# F6-dev-01 — Preston curve in the latest usable WDI year

# Question

Across countries/economies, how strongly is life expectancy at birth (LE, years) associated with the natural log of GDP per capita (PPP, constant 2021 intl $) in the most recent year with usable World Bank data? Estimand: slope β of the linear projection LE = α + β·ln(GDPpc) + ε, one equally weighted observation per country. The interpretation is descriptive/associational, not causal.

# Data and exclusions

Inputs: data/wb_indicators.csv (2349 rows) and data/wb_entities.csv (296 rows) (n_input_rows); SHA-256 hashes match data/README.md (see out/state/data_audit.md). Exclusion ledger (out/exclusions.csv):

1. Dropped 396 entity-year rows belonging to 44 aggregate entities (region = Aggregates) → 1953 rows, 217 countries/economies.
2. Analysis year: countries with both indicators number 199 in every year 2015–2022 and 197 in 2023 (complete_countries_by_year); the latest year with ≥150 is 2023 (analysis_year). Kept 217 country rows (n_countries_in_analysis_year).
3. Dropped 20 countries with a missing indicator in 2023 (n_excluded_missing); all 20 lack GDP (life expectancy is never missing). No imputation was done. The analysis uses 197 countries (n_analysed).

2349 − 396 − 1736 − 20 = 197. Data problems and how each was handled are listed in data_issues: the aggregates; the 20 missing countries (ASM, CHI, CUB, ERI, GIB, GUM, IMN, LIE, MAF, MCO, MNP, NCL, PRK, PYF, SSD, SYR, VEN, VGB, VIR, YEM); an implausibly volatile Central African Republic LE series in earlier years (its 2023 value of 57.4 looks plausible and was kept); and the drop in West Bank and Gaza LE in 2023 (kept as reported).

# Method

• Baseline: predict every country's LE with the cross-country mean. Its RMSE is baseline_metric.
• Main model: OLS of LE on ln(GDPpc) over the 197 countries, unweighted. Standard errors are heteroskedasticity-robust HC3 (se_type). HC3 was picked because it is the most conservative of HC0–HC3 in moderate samples. The 95% CI is β ± 1.96·SE (normal approximation). The fit metric is the RMSE of the in-sample residuals (model_metric).
• Conclusion rule as given in TASK.md: positive_effect if the 95% CI lies entirely above 0.
• Supplementary check, not part of the estimand: re-estimate after dropping the 5 countries with the largest Cook's distance (supp_sens_drop5_*). HC0–HC2 and classical SEs are also reported (supp_effect_se_by_type, supp_effect_se_classical).
• out/analysis.py regenerates everything deterministically. It uses no randomness (random_seed = null).
• Outputs: out/figure1.png (scatter plot with the OLS line) and out/table1.csv (means by income group).

# Results

• Slope: β = 5.27 years per log point of GDP per capita (effect_estimate = 5.267). Intercept 22.57 years (intercept). A 10% income difference therefore goes with about 0.50 more years of LE (5.267·ln 1.1, derived from effect_estimate). A doubling goes with about 3.65 years (5.267·ln 2).
• Strength: Pearson r = 0.861 (pearson_r), R² = 0.742 (r_squared).
• Fit: residual RMSE is 3.59 years (model_metric), against 7.08 years for the mean-only baseline (baseline_metric). ln(GDPpc) roughly halves the prediction error.
• Conclusion: positive_effect. In 2023 the Preston-curve association holds and is strong.
• Table 1 (out/table1.csv): mean LE rises with income group, from 63.2 years (Low income, n=20) to 67.8 (Lower middle income, n=46), 73.7 (Upper middle income, n=58) and 79.7 (High income, n=73).
• Largest negative residual: Nigeria, −15.3 years (supp_largest_neg_residual_iso3, supp_largest_neg_residual).

# Uncertainty

• HC3 SE = 0.211 (effect_se). 95% CI for β: [4.85, 5.68] (effect_ci_low, effect_ci_high), well away from zero.
• The choice among robust SEs makes little difference: HC0 0.207, HC1 0.208, HC2 0.209, HC3 0.211, classical 0.222 (supp_effect_se_by_type, supp_effect_se_classical).
• Dropping the 5 most influential countries (NGA, GRL, TCD, SLB, GUY; supp_sens_drop5_cooks_iso3) gives β = 5.32 (SE 0.183), inside the main CI (supp_sens_drop5_effect_estimate, supp_sens_drop5_effect_se).
• The CI covers sampling-style uncertainty only, treating countries as independent draws from a superpopulation. The 197 countries are close to the full population of economies, so this is a modelling convention. The CI does not cover measurement error, model misspecification or non-random missingness.

# Limitations and open issues

• No causal interpretation. β is an association. Confounders such as institutions, education, health technology, disease environment and conflict affect both income and LE, and causation can also run from health to income.
• Functional form. The linear-in-log line is a projection. The figure shows curvature at the top: LE flattens among high-income economies, and the low-income residuals are widely spread. A nonlinear or spline fit was not estimated, so β is an average slope over the income range, not a constant marginal effect.
• Non-random missingness. The 20 excluded economies are either small high-income territories (e.g. MCO, LIE, GIB) or conflict-affected/sanctioned states (SSD, SYR, YEM, VEN, PRK, CUB, ERI). Their direction of bias is unclear, and no bounding analysis was done.
• Measurement and data quality. GDP PPP figures for fragile states and small territories are imprecise, and errors in ln(GDPpc) would bias β toward zero (attenuation). LE estimates for countries without complete vital registration are modelled, not observed. The CAF series shows implausible year-to-year swings (18.8 years in 2022), which suggests vintage or modelling artefacts in the LE data. 2023 values were not validated against external sources.
• Single year. 2023 can still carry post-COVID mortality effects and conflict shocks (e.g. West Bank and Gaza). Stability across 2015–2022 was not checked because the task restricts the analysis to the selected year.
• Equal weighting. Each economy counts once whatever its population, so tiny territories weigh as much as India or China. A population-weighted slope would answer a different question and was not computed.
• What would invalidate the result: a misclassification of aggregates as countries (checked: all 44 aggregate entities in the file were removed via the region flag); a systematic revision of the 2023 WDI vintage; or leverage concentrated in a few points (partly checked with the leave-5-out sensitivity, which did not change the conclusion). Not checked: heavier-tailed inference (e.g. bootstrap), spatial or regional correlation of errors (regional clustering would probably widen the CI), and outliers beyond the 5 most influential points.