# Gravity equation: Is it always appropriate to use ppml?

I am interested in estimating countries' comparative advantage. Following Costinot et al. (2012) which is a many-sector version of Eaton and Kortum I have the following theoretical relationship between trade flows and productivities. $$\left( \frac{X_{ij}^{k}X_{i^{\prime}j}^{k^{\prime}}}{X_{ij}^{k^{\prime}% }X_{i^{\prime}j}^{k}}\right) =\left( \frac{A_{i}^{k}}{A_{i}^{k^{\prime}}% }\frac{A_{i^{\prime}}^{k^{\prime}}}{A_{i^{\prime}}^{k}}\right) ^{\theta }\left( \frac{d_{ij}^{k}}{d_{ij}^{k^{\prime}}}\frac{d_{i^{\prime}% j}^{k^{\prime}}}{d_{i^{\prime}j}^{k}}\right) ^{-\theta}%$$ where

• $X_{ij}^{k}$ are sector $k$ trade flows from country $i$ to country $j$
• $A_{i}^{k}$ is country $i$'s (labor) productivity in sector $k$
• $d_{ij}^{k}$ are the corresponding trade costs
• $\theta$ is an elasticity whose value I take from the literature.

The double-ratio $\left( A_{i}^{k}A_{i^{\prime}}^{k^{\prime}}\right) /\left( A_{i^{\prime}}^{k}A_{i}^{k^{\prime}}\right)$ is country $i$'s comparative advantage in sector $k$. Following Costinot et al., I derive the estimation equation from the theoretical relationship above in two steps. First, I take logs and control for the differences by adding appropriate fixed effects:

$$\log\left( X_{ij}^{k}\right) =\theta\log\left( A_{i}^{k}\right) +\delta_{j}^{k}+\delta_{ij}-\theta\log\left( d_{ij}^{k}\right) ,$$ where $\delta_{j}^{k}$ is an importer-sector fixed effects that controls for exporter-difference and $\delta_{ij}$ is an exporter-importer fixed effect that controls for differences over sectors. In the second step I replace $\theta\log\left( A_{i}^{k}\right)$ by an exporter-sector fixed effect $\left( \delta_{i}^{k}\right)$ and I view the trade costs $\left( d_{ij}^{k}\right)$ as the error term $\left( \varepsilon_{ij}^{k}\right)$ so that the estimation equation becomes $$\log\left( X_{ij}^{k}\right) =\delta_{i}^{k}+\delta_{j}^{k}+\delta _{ij}+\varepsilon_{ij}^{k}.\label{estimation equation}%$$ Costinot et al. estimate this using OLS.

Please correct me if I am wrong (and keep in mind that econometrics is not one of my strong points). Assuming that $\varepsilon_{ij}^{k}$ is orthogonal to the three fixed effects ($\delta_{i}^{k}$, $\delta_{j}^{k}$, $\delta_{ij}$), the problems pointed out by Santos-Silva and Tenreyro (SST) do not apply here. Using the notation from above, SST discuss the case where the estimation equation is of the form $$X_{ij}^{k}=\exp\left( \delta_{i}^{k}+\delta_{j}^{k}+\delta_{ij}\right) +\varepsilon_{ij}^{k}\label{estimation equation ppml}%$$ whereas my estimation equation above implies that $$X_{ij}^{k}=\exp\left( \delta_{i}^{k}+\delta_{j}^{k}+\delta_{ij}% +\varepsilon_{ij}^{k}\right) .\label{estimation equation 2}%$$ In this case, it seems to me, estimation by OLS is fine.

Related question: Given that I have many zeros in the data set, would it then be wrong to estimate my estimation equation with ppml? My own answer to this question is that I lose the tight connection between the theoretical model and my estimates. My variable of interest $δ_{i}^{k}$ does not have the same interpretation in both cases. Again, please correct me if I am wrong.