The structure of gravity equations

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While the conditional independence cannot be directly tested, we can examine how the results change as we move toward meeting it in terms of the econometric and economic model. To do so let us write the true value of the (log) Xx, Mm, and ϕxm each as a (different) function of some vector of observable variables, summarized by , and an error term for each {ϵx, ϵm, ϵxm}. Using this we write log exports in its conditional expectation form, where uxm is a random error.

(4)
Most gravity estimates adopt some form of (4) with different control variables or fixed effects, for different country and time samples. Moreover, the large majority adopts ordinary least squares in which case the estimated parameter on the PTA variable, , is equal to the average partial effect and a PTA selection bias term: the difference across members and nonmembers of the error terms for exporter capabilities, importer demand characteristics, and bilateral market access.

(5)
where . I wrote the partial effect as an average over all PTAs but several studies focus on specific PTAs. This is reasonable since some PTAs may be deeper or broader than others. The approach in (5) extends to these cases, if we take the expectation over each of the relevant PTAs, but at the potential cost of small sample bias.

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