Converging to Convergence — Interactive Lab

A pedagogical companion to Converging to Convergence: Understanding the Main Ideas of the Convergence Literature ↗ Back to the post

Has the world been converging?

For decades, poor countries grew slower than rich ones — a pattern called divergence. Then, around the year 2000, this flipped: unconditional convergence emerged. The β coefficient of 10-year forward growth on initial log income shifted from +0.53 in the 1960s to −0.76 by 2007. This app lets you see, slide, and decompose that flip yourself.

Across four tabs you will: watch the rolling β curve cross zero in real time; toggle between β (catch-up) and σ (dispersion) to see why beta convergence leads sigma convergence by a decade; pull the δ × λ levers to reproduce the omitted-variable-bias identity from the post's §8; and explore the actual Kremer–Willis–You (2021) decade-by-decade forest plot.

The headline animation — β crosses zero around the year 2000

Each year the world ran a regression of 10-year forward growth on log GDP per capita. The slope coefficient — β — is the convergence coefficient. The animation below traces it from 1960 to 2008. Watch it stay positive through the 1980s (richer countries growing faster — divergence) and then plunge below zero after the late 1990s (catch-up — convergence). The dashed orange line is the no-catch-up reference.

Tab 2

β & σ Explorer

Drag a year slider and watch the rolling β coefficient and the income-dispersion σ move together. Compare growth across the four income quartiles.

Tab 3

OVB Simulator

Pull the δ (income–correlate) and λ (growth–correlate) sliders. Watch the predicted gap δ·λ shrink to match the post's headline 91% reduction (0.44 → 0.04).

Tab 4

Forest Plot

The decade-by-decade β coefficient with 95% CIs, plus the absolute-vs-conditional gap. Hover any point for SEs, sample sizes, and CIs.

Glossary (open a card if a term is unfamiliar)

β-convergence (unconditional)
The slope of 10-year forward growth on log initial income, with no controls. Negative ⇒ poorer countries catch up. This is the rolling line in Tab 2.
β* (conditional convergence)
Same regression but with growth correlates included. Often negative even when β is not. The gap β − β* is the OVB. See Tab 4's twin lines.
σ-convergence
The cross-country standard deviation of log GDP per capita at year t. Tracks the width of the world income distribution, not the slope.
OVB identity
β − β* = δ × λ. The gap between unconditional and conditional convergence is exactly the product of two slopes. Tab 3 is built around this identity.
δ — correlate-on-income slope
How much richer countries have more of the correlate (e.g. democracy). Stable across decades — the paper's finding.
λ — growth-on-correlate slope
How much the correlate predicts growth (controlling for income). Has collapsed for short-run policy variables since the 1990s — the paper's central result.
Lambda flattening
The empirical fact that λ for short-run policy variables (democracy, inflation, fiscal policy) fell by ~80% from 1985 to 2005. The 1990s Washington-Consensus growth regressions failed their out-of-sample test.
Growth correlates
Right-hand-side variables in 1990s growth regressions: inflation, investment, schooling, openness, democracy, rule of law. The 1985–2015 trend is convergence in each one of them.

The rolling β coefficient with 95% CI band

The chart below plots β_t, the convergence coefficient estimated each year from a year-interacted panel regression. The dashed orange line is β = 0 (no catch-up). Drag the year slider to scrub a vertical cursor across decades; the numeric β estimate updates live. Below it, σ (income dispersion) and the four income-quartile growth lines tell the complementary story.

Slide to inspect β in any specific year — the dot follows the line.
β at cursor year
—
slope of 10-yr growth on log income
Sign
—
+ ⇒ divergence, − ⇒ convergence
β trend slope per year
−0.025
post-§4 estimate (p < 0.01)

σ-convergence — has income dispersion narrowed?

σ is the cross-country standard deviation of log GDP per capita each year. It rose steadily from 0.95 (1960) to a peak of 1.22 in 2000, then eased to 1.13 by 2015. Beta convergence (poorer countries growing faster) is necessary but not sufficient for sigma convergence — catch-up has to overcome random shocks that push countries apart.

Who drives convergence? Growth by income quartile

Convergence since 2000 is driven by both ends of the income distribution. In the 1960s, Q4 (richest) grew fastest at 3.49% per year and Q1 (poorest) at only 2.46%. By 2007 the ordering had completely reversed: Q1 at 3.02%, Q4 at 0.31%. The marathon's leaders slowed and the back-of-pack accelerated — the pack compressed from both ends.

What to look for

  • β crosses zero around 1999–2000. Drag the year slider to 1985 (β ≈ +0.32) then to 2005 (β ≈ −0.74) — the sign-flip is the whole story.
  • σ peaks ~10 years after β crosses zero. Beta convergence leads sigma convergence. Catch-up growth has to overcome the random shocks that push countries apart.
  • Q4 (richest) growth collapses from 3.5% to 0.3%. The rich didn't just slow; they nearly stopped growing.

OVB simulator — pull the δ and λ levers

The omitted-variable-bias identity is the analytical engine of the post's §8: β − β* = δ × λ. The gap between unconditional and conditional convergence equals the product of (a) how much richer countries have more of the correlate (δ), and (b) how much the correlate predicts growth (λ). Pull either lever and watch the predicted gap update. The default values are the Polity-2 worked example from §8.3.

1985 Polity-2: δ = 0.494. 2005: 0.216. The paper finds δ is stable across decades.
1985 Polity-2: λ = 0.891. 2005: 0.183. The paper finds λ flattened by 80%.
1985 Polity-2: β* = −0.111. 2005: −0.807. β* is the convergence rate after controlling for the correlate.

Predicted gap δ × λ

δ × λ (gap)—
δ (slope)—
λ (slope)—
β = β* + δλ—

The OVB formula β − β* = δ × λ is an algebraic identity — it holds exactly in any linear regression.

Two periods at a glance

1985 actual gap0.440
2005 actual gap0.040
% reduction91%
δ 1985 → 20050.494 → 0.216
λ 1985 → 20050.891 → 0.183

The collapse comes mostly from λ flattening — δ is comparatively stable.

What β looks like when you turn the levers

With β = β* + δλ, the OVB identity tells you the unconditional β directly: fix the conditional slope β* and the two component slopes, and you have the headline number. Move the sliders to reproduce 1985 (β = +0.33) and 2005 (β = −0.77).

Three preset narratives

  • 1985 (preset): δ = 0.49, λ = 0.89, β* = −0.11. Gap = 0.44. β ≈ +0.33 (divergence).
  • 2005 (preset): δ = 0.22, λ = 0.18, β* = −0.81. Gap = 0.04. β ≈ −0.77 (convergence).
  • What if λ stayed at 1985 level? Keep λ = 0.89, set δ to its 2005 value 0.22. Gap = 0.20. β ≈ −0.61. Even with stable λ the world would still have converged — but mostly because δ shrank by half.

Monte Carlo: 100 random (δ, λ) draws

To see how much of the gap is driven by δ versus by λ, run 100 random draws where both vary uniformly over the post's empirical range. The histogram shows how the implied gap δ·λ is distributed. The orange marker is the actual 1985 gap of 0.44 — your draws should cluster well below it.

The post's headline figures — interactively

These numbers come straight from convergence2_beta_by_decade.csv and the Polity-2 OVB worked example in results_report.md. Toggle decades and outcome panels to compare. Hover any point to see its SE, 95% CI, and sample size. The dashed line is the no-catch-up reference (β = 0).

Panels

Rows

Absolute vs conditional convergence over time

From convergence2_conditional_convergence.csv: in 1985 the unconditional β was +0.42 (divergence) while the conditional β* was −1.07 (strong catch-up given institutions) — a gap of 1.49. By 2000 the gap had narrowed to 0.15. The two lines below converge to convergence: the unconditional β finally catches up with what was always true conditionally.

What to look for

  • Untoggle the 2000s and 2007 decades on the β-by-decade panel. The remaining four decades cluster around zero. Convergence really is a post-2000 phenomenon.
  • Hover the OVB panel's Polity2 1985 row. The estimate is 0.440. Now hover Polity2 2005 — it falls to 0.040, a 91% reduction. That is the punchline of §11 in one number.
  • Compare the λ panel to the δ panel. δ shrank by 56% (0.494 → 0.216). λ shrank by 79% (0.891 → 0.183). The flattening of λ is the dominant driver.
  • Turn on the Correlate convergence panel. Every short-run policy variable converged: inflation (−3.07), investment (−2.98), Polity2 (−2.03). The exception is Barro-Lee education (−0.16), which is not significant.
  • The twin-line chart's blue line (β*) stays firmly negative throughout. Conditional convergence held all along; what changed was the world catching up to the model.