Submitted:
14 January 2026
Posted:
14 January 2026
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Abstract
Keywords:
1. Introduction
2. Data and Methods
2.1. Data Sources
2.2. Methods and Preprocessing
- (1)
- Data preprocessing and analytic sample
- (2)
- Restricted cubic splines for non-linear GDP–satisfaction associations
- (3)
- Model specification: unadjusted vs adjusted
- (4)
- Inference on non-linearity: nested-model F test
- (5)
- Reporting and visualization
3. Results
3.1. Descriptive AssociationBbetween GDP anC ciLy-leSel Satisfaction
3.2. Covariate Screening and CcollinearitD Diagnostics
3.3. Non-LinearRrelationshiA Assessment
3.3.2. RCS Estimates of the GDP–satisfactionRrelationship
3.3.3. Robustness and Specification Ssensitivity
| Robustness check | Alternative specification(s) | Purpose | Key finding | Implication |
|---|---|---|---|---|
| Scale transformation | GDP (raw) vs. log(GDP) | Reduce right-skewness and leverage of extreme GDP values; improve interpretability. | Positive association persists after log transformation; log scale yields a more even spread of cities across the x-axis. | Use log(GDP) in all regression models. |
| Spline flexibility (df) | RCS df = 3, 4, 5 | Assess sensitivity of curve shape to spline complexity (under-/over-fitting). | Overall increasing pattern is stable across df; df=5 shows more tail fluctuation; df=3 approximates near-linearity; df=4 balances flexibility and stability. | Adopt df=4 as main specification; report df=3 and df=5 as sensitivity. |
| Formal non-linearity test | Nested F-test: Linear vs RCS (df=4) | Test whether spline terms add explanatory power beyond linearity. | Unadjusted: F=0.21, p=0.811 (N=47); Adjusted: F=0.08, p=0.919 (N=45). | Non-linearity is not statistically supported; linear model remains a parsimonious benchmark. |
| Tail uncertainty | Inspect 95% CI width along log(GDP) | Evaluate stability where observations are sparse (distribution tails). | CI widens at low and high log(GDP), indicating higher uncertainty at distribution tails. | Avoid over-interpreting local curvature at extremes; emphasize overall trend. |
4. Discussion
4.1. Main Findings and Iinterpretation oC curSe Shape
4.2. Comparison with Prior Lliterature
4.3. Policy Implications
5. Conclusions
5.1. Summary
5.2. Future Work
References
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| Group | Variable | N | Mean±SD | Min–Max |
|---|---|---|---|---|
| Dependent variable | GDP (raw) | 47 | 8635.83±8641.56 | 360.00–41611.00 |
| log(GDP) | 47 | 8.52±1.17 | 5.89–10.64 | |
| Explanatory variables | Overall city satisfaction | 47 | 75.43±3.76 | 69.75–84.65 |
| Control variables | log(registered population) | 47 | 15.42±0.82 | 13.24–17.35 |
| Population density (persons/km²) | 47 | 520.47±324.64 | 18.97–1283.29 | |
| Tertiary industry share of GDP (%) | 47 | 57.58±10.17 | 38.82–83.52 | |
| log(fiscal expenditure per capita) | 47 | 9.62±0.61 | 7.06–11.09 | |
| Mechanism variables (optional) | Hospital beds (per 10,000 persons) | 47 | 68.97±22.10 | 32.17–128.21 |
| Wastewater treatment rate (%) | 47 | 96.14±3.45 | 84.19–107.29 | |
| Harmless domestic waste treatment rate (%) | 47 | 98.83±4.20 | 73.25–100.00 |
| Variable | VIF | Tolerance |
|---|---|---|
| log(GDP) | 7.48 | 0.13 |
| log(registered population) | 6.7 | 0.15 |
| Hospital beds (per 10,000 persons) | 2.64 | 0.38 |
| Population density (persons/km²) | 2.23 | 0.45 |
| log(fiscal expenditure per capita) | 1.96 | 0.51 |
| Tertiary industry share of GDP (%) | 1.63 | 0.61 |
| Wastewater treatment rate (%) | 1.31 | 0.76 |
| Harmless domestic waste treatment rate (%) | 1.19 | 0.84 |
| Model | n | R2 | Adj_R2 | Nonlinearity_F | Nonlinearity_p |
|---|---|---|---|---|---|
| Model0_RCS_only | 47 | 0.1987 | 0.1427 | 0.2102 | 0.8112 |
| Model1_RCS+controls | 45 | 0.4063 | 0.2317 | 0.0848 | 0.9189 |
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