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Disdier & Head (2008, Review of Economics and Statistics 90(1)) ran a meta-analysis of 1,467 distance elasticities from 103 gravity studies and found the puzzle that the coefficient had, if anything, risen over the post-war period despite the container revolution and the Internet. The obvious question, does that still hold in the BACI + digital era?, is what we re-run here on one consistent pipeline: CEPII BACI 202601 (retrieved 2026-06-01), Legacy gravity_bilateral table (exact release provenance unverified), and WDI, from 1996 through 2024.
Method note. Our specification is the simple Tinbergen (1962) log-linear cross-section, ln Xij,t = αt + βt · ln distij + γt · ln(Yi,t · Yj,t) + εij,t, estimated separately by year. It is deliberately descriptive: no origin / destination fixed effects, no pair fixed effects, no multilateral-resistance iteration (Anderson & van Wincoop 2003), and therefore no identification of a structural trade-cost elasticity. Zero-trade pairs are dropped by the log and heteroskedasticity makes log-OLS inconsistent (Silva & Tenreyro 2006, REStat 88(4)). Figure 1b adds the modern PPML fix with exporter-year and importer-year fixed effects on a zero-inclusive panel. Standard errors in Figure 1 are cluster-robust on exporter ISO3 (CR1), computed in closed form via the Frisch-Waugh-Lovell partialled regressor. Our βtvalues should be read as the time-series evolution of a reduced-form correlation, directly comparable to Disdier & Head's meta-range but not to FE-PPML structural estimates.
Each dot in Figure 1 is one annual cross-section. The 95% interval uses cluster-robust standard errors on exporter, so that the within-exporter correlation of residuals (every exporter trades with many partners) does not understate uncertainty (Cameron & Miller 2015, Journal of Human Resources 50(2)). If containerisation and digital commerce were hollowing out distance, we would expect β̂t to drift toward zero. The opposite shows up in the raw data.
-- per-year β̂(ln dist) with CR1 SE on exporter, via Frisch-Waugh-Lovell
WITH base AS (
SELECT y.year AS yr, co.iso3 AS iso3_o,
LN(y.total_value * 1000) AS yv, LN(gd.dist) AS xv, LN(go.gdp * gi.gdp) AS gv
FROM 'bilateral_year/**/*.parquet' y
JOIN countries co ON co.code = y.exporter_code
JOIN countries cd ON cd.code = y.importer_code
JOIN gravity_bilateral gd ON gd.iso3_o = co.iso3 AND gd.iso3_d = cd.iso3
JOIN wdi go ON go.iso3 = co.iso3 AND go.year = y.year
JOIN wdi gi ON gi.iso3 = cd.iso3 AND gi.year = y.year
WHERE y.total_value > 0 AND gd.dist > 0 AND co.iso3 <> cd.iso3
)
-- closed-form OLS (yv ~ xv + gv) per year, plus cluster-robust variance:
-- V_CR(β) = (G/(G-1)) · ((n-1)/(n-k)) · (Σ_g (Σ_i x̃_i · ê_i)²) / (Σ x̃²)²
-- where x̃ = xv partialled on constant and gv.If container shipping matters asymmetrically, a ton of soybeans moves more cheaply than a ton of medical devices, then distance elasticities should differ by sector. BACI is HS6-coded but our bilateral extract is aggregated across products, so we approximate by classifying each exporter into its dominant HS Section in 2024exports and running the same gravity within each exporter-specialization group. This is not the cleanest cut (a commodity exporter also sells some machinery), but it preserves the gravity identity. A fully bilateral-by-HS6 PPML would be the right fix; see Head & Mayer (2014) §4 and Yotov et al. (2016) ch. 2 for the recipe.
A cleaner test of the digital-trade hypothesis would track services flows: software, streaming, cloud, professional services. BACI tracks goods only, so we use a goods-side proxy: exporters whose largest HS Section in 2024 falls in chapters 49 (printed matter), 84 (computers and machinery), or 85 (electrical and telecom). This is the ICT hardware and information-goods bloc, a chapter-level approximation rather than a paper-derived classification. If digital infrastructure were genuinely lowering the distance wedge in goods, we would expect the distance coefficient on trade fromthese exporters to be less negative than the all-exporter benchmark.
Figure 2 pooled HS Sections into four groups; Figure 3b is the full heterogeneity read at the WCO HS Section level. Each bar is the estimated β̂dist in the 2024 cross-section for exporters whose dominant HS Section equals that Section, with CR1 cluster-robust SE on exporter iso3. Sections with fewer than 300 bilateral pairs are dropped. Sorted from steepest decay (most negative β) at the top to shallowest at the bottom. Blue = Sections I-V (commodities / animal / vegetable / food / mineral), green = Sections VI-XV (intermediate manufactures), orange = Sections XVI-XXI (machinery, transport, instruments, arts). This is the sector-level version of the Feyrer (2019, AEJ: Applied Economics) finding that bulk commodities have fallen in per-unit freight cost more than high-value manufactures, if shipping-cost declines were uniform, β̂ should look flat across sections; it does not.
Figure 4 collapses the 21-section heterogeneity in Figure 3b into the three UN Broad Economic Categories (Rev.5, United Nations Statistics Division 2016) that matter most for the distance-trade literature: consumption goods, capital goods, and intermediates, with every HS6 line assigned by the official UNSD HS-BEC correspondence (dual-use “mixed” lines excluded from the classification vote). This is the Johnson & Noguera (2012, Journal of International Economics86(2)) intermediate-vs-final split at full HS6 granularity, but with an additional capital-goods cut because Bown & Crowley (2016, Handbook of Commercial Policy) and Antràs & Chor (2022, Handbook of International Economics vol. 5) find capital goods to behave distinctly: high value-to-weight, long investment horizons, and tariff regimes that systematically treat capital equipment more leniently than consumer or intermediate goods. If trade composition drives the distance coefficient (rather than the technology of shipping), we would expect the three blocks to sort in a predictable order on β̂.
Table 1 places our workbench estimate next to the three main reference points in the distance-gravity literature. Comparability is not one-for-one because each study uses a different regressor set, fixed-effect structure, and time window, but the ordering and range are instructive.
Three decades of container shipping scaling up, freight costs falling, and digital infrastructure expanding have not visibly pulled the distance coefficient in a standard cross-sectional gravity toward zero. Our trend per decade on β̂t is -0.105 log points, inside noise. The sector decomposition suggests where the action is: complex manufacturing, the category most exposed to containerised and air-freight logistics, shows the shallowest distance decay, while bulk-commodity specialisation shows the steepest. The digital-intensive-exporter cut does not reveal a pro-digital flattening in goods, consistent with the prior that digital kills distance in services rather than in BACI-tracked physical goods, a test we cannot run without bilateral services-trade data of comparable quality.
Figure 1 remains descriptive gravity without fixed effects or MR iteration, and its log transform drops zero-trade pairs. Figure 1b addresses both limitations at the aggregate level: with exporter-year and importer-year FE, pooled PPML is -0.637, versus -1.308 for positive-flow log-OLS under the same fixed effects and covariates. The estimator materially changes the level, but the PPML series does not flatten: it moves from -0.54 in 1996 to -0.64 in 2020, mildly steepening. The puzzle is still a puzzle, subject to the disclosed missing-distance country hole.
| β̂(ln dist) |
|---|
| SE |
|---|
| PPML | exporter-year + importer-year | 86,572 | -0.637 | 0.030 |
| log-OLS, positive flows | exporter-year + importer-year | 75,323 | -1.308 | 0.025 |
| PPML | exporter-year + importer-year + pair | 85,370 | not identified (absorbed by pair FE) | |
dist and distcap are null for 13 of 243 origins in gravity_bilateral. Six are in the pinned top-120 GDP universe: Germany (DEU), Malaysia (MYS), Vietnam (VNM), Pakistan (PAK), Ethiopia (ETH), and Yemen (YEM), so all six are absent from this 114-country estimation sample. The page's existing log-OLS drops them identically, so this missing-distance comparison is like-for-like. The sign of the omission is unknown: an aggregate gravity estimate missing Germany is not biased in a direction anyone can name. PPML drops observations separated by the fixed effects, following Correia, Guimarães & Zylkin; table N is the estimated rather than built sample.SELECT spec, term, year, estimate, std_error, n_obs,
n_pairs, n_countries, zero_share, estimator, fixed_effects
FROM read_parquet('research_gravity_ppml_fe.parquet')
ORDER BY spec, year NULLS FIRST, term;-- β̂(dist) per HS Section, LATEST cross-section, CR1 SE on exporter -- Classify each exporter by its dominant HS Section in LATEST-year exports, -- then run yv = α + β·xv + γ·gv within each section. Section dropped if n<300.
-- β̂(dist) per BEC end-use bucket, LATEST cross-section, CR1 SE on exporter -- Buckets from UNSD HS-BEC Rev.5 correspondence (hs_bec.parquet): -- intermediate / consumption / capital; 'mixed' excluded. Classify each -- exporter by its dominant bucket in LATEST year exports, run -- yv = α + β·xv + γ·gv within each bucket.
-- trade-weighted mean great-circle distance per year
WITH gd AS (
SELECT iso3_o, iso3_d, AVG(dist) AS dist
FROM 'gravity_bilateral/**/*.parquet'
WHERE dist IS NOT NULL AND iso3_o <> iso3_d
GROUP BY iso3_o, iso3_d)
SELECT y.year,
SUM(y.total_value * gd.dist) / NULLIF(SUM(y.total_value), 0) AS d_bar
FROM 'bilateral_year/**/*.parquet' y
JOIN countries co ON co.code = y.exporter_code
JOIN countries cd ON cd.code = y.importer_code
JOIN gd ON gd.iso3_o = co.iso3 AND gd.iso3_d = cd.iso3
WHERE y.total_value > 0 AND gd.dist > 0 GROUP BY y.year ORDER BY y.year;| −1.31 to −1.57 by year, −1.52 in 2024 |
| Descriptive, no multilateral-resistance terms. Larger magnitude than structural estimates because no origin/destination FE; drops zeros so Silva-Tenreyro PPML bias applies. See Figure 1 note. |
| This page (workbench), PPML + exporter-year and importer-year FE | PPML, zero-inclusive flows, CRV1 clustered on country pair | 114 countries, 1996-2020 in 4-year steps, estimated N = 86,572 | -0.64 (s.e. 0.03) | Offline pyfixest estimate on the pinned top-120 GDP universe; perfect-separation observations removed. |