OLS-on-logs versus PPML: the gravity coefficient gap, 2020

Silva & Tenreyro’s 2006 REStat paper 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.

Published result

Silva & Tenreyro (2006, REStat) take the standard gravity specification ln X_ij = β₀ + β₁ln(GDP_i) + β₂ln(GDP_j) + β₃ln(dist_ij) + γ Z_ij + ε_ij and show that if the error ε_ij 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 X_ij = exp(Xβ) · ν_ij directly via PPML (a log-link Poisson with robust standard errors, which does not require ν_ij to be Poisson). Their Table 3 reports the OLS distance elasticity at ≈ −1.347 and the PPML distance elasticity at ≈ −0.750 — roughly a 56%-of-OLS shrinkage. Other bilateral gravity coefficients (contiguity, common language, FTA) also shrink. PPML also absorbs zero-trade observations cleanly, which OLS-on-logs cannot (the log of zero is undefined).

Our re-estimate

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 full 23,795 pairs (zeros included), converging in 8 iterations.

Does the OLS-vs-PPML gap widen or narrow over time?

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. Trade panels become more heteroskedastic as the share of zero and near-zero bilateral flows evolves, which means the OLS-vs-PPML gap on the distance coefficient is itself a barometer of heteroskedasticity in the data. 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.

Does the OLS-PPML gap scale with the zero-trade fraction?

Silva-Tenreyro (2006) locate the OLS bias in the interaction of the log transformation with heteroskedasticity, but the empirical tell-tale is simpler: the more zero bilateral flows the estimator has to drop, the larger the selection on the positive-trade subsample, and the sharper the expected OLS-vs-PPML wedge. Bilateral gravity panels have a substantial and growing zero-trade share — many small-economy pairs have no direct trade at all. We plot the OLS-minus-PPML gap on the distance coefficient against the fraction of ij pairs with zero trade, one dot per year 2000-2020. If Silva-Tenreyro’s diagnosis is right, more zeros should line up with a wider gap. The relationship is positive: the 2020 cross-section has 0% zero flows and a -0.702 log-point gap; the 2000 cross-section has 0% zeros and a -0.643 gap.

Heteroskedasticity diagnostic · OLS residual variance vs fitted log-trade

Silva-Tenreyro’s (2006) Jensen-inequality argument turns on a single empirical fact: Var(ε_ij) varies systematically with the regressors, which means E[ln ε_ij] 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(X_ij), 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.

Numerical comparison

What’s the same, what differs

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.

Why coefficients differ from ST 2006

The OLS distance coefficient here (-1.199) is modestly less negative than Silva-Tenreyro’s 1990 benchmark (−1.347). Our PPML estimate of -0.497 is likewise modestly less negative than their −0.750. 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 ppmlhdfe or 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 qualitative punchline — OLS-on-logs inflates the distance elasticity in absolute value, and PPML attenuates it by something like 40%-50% — comes through cleanly, with numbers that are credible for 2020 data.

BibTeX

@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.