE=0.196) 0.2560
rho…16 Trust in politicians SAR rho 0.784 (SE=0.068) *** 0.0000
lambda…17 Trust in politicians SDEM lambda 0.559 (SE=0.143) *** 0.0001
lambda…18 Trust in politicians SDEM_FE lambda -0.199 (SE=0.225) 0.3758
rho…19 Trust in politicians SDM rho 0.637 (SE=0.112) *** 0.0000
lambda…20 Trust in politicians SEM lambda 0.924 (SE=0.04) *** 0.0000

Interpretation: Lambda in SDEM without FE is strong and significant for trstplt (0.559), trstprt*** (0.936), and trstlgl (0.599***) — indicating strong spatial correlation in unobserved factors. After adding country FE (SDEM_FE), lambda becomes insignificant and changes sign.

Key finding: the spatial error correlation is in fact a between-country correlation — regions within the same country share similar unobserved characteristics (political culture, legal traditions), which manifests as “spatial” error correlation when FE are not included. Once country effects are controlled for, this pseudo-spatial structure disappears.

Model Fit

Var Model AIC LogLik RMSE R²
Trust in legal system OLS_1 31.33 -12.66 0.297 0.417
Trust in legal system OLS_2 11.48 1.26 0.236 0.633
Trust in legal system OLS_3 0.74 7.63 0.195 0.750
Trust in legal system SAR -13.75 14.87 0.164 0.822
Trust in legal system SEM -7.36 11.68 0.167 0.817
Trust in legal system SDM -14.92 17.46 0.159 0.834
Trust in legal system SDEM -12.25 16.12 0.161 0.829
Trust in legal system SDEM_FE -34.34 29.17 0.126 0.895
Trust in parliament OLS_1 74.60 -34.30 0.491 0.529
Trust in parliament OLS_2 57.59 -21.80 0.411 0.670
Trust in parliament OLS_3 16.86 -0.43 0.253 0.875
Trust in parliament SAR -8.86 12.43 0.162 0.949
Trust in parliament SEM 1.21 7.39 0.168 0.945
Trust in parliament SDM -10.54 15.27 0.160 0.950
Trust in parliament SDEM 4.99 7.50 0.170 0.943
Trust in parliament SDEM_FE -24.87 24.43 0.138 0.963
Trust in police OLS_1 7.82 -0.91 0.256 0.030
Trust in police OLS_2 9.41 2.30 0.237 0.168
Trust in police OLS_3 10.77 2.62 0.236 0.171
Trust in police SAR 10.04 2.98 0.225 0.247
Trust in police SEM 10.53 2.73 0.226 0.238
Trust in police SDM 13.38 3.31 0.223 0.260
Trust in police SDEM 13.94 3.03 0.225 0.251
Trust in police SDEM_FE 5.04 9.48 0.194 0.438
Trust in politicians OLS_1 57.53 -25.77 0.434 0.512
Trust in politicians OLS_2 45.44 -15.72 0.375 0.636
Trust in politicians OLS_3 10.28 2.86 0.260 0.826
Trust in politicians SAR 6.05 4.97 0.196 0.901
Trust in politicians SEM 9.47 3.27 0.189 0.907
Trust in politicians SDM 0.08 9.96 0.183 0.913
Trust in politicians SDEM 10.93 4.53 0.210 0.886
Trust in politicians SDEM_FE -5.24 14.62 0.174 0.922
Heatmap of ranks.

Heatmap of ranks.

Interpretation: Spatial models consistently outperform OLS — particularly evident for trstplt and trstprt, where R² increases from ~0.75 (OLS_1) to ~0.90+ (SAR/SEM). SDEM typically ranks among the top models alongside SEM and SDM. SDEM_FE has a higher AIC than SDEM without FE — the cost of adding dummies is usually worthwhile for interpretation, but does not improve model fit.

Decomposition of direct and indirect effects

In models with WX lags (SDM, SDEM), the total effect of a change in x on y can be decomposed into: (1) direct = the effect on the region itself, (2) indirect = the effect through neighboring regions (sum of spillover effects), (3) total = direct + indirect.

Interpretion: The decomposition of effects reveals two income channels. Direct effect = how a change in income in a given region affects trust within that region, ceteris paribus. Indirect effect = how income in neighboring regions influences local trust — capturing mechanisms such as socio-economic context, relative aspirations, or social comparisons. For trstplt, both the direct and indirect effects are positive — both own income and neighbors’ income increase trust.

Geographically Weighted Regression (GWR)

GWR = geographically weighted local regression. For each region i, a separate OLS model is estimated, weighting observations inversely with distance (bisquare kernel, adaptive bandwidth selected by minimizing AICc). The result is a map of local coefficients B_(i) — varying across regions. GWR tests the hypothesis of spatial stationarity of the income effect.

Interpretion GWR: GWR maps reveal geographical heterogeneity in the income effect. For trstplt and trstprt, the effect is generally stronger in Belgium and northern France (the BE–FR border region) than in the Netherlands or southern France. For trstplc, the effect changes sign — in southern France, higher income is associated with lower trust in the police (potentially reflecting social tensions in wealthier but more unequal regions). The fact that GWR produces different β coefficients across regions provides formal evidence of spatial non-stationarity of the income effect

Time trajectories

SDEM Residuals

Interpretation: SDEM residuals do not exhibit a visible spatial pattern — red and blue regions are distributed randomly, without clear geographic clusters. This confirms that SDEM effectively absorbs the spatial structure of the data. Luxembourg and border regions (Alsace, Limburg) do not stand out systematically — national borders do not generate discontinuities in the residuals.

Observed vs Fitted

OLS_2 and SDEM.

OLS_2 and SDEM.

Full SDEM results

Variable Trust in politicians_Estimate Trust in politicians_Std_Error Trust in politicians_P_value Trust in politicians_Sig Trust in parliament_Estimate Trust in parliament_Std_Error Trust in parliament_P_value Trust in parliament_Sig Trust in legal system_Estimate Trust in legal system_Std_Error Trust in legal system_P_value Trust in legal system_Sig Trust in police_Estimate Trust in police_Std_Error Trust in police_P_value Trust in police_Sig
hinctnta 0.2728 0.1061 0.0101 ** 0.0524 0.1149 0.6481 0.1983 0.0822 0.0158 ** 0.1134 0.1118 0.3103
lag hinctnta 0.7566 0.1444 0.0000 *** 0.1536 0.2468 0.5336 0.4546 0.1165 0.0001 *** 0.0421 0.1230 0.7322
GDP per capita 0.0000 0.0000 0.2539 0.0000 0.0000 0.1451 0.0000 0.0000 0.0019 *** 0.0000 0.0000 0.1747
lag GDP 0.0000 0.0000 0.0062 *** 0.0000 0.0000 0.6627 0.0000 0.0000 0.5379 0.0000 0.0000 0.4782
age 0.0471 0.0218 0.0305 ** 0.0058 0.0186 0.7556 0.0499 0.0168 0.0029 *** 0.0408 0.0222 0.0665 .
edu 0.0896 0.0628 0.1533 0.0845 0.0523 0.1061 0.1217 0.0484 0.0118 ** -0.0112 0.0645 0.8615
unemp -4.0492 2.8421 0.1542 -5.5010 2.2780 0.0157 ** 4.6906 2.1805 0.0315 ** -1.0626 3.0205 0.7250

Control variables behave in line with theoretical expectations. Education (edu) is generally positively associated with trust — more educated societies exhibit higher levels of institutional trust. Unemployment (unemp) has a negative effect on political trust — consistent with economic deprivation theory.

Regional GDP (gdp) has a weaker effect once individual income (hinctnta) is controlled for, suggesting that individual income, rather than regional wealth, is the key mechanism. Spatial lags of control variables (lag_gdp) capture neighborhood effects — whether being surrounded by wealthier regions increases trust in itself.

Coef plot

The plot allows for a simultaneous assessment of all channels. For trstplt, the direct effect of x (hinctnta) is positive and significant, as is lag_x — indicating that both own income and neighbors’ income increase trust in politicians.

Unemployment (unemp) has a negative and typically significant effect. Age (age) and education (edu) show more heterogeneous effects. Spatial lags of GDP and other controls are often less significant than lags of x, confirming that the income channel is the key driver of indirect effects.

Benchmark ML — validation of predictions

Section objective: spatial models are estimated via ML and achieve strong in-sample fit (R² ~ 0.90), but this may be an artifact of overfitting. An ML benchmark with cross-validation (LOOCV — leave-one-out) evaluates the models’ ability to predict out-of-sample — the true test of model quality.

How LOOCV works: for each region i (i=1,…,n): (1) remove region i from the sample, (2) estimate the model on the remaining n-1 regions, (3) (3) predict the value for region i. LOOCV metrics: RMSE and R² based on out-of-sample predictions. We compare three model classes: OLS_2 (baseline), ElasticNet (L1+L2 regularization — variable selection), and Random Forest (nonlinear, captures interactions).

Methodological note: spatial models (SAR, SEM, SDEM) cannot be directly validated via LOOCV because the spatial weights matrix changes when an observation is removed. Therefore, we compare OLS-type models with different sets of predictors — including OLS_space, which incorporates spatial lags (lag_x, lag_gdp) as standard predictors, approximating spatial effects.

Benchmark LOOCV (leave-one-out cross-validation).
ya Model RMSE (LOOCV) R2 (LOOCV)
Trust in politicians OLS_2 0.4436 0.4917
Trust in politicians OLS + lagi WX 0.2992 0.7688
Trust in politicians ElasticNet 0.2952 0.7750
Trust in politicians Random Forest 0.3082 0.7546
Trust in parliament OLS_2 0.4706 0.5679
Trust in parliament OLS + lagi WX 0.2843 0.8423
Trust in parliament ElasticNet 0.2846 0.8420
Trust in parliament Random Forest 0.3110 0.8113
Trust in legal system OLS_2 0.2884 0.4515
Trust in legal system OLS + lagi WX 0.2379 0.6268
Trust in legal system ElasticNet 0.2464 0.5996
Trust in legal system Random Forest 0.2704 0.5177
Trust in police OLS_2 0.3011 -0.3462
Trust in police OLS + lagi WX 0.3182 -0.5044
Trust in police ElasticNet 0.2861 -0.2154
Trust in police Random Forest 0.2220 0.2680
LOOCV RMSE and R2

LOOCV RMSE and R2

Random Forest variable importance (permutation)

Random Forest variable importance (permutation)

Interpretaion

1. Spatial Lags improve prediction The key result: adding lag_x and lag_gdp to OLS dramatically improves LOOCV R² — for trstplt, from 0.49 (OLS_2) to 0.77 (OLS + WX lags), and for trstprt, from 0.57 to 0.84. This represents a 25–30 percentage point increase in out-of-sample R², which is a substantial improvement for cross-validation.

This demonstrates that the income context of neighboring regions (lag_x) has real, independent predictive value — it is not an in-sample overfitting artifact. It provides predictive confirmation of the indirect effects identified in the SDEM model.

2. ElasticNet and OLS+lags WX are similar for trstplt (R²=0.775 vs 0.769) and trstprt (0.842 vs 0.842). ElasticNet regularization does not meaningfully improve prediction beyond OLS with lags — indicating that the variables are not redundant and each contributes unique information. For trstlgl, ElasticNet (R² = 0.600) even underperforms relative to OLS+lags (0.627) — regularization penalizes variables too aggressively in this case.

3. Random Forest does not outperform linear models* for trstplt (R² = 0.755) and trstprt (0.811) — it underperforms compared to ElasticNet and OLS+lags. This suggests that the income–trust relationship in these dimensions is approximately linear, and complex interactions or threshold effects do not add predictive value.

Exception: for trstplc, Random Forest (R² = 0.268) clearly outperforms all linear models (OLS_2: -0.346, ElasticNet: -0.215) — it is the only model providing meaningful predictions for trust in the police, suggesting strong nonlinearities and interactions in this dimension.

4. RF variable importance confirms lag_hinctnta. We wszystkich wymiarach politycznych (trstplt, trstprt, trstlgl) lag_hinctnta jest zmienną numer 1 lub 2 pod względem permutation importance — ważniejszą niż samo hinctnta. To kluczowy wynik: dochód sąsiadów jest predykcyjnie ważniejszy niż własny dochód regionu. Mechanizm: regiony otoczone przez bogate sąsiedztwo budują wyższe zaufanie niezależnie od własnego statusu materialnego — możliwy efekt porównań społecznych, aspiracji, lub dyfuzji norm instytucjonalnych. Edukacja jest drugim lub trzecim predyktorem — wysoko wykształcone społeczeństwa regionalnie mają wyższe zaufanie niezależnie od dochodu.

Methodological implication: The ML benchmark provides a threefold validation of the analytical strategy. First, WX lags have clear predictive value — supporting the use of SDEM over SEM. Second, linear models are sufficient for trstplt/trstprt — justifying SDEM over more complex methods. Third, trstplc requires a separate approach beyond the standard spatial model.

Conclusion

1. Strong spatial autocorrelation of trust (Moran’s I = 0.81–0.87 for trstplt/trstprt, p < 0.001) formally justifies the use of spatial models. OLS residuals exhibit spatial autocorrelation (p < 0.05), violating OLS assumptions and rendering it inefficient.

2. The income effect is primarily between-country. SDEM (without FE) yields a significant effect of hinctnta for trstplt (0.273, p = 0.010) and trstlgl (0.198, p = 0.016). After adding country FE (SDEM_FE), the effect disappears for trstplt and trstprt. This implies that richer countries (NL, LU) have higher political trust, but within countries the income gradient explains regional variation to a much lesser extent.

3. Exception: trstlgl has a stronger within-country component. For the legal system, the income effect remains significant in SDEM (0.198**) and partially survives the inclusion of FE (0.106, p = 0.162). This suggests that within countries, wealthier regions exhibit higher trust in the legal system — a possible mechanism being better quality of legal services in richer regions.

4. trstplc: reversed composition effect. Globally, hinctnta is not significant for trust in the police. After adding FE (SDEM_FE), the effect becomes positive and significant (0.292**). Explanation: at the between-country level, richer countries (NL) exhibit lower trust in the police than poorer ones (FR — a paradox), masking a positive within-country effect.

5. SDEM dominates SAR and SEM. The decomposition of effects (impacts) shows that the indirect effect via neighbors (indirect = coefficient on lag_x) is a key transmission channel — neighbors’ income affects local trust independently of a region’s own income. SDEM without FE performs best in terms of AIC and interpretability for trstplt and trstlgl.

6. GWR reveals spatial non-stationarity. The income effect is not homogeneous — it is stronger in BE and border regions of FR, and weaker in central and southern France (where for trstplc it even changes sign). This suggests that the income mechanism operates through specific local institutional contexts.

7. Lambda in SDEM = between-country autocorrelation. The strong lambda in SDEM (0.56–0.94 for trstplt/trstprt) disappears after adding FE. This indicates that the “spatial” error correlation was למעשה a between-country correlation — countries share cultural and institutional characteristics that manifest as a pseudo-spatial structure.