GLS Estimation in Eaton-Kortum (2002): Two-componet error and importer- and exporter- FE interpretation

[Noé J Nava]

Dear Stata users,

I am using Eaton and Kortum (2002) General Equilibrium model, so I have to estimate a model by Generalized Least Square to recover a set of structural parameters. For your reference, the paper is included and my purpose is to estimate equation (30) in page 1761. Two issues arise and I will describe them as presented in the title.

The data I am using is the following:

clear
input str14(exporter importer) double contiguity float(dist_1 dist_2) double value
"japan" "japan"       0 0 0  1.872360972809597
"japan" "uk"      0 0 1  72.76820000000001
"japan" "usa"        0 0 0 157.77328270894705
"japan" "france" 0 1 0                  0
"japan" "brasil"  0 0 1  6.602928528463685
end

1) Because of heteroskedasticity associated with trade data, EK employed a GLS estimator. I understand that under small number of observations, GLS is different to OLS beyond Standard Errors (SE). EK assume an orthogonal error that consists of two components: country-pair specific that affects two-way trade, and another one that affects one-way trade. Each component has its own variance. EK further discuss the the implication of such assumptions for the variance-covariance matrix.

I would like to have a clarification to what this means. The model I have ran is the following. While this is, perhaps, GLS simplest form I do not completely understand what EK mean by their assumption about the structure of the errors described above. How would the assumptions translate into code?

reg value contiguity dist_* i.exporterID i.importerID
predict e, residual
gen e_sq = e^2
reg e_sq contiguity dist_* i.exporterID i.importerID

2) I post this question at the end since I expect fewer people will know about this topic. To estimate equation (30) in page 1761, I know it is used importer and exporter Fixed Effects such as the model estimated in the code above. The difference is that the estimation is more convoluted. My actual estimation for equation (30) without GLS estimator and using linear restrictions is shown below. Notice that constant is dropped to be able to estimate all Fixed Effects and the command collinear is enforced.

constraint 1 EXPO_FE1 + EXPO_FE2 + EXPO_FE3 + EXPO_FE4 + EXPO_FE5 + EXPO_FE6 + EXPO_FE7 + EXPO_FE8 + EXPO_FE9 + EXPO_FE10 + EXPO_FE11 + EXPO_FE12 + EXPO_FE13 + EXPO_FE14 + EXPO_FE15 + EXPO_FE16 + EXPO_FE17 = -EXPO_FE18

constraint 2 IMPO_FE1 + IMPO_FE2 + IMPO_FE3 + IMPO_FE4 + IMPO_FE5 + IMPO_FE6 + IMPO_FE7 + IMPO_FE8 + IMPO_FE9 + IMPO_FE10 + IMPO_FE11 + IMPO_FE12 + IMPO_FE13 + IMPO_FE14 + IMPO_FE15 + IMPO_FE16 + IMPO_FE17  = - IMPO_FE18

cnsreg dep_value contiguity dist_* EXPO_FE* IMPO_FE* , constraints(1-2) nocons collinear

Equation (30) shows the S's for exporter which were previously defined as "source-country dummies". Later, the importer fixed effects is included as "an overall destination effect". Despite this distiction, EK kept the S's for importer in Equation (30) despite that it is not estimated. Why is that?