Model diagnostics and robustness checks
| Test | Statistic | p-value | Interpretation |
|---|---|---|---|
| Goodness-of-fit (quadratic model) | logLik = 68.186; AIC = −128.37; BIC = −121.62; Pseudo-R2 (FC) = 0.946 | Overall fit of the quadratic (education-only) specification | |
| Hosmer-Lemeshow test | N/a for beta regression (binary-outcome test only) | Not applicable; outcome is a continuous proportion, not binary | |
| Residual analysis (Pearson) | SD = 1.03; |res| > 2:1 obs; |res| > 3:0 obs; Shapiro p = 0.835 | No concerning outliers; dispersion is reasonable | |
| Linear vs Quadratic (LR test) | LR = 67.17 (df = 1) | <0.001 | Quadratic term improves fit, supporting convexity |
| Quadratic vs Cubic (LR test) | LR = 31.85 (df = 1) | <0.001 | Cubic term improves fit; consider higher-order shape |
| Wild cluster bootstrap (quadratic coeff.) | Estimate = 2.8845; 95% CI = [−0.2817, 6.0507]; B = 999; Webb weights; G = 5 | 0.0791 | Marginally significant (p < 0.10); interpret curvature with caution |
| Test | Statistic | Interpretation | |
|---|---|---|---|
| Goodness-of-fit (quadratic model) | logLik = 68.186; AIC = −128.37; BIC = −121.62; Pseudo- | Overall fit of the quadratic (education-only) specification | |
| Hosmer-Lemeshow test | N/a for beta regression (binary-outcome test only) | Not applicable; outcome is a continuous proportion, not binary | |
| Residual analysis (Pearson) | SD = 1.03; |res| > 2:1 obs; |res| > 3:0 obs; Shapiro | No concerning outliers; dispersion is reasonable | |
| Linear vs Quadratic ( | LR = 67.17 (df = 1) | <0.001 | Quadratic term improves fit, supporting convexity |
| Quadratic vs Cubic ( | LR = 31.85 (df = 1) | <0.001 | Cubic term improves fit; consider higher-order shape |
| Wild cluster bootstrap (quadratic coeff.) | Estimate = 2.8845; 95% | 0.0791 | Marginally significant ( |
Beta regression with logit link on FLFP proportion (education-only, no year fixed effects). The log-likelihood value reported here (68.186) differs from that in Table 3 (75.482) because this diagnostic specification excludes year fixed effects; the difference reflects the reduced parameter count rather than any inconsistency in estimation. Pseudo-R2 (FC) = Ferrari and Cribari-Neto (2004). LR tests compare nested polynomial specifications. The Hosmer-Lemeshow test does not apply to continuous outcomes. Wild cluster bootstrap: Webb weights (B = 999), clustered by education group (G = 5), via betareg working-response linearisation. Cameron et al. (2008); MacKinnon et al. (2023)
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