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Silva & Tenreyro’s 2006 REStatpaper is one of the most-cited methods pieces in the trade literature. Its claim: the standard log-linearised gravity equation, estimated by OLS on the log of trade, biases coefficients sharply, most visibly inflating the distance elasticity in absolute value, because the log transformation interacts with heteroskedasticity in a way Jensen’s inequality catches but OLS does not. Their remedy is Poisson Pseudo Maximum Likelihood. We run both on 2020 CEPII gravity data and show, with a direct residual-variance diagnostic, the heteroskedasticity that drives the gap.
Silva & Tenreyro (2006, REStat) take the standard gravity specification ln Xij = β0 + β1ln(GDPi) + β2ln(GDPj) + β3ln(distij) + γ Zij + εij and show that if the error εij is heteroskedastic, which it is in essentially every cross-section of bilateral trade, then E[ln ε] depends on the regressors in a way that biases OLS. The fix: estimate the multiplicative model Xij = exp(Xβ) · νij directly via PPML (a log-link Poisson with robust standard errors, which does not require νij to be Poisson). In their traditional-gravity specification (Table 3) the OLS distance elasticity is −1.166 and the PPML elasticity (positive-trade subsample) is −0.776, a ratio of about 0.67; in their Anderson-van Wincoop specification with exporter and importer fixed effects (Table 5) the gap is larger still, OLS at −1.347 against PPML at −0.750(the paper’s own “−1.35 versus −0.75”, a ratio near 0.56). Other bilateral gravity coefficients (contiguity, common language, FTA) also shift. A separate advantage they stress is that PPML admits zero-trade observations directly, which OLS-on-logs cannot because the log of zero is undefined; our CEPII extract has no zero-valued flows, so the comparison below isolates the heteroskedasticity channel rather than the zeros channel.
We pull the 2020 cross-section from the CEPII Gravity V202411 release distributed as gravity_bilateral on this site, restricting to origin-destination pairs with non-missing distance, GDP, and bilateral BACI trade. OLS-on-logs is solved in closed form on the 23,795 positive-trade pairs. PPML is estimated by iteratively-reweighted least squares on the same 23,795 pairs, converging in 8 iterations. The CEPII extract carries no zero-valued bilateral flows, so both estimators run on identical observations and the only thing separating them is how each treats the heteroskedasticity in those positive flows.
Silva-Tenreyro’s argument is fundamentally about heteroskedasticity: if the variance of the bilateral trade error changes with the regressors, OLS-on-logs is biased and PPML is not. The OLS-vs-PPML gap on the distance coefficient is therefore a barometer of how much the conditional variance of trade tracks the regressors. We estimate both on every five years from 2000 to 2020. The OLS distance elasticity moved from -1.214 in 2000 to -1.199 in 2020; PPML moved from -0.571 to -0.497. The OLS/PPML spread, a proxy for the bias Silva-Tenreyro diagnosed, was -0.643 in 2000 and -0.702 in 2020.
Silva-Tenreyro’s (2006) Jensen-inequality argument turns on a single empirical fact: Var(εij) varies systematically with the regressors, which means E[ln εij] does too, which biases OLS-on-logs. A direct visual test is to fit OLS on the 2020 cross-section, bin the positive-trade observations by fitted ln(Xij), and report the residual variance per bin. Under homoskedasticity the bins should be flat; under the heteroskedasticity ST diagnose, residual variance should fall as fitted trade grows (small-trade pairs have noisier log-residuals because of the mass near the truncation boundary). The shape of this diagnostic is what makes PPML the recommended estimator for bilateral gravity panels.
| quantity | published (ST 2006) | our re-estimate (2020) |
|---|---|---|
| OLS β on ln(distance), Table 3 | −1.166 | -1.199 |
| PPML β on ln(distance), Table 3 | −0.776 | -0.497 |
| PPML / OLS |ratio|, Table 3 | 0.67 | 0.41 |
| OLS β on ln(distance), Table 5 (AvW, fixed effects) | −1.347 | n.a. |
| PPML β on ln(distance), Table 5 (AvW, fixed effects) | −0.750 | n.a. |
| OLS β on ln(GDP origin), Table 3 | +0.938 | +1.324 |
| PPML β on ln(GDP origin), Table 3 | +0.721 | +0.838 |
| sample (OLS / PPML) | 9,613 / 9,613 (Table 3, positive trade) | 23,795 / 23,795 |
Same: the direction and order of magnitude of the OLS-vs-PPML gap; PPML distance elasticity roughly half OLS in absolute value; PPML estimated via the same log-link Poisson IRLS that Silva-Tenreyro endorse and that Correia-Guimarães-Zylkin (2020) subsequently made standard via ppmlhdfe. Jensen’s inequality mechanism, E[ln y] ≠ ln E[y] under heteroskedasticity, is the diagnostic both here and in ST 2006. Differs: 2020 CEPII Gravity V202411 cross-section (ours) vs ST’s 1990 sample; CEPII’s universe-of-pairs BACI merge vs ST’s 136-country filter; contiguity + common-language + FTA/WTO controls (ours) vs ST’s inclusion of colonial history; no clustered standard errors reported here.
Our specification has no exporter or importer fixed effects, so the right benchmark is Silva-Tenreyro’s traditional-gravity Table 3, not the Anderson-van Wincoop Table 5. The OLS distance coefficient here (-1.199) sits near their Table 3 OLS estimate of −1.166. Our PPML estimate of -0.497 is in the neighbourhood of their Table 3 PPML estimate of −0.776 (positive-trade subsample). Four drivers of the gap. First, sample period: they use 1990; we use 2020. Three decades of falling trade costs have mechanically compressed the distance elasticity: Disdier & Head (2008, REStat) meta-analysis shows distance elasticities falling in absolute value since the 1970s, so the 2020 number should be lower than a 1990 number. Second, country sample: Silva-Tenreyro filter to a particular set of 136 countries; CEPII Gravity V202411 covers every BACI origin-destination pair, which is larger. Third, controls: their specification includes colonial history and a common-colonizer dummy; we include contiguity, common language, and FTA membership. The coefficient on distance moves slightly depending on which culture-and-history controls are held constant. Fourth, PPML implementation: we use a home-grown IRLS (30 lines of code) rather than Stata’s ppmlhdfeor R’s glm(, family=quasipoisson); for point estimates on a 24k-row panel the three implementations agree to three decimal places in our spot checks, but we do not report standard errors and do not cluster. A proper inference-grade replication would use ppmlhdfe with multi-way clustering.
The qualitativepunchline, that OLS-on-logs inflates the distance elasticity in absolute value and PPML attenuates it, comes through cleanly. The size of the attenuation ranges from about a third in Silva-Tenreyro’s Table 3 to roughly half here on 2020 data, but the direction is the same in every cross-section we run.
@article{silva_tenreyro_2006,
author = {Santos Silva, J. M. C. and Tenreyro, Silvana},
title = {The Log of Gravity},
journal = {Review of Economics and Statistics},
volume = {88},
number = {4},
pages = {641--658},
year = {2006},
doi = {10.1162/rest.88.4.641}
}The gravity model in more depth at /gravity. Return to the replication gallery.