Eleven Trade Equations, One Global Shock

U.S. exports to Canada do not live in a Canadian vacuum. Nor do exports to Germany, Japan or Mexico. A pandemic, a dollar move, a shipping disruption or a synchronized manufacturing cycle can hit many trade relationships at once. Eleven separate ordinary least-squares regressions see eleven separate error terms. Seemingly unrelated regression sees the system.

11trade equations

353common monthly observations

0.300average residual correlation

0.716maximum residual correlation

The question

For each partner, RainbowStats regresses the monthly log change in U.S. exports on the monthly log change in U.S. imports:

Δlog(Exports_i,t) = α_i + β_i Δlog(Imports_i,t) + ε_i,t

Because both variables are in log differences, β is a short-run co-movement elasticity. A coefficient of 0.80 means that a 1% monthly increase in imports is associated with roughly a 0.8% monthly increase in exports for that partner. It is not a causal import multiplier, and it is not a model of the trade balance.

Why SUR?

Each bilateral equation has its own intercept, slope and economic history. That is why the equations are “seemingly unrelated.” Their residuals, however, are related. The system’s average residual correlation is 0.300 and its maximum is 0.716—enough shared disturbance for feasible generalized least squares to usefully reweight the information across equations.

The procedure is straightforward: estimate the eleven equations, combine the regression objects into a simultaneous system, estimate the SUR covariance structure, and display the result. RainbowStats performs all four steps in one expression.

ExpImpReg=slideshow(SUR_REGRESSION(SIMU_REGRESSION(LOOP_OBJECTS(REGRESSION(LOGDIFF(LIST(EXP$$,IMP$$))),CA,MX,JP,UK,GE,FR,4759,KR,6021,0003,0004))))

Run the complete analysis in RainbowStats

The elasticity drop is the result

Canada and Mexico remain tightly linked after the system correction: their SUR elasticities are 0.795 and 0.850. For several other partners, the coefficient falls sharply once common shocks are recognized. The U.K. estimate drops from 0.158 to essentially zero; Italy falls from 0.136 to 0.044; Japan from 0.314 to 0.177; and the European Union from 0.375 to 0.185.

OLS versus SUR co-movement elasticitiesMonthly log changes, February 1997–June 2026 common SUR sample. Full bar length equals 1.0.

OLSSUR

Canada

Mexico

Japan

U.K.

Germany

France

Italy

South Korea

Australia

World

0.0 ← coefficient → 1.0

The key interpretation: OLS can attribute globally synchronized trade movements to a bilateral import-export relationship. SUR asks how much of that apparent relationship survives after the covariance of the eleven equation errors is recognized.

Efficiency improves even when fit does not

SUR reduces every reported slope standard error in the system. The largest gain is for the European Union, where the OLS standard error of 0.037 falls to 0.024—a 1.549 efficiency ratio. Germany improves from 0.043 to 0.034, a ratio of 1.276. The gain is not cosmetic: it changes the economic reading of several coefficients even when statistical significance remains.

Partner	OLS β	SUR β	OLS SE	SUR SE	SE gain
Canada	0.944	0.795	0.034	0.030	1.139×
Mexico	0.821	0.850	0.027	0.026	1.055×
Japan	0.314	0.177	0.044	0.038	1.155×
United Kingdom	0.158	0.003	0.056	0.047	1.192×
Germany	0.393	0.241	0.043	0.034	1.276×
France	0.245	0.117	0.050	0.042	1.188×
Italy	0.136	0.044	0.061	0.053	1.143×
South Korea	0.246	0.215	0.068	0.060	1.124×
Australia	0.002	0.013	0.035	0.031	1.124×
European Union	0.375	0.185	0.037	0.024	1.549×
World	0.442	0.395	0.049	0.041	1.187×

A useful SUR nuance: equation-by-equation OLS minimizes the ordinary unweighted sum of squared residuals. SUR minimizes a covariance-weighted objective. Therefore, the reported ordinary SSR can rise slightly—from 36.655 under OLS to 37.249 under SUR—without contradicting an efficiency gain. The 0.984 figure in the summary is the OLS-to-SUR SSR ratio, not evidence that SUR should mechanically produce a smaller raw SSR.

What the system says

North American supply chains stand out. Canada and Mexico retain the strongest co-movement elasticities after the system correction.
Shared shocks explain part of the European result. The U.K., France, Italy, Germany and aggregate EU coefficients generally shrink under SUR.
Australia is effectively unrelated in this specification. Both OLS and SUR estimates are near zero.
The world equation remains meaningful. Its SUR elasticity is 0.395 with a t-statistic of 9.526, consistent with broad import-export co-movement even after accounting for correlated disturbances.

One line, a complete system

The econometrics are not new. The compression is. A conventional workflow would require data retrieval, alignment, transformation, eleven separate regressions, residual extraction, covariance estimation, feasible GLS and output formatting. Here the entire system is reproducible from one line.

That matters because sophisticated methods become more useful when the path from question to result is short enough to invite experimentation. Add a partner, remove an aggregate, change the transformation or inspect the residual matrix. The analysis is not trapped in a spreadsheet or a private notebook. It is live, inspectable and repeatable.

Method note. The SUR system uses the 353 observations common to all eleven monthly equations, from February 1997 through June 2026. Individual series may have longer histories outside the common system sample. Estimates were verified from the live RainbowStats result on August 26, 2026. Country and aggregate codes: CA (Canada), MX (Mexico), JP (Japan), UK (United Kingdom), GE (Germany), FR (France), 4759 (Italy), KR (South Korea), 6021 (Australia), 0003 (European Union), and 0004 (World).