Common mistake 1 with Gravity model of trade

Taking logs and running OLS. It throws away every zero flow and it is biased whenever the error variance depends on the regressors, which in trade data it always does.

Common mistake 2 with Gravity model of trade

Omitting exporter-time and importer-time effects. Baldwin and Taglioni called this the gold-medal error, and it was standard practice for thirty years.

Common mistake 3 with Gravity model of trade

Averaging the panel over time to "deal with" the zeros. That destroys the variation a trade agreement is identified from.

Common mistake 4 with Gravity model of trade

Reading the trade-agreement coefficient as causal without pair effects. Countries sign agreements with partners they already trade heavily with.

Common mistake 5 with Gravity model of trade

Adding country-pair fixed effects and then wondering why the distance coefficient has vanished. Distance does not change; the effects absorbed it.

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Gravity model of trade

The workhorse of empirical trade, estimated the way the field actually estimates it: Poisson pseudo-maximum likelihood, which keeps the zero flows and stays consistent when the error variance moves with the regressors. Reports the log-linear regression beside it so you can see what taking logs costs.

What it needs

Trade flow - In levels, including the zeros. Do not log it - the model does that where it belongs.
Exporter
Importer
Year
Gravity variables - Distance, GDP each side, a shared border, a shared language, a trade agreement.

The intuition

Trade between two countries behaves like Newton's law: proportional to the product of their economic masses, inversely proportional to the distance between them. That analogy was noticed in 1962 and dismissed as atheoretical for twenty years, until Anderson and others derived it from proper trade models. The modern estimation problem is not the specification, which is agreed, but the fact that the obvious way to fit it - take logs, run OLS - is wrong twice over.

When to use it

Any question about what moves trade between places: does a trade agreement raise it, how much does distance still cost in an age of container ships, what would leaving a customs union do. The same model is used for migration flows, foreign direct investment and commuting.

Where the estimator comes from

The structural gravity equation is X_{ij}=\frac{Y_iE_j}{Y}\left(\frac{\tau_{ij}}{\Pi_i P_j}\right)^{1-\sigma}, where \tau_{ij} is bilateral trade cost, \sigma the elasticity of substitution, and \Pi_i, P_j the OUTWARD and INWARD multilateral resistance terms - how hard it is for i to sell anywhere and for j to buy from anywhere. Those terms are unobservable and vary by country and year, which is exactly what exporter-time and importer-time fixed effects absorb. Estimation is by PPML: maximise \sum_{ij}\left[y_{ij}\,x_{ij}'\beta-\exp(x_{ij}'\beta)\right], whose first-order condition is \sum_{ij}\left(y_{ij}-e^{x_{ij}'\beta}\right)x_{ij}=0 - a moment condition on the MEAN alone, which is why nothing about the variance or the distribution needs to hold for consistency.

The model

X_{ijt}=\exp\!\big(\beta_0+\beta_1\ln Y_{it}+\beta_2\ln Y_{jt}+\beta_3\ln d_{ij}+\beta_4 \mathrm{RTA}_{ijt}+\eta_{it}+\theta_{jt}+\mu_{ij}\big)\varepsilon_{ijt}

What it assumes

The conditional MEAN is exponential. That is all PPML needs - not that the data are counts, not that the variance equals the mean, not normality of anything.
The zeros are real zeros: pairs that genuinely did not trade, not missing data recorded as nought. Those are different things and only one of them belongs in the model.
Exporter-time and importer-time effects are included, or the estimates are biased by multilateral resistance - how attractive all the OTHER partners were that year.
Standard errors are clustered by country pair, because the same pair appears once a year for the whole sample and its shocks persist.
A trade agreement is not assigned at random. Countries that sign are countries that already trade, so the coefficient is a correlation until pair effects and timing are used to argue otherwise.

Statistical properties

PPML is consistent for beta whenever E[y|x] = exp(x'b), whatever the variance does and whether or not y is a count. "Poisson" names the likelihood being maximised, not a claim about the data.
Log-linear OLS is biased under heteroskedasticity because E[ln y] is not ln E[y]. With a multiplicative error whose variance depends on x, the bias term is -sigma-squared(x)/2 and it MOVES with the regressors, which is what makes it a bias rather than a shifted intercept.
PPML uses the zeros. Roughly half of all country pairs do not trade in a given year, and their absence is informative about exactly the small, distant, unconnected pairs the model is supposed to explain.
The first-order condition makes PPML fit the totals exactly when fixed effects are included: predicted exports for each exporter-year equal actual exports. Log-OLS has no such property.
Distance elasticities cluster around -1 across hundreds of studies and a century of falling freight costs, which Disdier and Head documented and nobody has fully explained.
With pair fixed effects the trade-agreement coefficient is identified only from pairs that changed status inside the sample. That is the right comparison and a much smaller one than the row count suggests.

How to read the output

Coefficients on logged variables are ELASTICITIES. A distance coefficient of −1 means a 1% longer route goes with a 1% smaller flow.
Coefficients on 0/1 variables are log points. exp(b) − 1 turns them into a percentage: 0.30 is not 30% but 35%.
The distance elasticity is around −1 in almost every study ever run, across a century in which freight costs collapsed. That stability is one of the genuine puzzles of the field.
GDP elasticities near 1 mean trade scales roughly proportionally with size, which is what the theory predicts.
Compare the PPML and log-OLS columns. The gap is not noise; it is the bias, and it is usually largest on the size elasticities.

A worked reading

On 13,464 pair-years with the elasticities built in, PPML returns distance -1.05, GDP 0.91 and 0.90, an agreement worth 0.33 - against a truth of -1.00, 0.90, 0.90 and 0.30. The log-linear regression on the same data returns -0.90 and 0.67, missing the size elasticity by a quarter, and it does so having discarded 4,416 rows - a third of the sample - because log(0) has no value. The Park-style test gives lambda = 1.57, between Poisson and Gamma, which is the sense in which the variance was "wrong" for logs all along. Nothing about this depends on the data being simulated; Santos Silva and Tenreyro found the same pattern in real trade flows.

Symptom → diagnosis → what to do

If you see this	It means	Do this
PPML and log-OLS far apart	Heteroskedasticity, as theory predicts	Report PPML; the gap is the bias, not noise
Distance coefficient vanishes with pair effects	Distance does not vary within a pair	Correct. It has been absorbed, not disproved
GDP coefficients vanish with exporter-time effects	GDP varies only by country and year	Same reason; the effects contain it
Trade-agreement effect implausibly large	Selection: countries sign with partners they already trade with	Pair effects and lead terms, or treat it as a correlation
PPML will not converge	Perfect separation, or a regressor on a wild scale	Drop separated fixed effects; scale the regressors
Standard errors tiny on 100,000 rows	Not clustered by pair	The same pair recurs every year and its shocks persist

Common mistakes

Taking logs and running OLS. It throws away every zero flow and it is biased whenever the error variance depends on the regressors, which in trade data it always does.
Omitting exporter-time and importer-time effects. Baldwin and Taglioni called this the gold-medal error, and it was standard practice for thirty years.
Averaging the panel over time to "deal with" the zeros. That destroys the variation a trade agreement is identified from.
Reading the trade-agreement coefficient as causal without pair effects. Countries sign agreements with partners they already trade heavily with.
Adding country-pair fixed effects and then wondering why the distance coefficient has vanished. Distance does not change; the effects absorbed it.

Where to read this properly

Santos Silva and Tenreyro (2006), "The Log of Gravity", Review of Economics and Statistics 88(4).
Anderson and van Wincoop (2003), "Gravity with Gravitas", American Economic Review 93(1).
Head and Mayer (2014), "Gravity Equations: Workhorse, Toolkit, and Cookbook", Handbook of International Economics 4.
Correia, Guimaraes and Zylkin (2020), "Fast Poisson Estimation with High-Dimensional Fixed Effects", Stata Journal 20(1).
Yotov, Piermartini, Monteiro and Larch (2016), An Advanced Guide to Trade Policy Analysis, WTO/UNCTAD.

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Gravity model of trade

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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