The US-Canada border puzzle, restated for 2019 trade

Anderson & van Wincoop’s 2003 AER paper solved one of the most-cited puzzles in international trade: McCallum’s (1995) finding that Canadian provinces traded 22 times more with each other than with US states of comparable size and distance. Anderson & van Wincoop showed that the right theoretical framework — gravity with multilateral resistance terms — reduces the puzzle to about 10×, still economically meaningful but no longer a paradox. We cannot re-solve the MR system here (it requires intranational Canadian and US interstate trade, plus iterative decomposition). What we can do is show the raw size-normalized bilateral intensity of US-Canada trade in 2019 and compare it to the world distribution of country pairs.

Published result

McCallum (1995) regressed log bilateral trade among Canadian provinces and US states on log GDP, log distance, and a Canada-Canada border dummy, and found the border dummy worth exp(3.09) ≈ 22 — provinces trade 22 times more with each other than comparable US-Canada pairs. Anderson & van Wincoop (2003, AER) show this is partly an artefact of omitting multilateral-resistance terms: the price-index-like objects P_i^1-σ and P_j^1-σ that come out of the CES consumer problem and that a structural gravity estimator must include. Re-estimating the gravity equation with implicit multilateral resistance terms (solved by iteration on the fixed-point system) reduces the US-Canada border effect to about 10.7× for Canadian provinces and about 1.5× for US states — still large, no longer absurd. The paper’s method — “gravity with gravitas” — has become the workhorse specification for bilateral trade-cost estimation; every modern gravity paper cites it.

Our re-estimate

We cannot compute the multilateral-resistance correction from BACI alone: BACI has only cross-border flows, not intranational (province-province, state-state) trade. What we can compute is the raw intensity ratio that motivated McCallum. Define bilateral intensity as I_ij = (X_ij × World) / (X_i × M_j), where X_i and M_j are country-level totals. If trade flowed in proportion to size with no frictions, I_ij would equal 1 for all pairs. Observed values capture the bilateral-specific intensity after size-normalization — not a clean MR-corrected border effect, but a diagnostic.

For 2019 (pre-COVID, pre-USMCA-ratification) the raw CAN→USA bilateral intensity is 5.30×, and USA→CAN is 6.20×. The world-wide mean intensity over 32,506 directed pairs is 9.38× (the mean is inflated by a long right tail of bilateral pairs with tight regional ties); the median is 0.17×. US-Canada trade is thus roughly 33.60× as intense as the median world pair but only 0.61× as intense as the mean world pair — the raw “puzzle ratio” without MR correction.

Has NAFTA closed the wedge? US-Canada intensity, 1995-2022

AvW’s headline point is static: the MR-corrected US-Canada border effect is 10.7× for provinces, 1.5× for states, at a single cross-section. A natural dynamic question: under NAFTA (1994), then USMCA (2020), has the raw cross-country size-normalised intensity of US-Canada trade been falling toward the world distribution? If continental integration worked, the ratio Iij(t) / medianworld(t) should decline monotonically.

NAFTA → USMCA · all three directed North American borders

AvW’s published analysis is US-Canada specific (McCallum’s 1988 Canadian-province data). NAFTA (1994) brought Mexico into the same continental free-trade area, and USMCA (2020) renegotiated it with stronger rules-of-origin. If continental integration tightened the North American wedge, we should see USA-MEX and CAN-MEX intensity rising along the same path as USA-CAN. Below, all six directed North American pairs on the same axis, 1995-2022.

EU Single Market · intra-EU border effect over time

AvW’s US-Canada border puzzle has a European analog: the Single European Market (completed 1993), then successive enlargements to 28 members and the Brexit exit (2020). If SEM integration closed the intra-EU wedge, mean intra-EU intensity should fall toward the world median; if SEM is a preferential zone rather than a de-facto single market, the wedge should persist. We label every directed pair as “both-EU” or “not-both-EU” using gravity_bilateral’s time-varying eu_o, eu_d flags (EU-15 through 2003, EU-25 from 2004, EU-27 from 2007, EU-28 from 2013, EU-27 again from 2020) and track mean intra-EU intensity year by year.

Distance gradient · pair-level intensity vs log(distance), 2019

AvW (2003) frame the border puzzle as “trade is more sensitive to distance than McCallum’s regression alone implies, once you control for multilateral resistance.” The cleanest visual diagnostic is a scatter of log(intensity) against log(distance) for every directed country pair. Without origin and destination fixed effects (i.e., without the AvW MR correction), the raw slope is an upper bound on the true distance elasticity. Where each pair sits relative to the regression line is what shows the McCallum-style border bonus for contiguous, integrated partners (USA-CAN, intra-EU) and the deficit for distant or sanction-affected pairs.

Numerical comparison

Multilateral-resistance proxies (σ = 5, δ = 0.3)

AvW’s structural gravity has a fixed-point system in log form: P_j1-σ = Σ_i (tij/Πi)1-σ Yi/YW and Π_i1-σ = Σ_j (tij/Pj)1-σ Ej/YW. A defensible first-order Taylor expansion (Baier-Bergstrand 2009, J Int Econ) replaces the implicit fixed point with a single pass: assume tij = dijδ with δ ≈ 0.3 (Disdier-Head 2008 meta on distance elasticities) and σ = 5 (AvW’s preferred value), then weight by observed trade shares. Π_i^1-σ = Σ_j s^exp_ij · d_ij^(1-σ)δis the outward resistance index; P_j^1-σ uses import shares. These are proxies — the true fixed point requires intranational X_ii and iteration — but they rank countries by how geographically isolated their trade basket is.

Baier-Bergstrand (2009) show this linearisation approximates the exact AvW fixed point with error <5% for OECD samples. The absolute level of Π and P has no meaning (units depend on the distance power); only cross-country ranking does.

Why it might differ

This is a diagnostic, not a replication in the strict sense. Three important caveats on this page. First, no MR decomposition: Anderson-van Wincoop’s core contribution is the fixed-point system P_i^1-σ = Σ_j(t_ij/P_j)^1-σ Y_j / Y_W that turns McCallum’s regression into a structural gravity equation. Solving that system requires (a) an assumption about σ (they use σ = 5), (b) bilateral trade-cost terms t_ij, and (c) intranational trade flows X_ii, which are not in BACI. Our page flags but does not execute this step. Second, unit of comparison: McCallum compared Canadian provinces to US states; we compare the US and Canada as countries against the full world cross-section. Our “puzzle ratio” of 33.60× is therefore not directly comparable to McCallum’s 22×. Third, time: McCallum used 1988 data; we use 2019, more than three decades later. NAFTA (1994), its replacement USMCA (2020), and a generation of continental supply-chain integration all sit between the two vintages.

The qualitative lesson that survives the ambiguity: US-Canada trade is strikingly intense relative to the world median, even in 2019, even without a structural border coefficient. A proper MR-corrected Anderson-van Wincoop (2003) replication would need CFIB-Census-style intranational trade panels — outside this site’s remit.

BibTeX

@article{anderson_vanwincoop_2003,
  author  = {Anderson, James E. and van Wincoop, Eric},
  title   = {Gravity with Gravitas: A Solution to the Border Puzzle},
  journal = {American Economic Review},
  volume  = {93},
  number  = {1},
  pages   = {170--192},
  year    = {2003},
  doi     = {10.1257/000282803321455214}
}

Structural gravity estimation on this site at Silva-Tenreyro (2006). The distance-bin version at Eaton-Kortum (2002). Return to the replication gallery.