Beta- and Sigma-Convergence of Real GDP per Capita, 1990–2024: Cross-Country Estimates and an Ornstein–Uhlenbeck Reconciliation

Using World Bank WDI GDP per capita at PPP (constant 2021 international dollars, NY.GDP.PCAP.PP.KD; source update 8 October 2026) for 183–194 economies, I estimate unconditional beta-convergence and track sigma-convergence. The cross-country convergence speed is small but statistically different from zero: λ = 0.44% per year for 2000–2024 (β = −0.00419, HC1 s.e. 0.00094, N = 189), against the roughly 2% per year typical of conditional estimates. The dispersion of log income in a balanced panel of 183 economies fell from 1.202 (2000) to 1.150 (2024), a trend of −0.0022 per year (HAC s.e. 0.0004). I derive the stochastic Solow–OU mapping between the two concepts and prove that beta-convergence is necessary but not sufficient for sigma-convergence. Calibrated to the estimates, the mapping implies that cross-country dispersion is still above its stationary level. Within-income-group β estimates are larger, but they are biased toward convergence because group membership is defined ex post.