# F6-dev-01 — Preston curve in World Bank WDI data

# Question

Does the cross-country relationship between real income per capita and life expectancy (the Preston curve) hold in the most recent usable year, and how strong is it? The estimand is the slope β of the equal-weighted linear projection LE_i = α + β·ln(GDPpc_i) + ε_i across countries/economies. LE is life expectancy at birth in years; GDPpc is GDP per capita in PPP constant 2021 international $. β is descriptive (associational), not causal.

# Data and exclusions

Inputs: data/wb_indicators.csv (2,349 entity-year rows) and data/wb_entities.csv (296 entities) (n_input_rows). The SHA-256 hashes match data/README.md (see out/state/data_audit.md). Exclusion ledger (out/exclusions.csv):

1. Dropped the 79 entities with region Aggregates, i.e. 396 rows. That leaves 1,953 rows covering 217 countries/economies.
2. Kept the analysis year. The number of countries/economies with both indicators is 199 in every year 2015–2022 and 197 in 2023 (complete_countries_by_year). The latest year with ≥150 is therefore 2023 (analysis_year), which has 217 countries/economies (n_countries_in_year).
3. Dropped the 20 countries/economies with a missing indicator (n_excluded_missing). All 20 are missing GDP, none is missing LE. There is no imputation. This leaves 197 analysed countries (n_analysed).

The data problems found and how each was handled are listed in data_issues:

• Aggregates mixed into the indicator file.
• GDP missingness is non-random. The excluded economies are mostly small high-income territories plus conflict or sanctioned states (CUB, ERI, PRK, SSD, SYR, VEN, YEM).
• The Central African Republic's LE series is erratic before 2023 (18.8 years in 2022). Its 2023 value of 57.4 is plausible and is kept.
• LE in West Bank & Gaza fell by about 11.5 years in 2023. Kept as reported.

# Method

• Baseline: predict every country's LE by the sample mean. Its RMSE is baseline_metric.
• Main model: OLS of LE on ln(GDPpc), one equally weighted observation per country. I report the in-sample residual RMSE (model_metric), R² (r_squared) and Pearson r (pearson_r).
• Uncertainty: heteroskedasticity-robust standard error, HC3 (effect_se_type). I chose HC3 because it is the most conservative of the HC variants at this sample size. The 95% CI is β ± t₀.₉₇₅(n−2)·SE (ci_critical_value_t).
• Decision rule: the conclusion code follows from whether the CI excludes zero.
• Sensitivity: SEs under HC0, HC1, HC2 and the classical formula, and a re-fit after dropping the 5 largest absolute residuals.
• Implementation: out/analysis.py reads only data/, uses no randomness (random_seed = null), and generates figure1.png and table1.csv.

# Results

• Fit: LE = 22.57 (intercept) + 5.267 (effect_estimate) × ln(GDPpc), with n = 197 (n_analysed).
• What the slope means: a 1-log-point difference in GDP per capita (about 2.7×) goes with 5.27 more years of life expectancy. A doubling goes with about 5.267 × ln 2 ≈ 3.65 years (arithmetic on effect_estimate).
• Strength: Pearson r = 0.861 (pearson_r) and R² = 0.742 (r_squared).
• Against the baseline: residual RMSE is 3.59 years (model_metric), versus 7.08 years for the mean-only baseline (baseline_metric). That is roughly half the prediction error. The mean LE is 73.5 years (mean_life_expectancy).
• Conclusion: positive_effect (conclusion). The Preston-curve association holds clearly in 2023.
• Table 1 (out/table1.csv) shows mean LE rising steadily with income group:

• Figure 1 (out/figure1.png) shows the scatter with the fitted line.

# Uncertainty

• Main interval: HC3 SE = 0.211 (effect_se); 95% CI [4.85, 5.68] (effect_ci_low, effect_ci_high); t ≈ 25 (effect_t_stat).
• Other SE formulas: HC0 0.207 (sensitivity_se_hc0), HC1 0.208 (sensitivity_se_hc1), HC2 0.209 (sensitivity_se_hc2), classical 0.222 (sensitivity_se_classical). None of these changes the conclusion.
• Influence of outliers: dropping the 5 largest residuals (NGA, GRL, NRU, GNQ, GUY; sensitivity_trim5_dropped_iso3) gives β = 5.32 (sensitivity_trim5_effect), well inside the main CI.
• What the CI means: the sample is nearly the whole population of economies. The interval therefore describes sampling uncertainty under a superpopulation view of countries, not uncertainty about which countries are included.

# Limitations and open issues

• What would invalidate the result:
  • The 2023 values may be revised later, or may be WDI model-based estimates/projections rather than measurements. This is not verifiable from the files.
  • Missingness may be non-random. The 20 excluded economies include rich microstates (e.g. Monaco, Liechtenstein) and low-LE conflict states (South Sudan, Yemen). Including them could move β in either direction; I did not quantify this.
  • The definition of the aggregates could be wrong. I relied on the Aggregates flag.
• Specification threats:
  • The linear-in-log form is a summary. The curve visibly flattens and spreads at high income, where several rich economies sit well below the line.
  • β is an equal-weighted cross-country association. It is neither population-weighted nor causal. Confounders such as health systems, disease environment and institutions, and reverse causality from health to income, are not addressed.
  • Countries are not independent (regional clustering, spatial correlation). The robust SE does not account for this, so the CI may be too narrow.
• Not checked:
  • Other years. The year is fixed by the rule; I estimated no other year.
  • Nonlinear or regional specifications.
  • Measurement error in PPP GDP, which would attenuate β.
  • Leverage diagnostics beyond the trim-5 refit.
• Data quirks: the Central African Republic's erratic pre-2023 LE series and the West Bank & Gaza 2023 drop indicate data-quality and shock issues. In 2023 they affect single observations only.