Dear Lee Squirrell

I totally agree to your concern due to the problem of panel data, I guess that your independent variables should be following the traditional gravity variables such as cultural distance factors, parent-country per capita GDP, relative labor endowments, and regional trade agreements...in which honorly I suggest that you should check the endonenous problem of panel data and you can use GMM or GMM estimator or LSDV of IV estimator to solve this problem. I attach some papers below in the case of your consideration. If you have problem with the GMM estimators, I may help in some case.

Please don't hesitate to contact me

Cheers

when you have different countries ,I think you have to try with Choosing Between Fixed and Random Effects: Here is a STATA guide to apply them: (enjoy ;) ) 

Fixed Effects Regression

Fixed effects regression is the model to use when you want to control for omitted variables that differ between cases but are constant over time. It lets you use the changes in the variables over time to estimate the effects of the independent variables on your dependent variable, and is the main technique used for analysis of panel data.

The command for a linear regression on panel data with fixed effects in Stata is xtreg with the fe option, used like this:

xtreg dependentvar independentvar1 independentvar2 independentvar3 ... , fe

If you prefer to use the menus, the command is under Statistics > Cross-sectional time series > Linear models > Linear regression.

This is equivalent to generating dummy variables for each of your cases and including them in a standard linear regression to control for these fixed "case effects". It works best when you have relatively fewer cases and more time periods, as each dummy variable removes one degree of freedom from your model.

Between Effects

Regression with between effects is the model to use when you want to control for omitted variables that change over time but are constant between cases. It allows you to use the variation between cases to estimate the effect of the omitted independent variables on your dependent variable.

The command for a linear regression on panel data with between effects in Stata is xtreg with the be option.

Running xtreg with between effects is equivalent to taking the mean of each variable for each case across time and running a regression on the collapsed dataset of means. As this results in loss of information, between effects are not used much in practice. Researchers who want to look at time effects without considering panel effects generally will use a set of time dummy variables, which is the same as running time fixed effects.

The between effects estimator is mostly important because it is used to produce the random effects estimator.

Random Effects

If you have reason to believe that some omitted variables may be constant over time but vary between cases, and others may be fixed between cases but vary over time, then you can include both types by using random effects. Stata's random-effects estimator is a weighted average of fixed and between effects.

The command for a linear regression on panel data with random effects in Stata is xtreg with the re option.

Choosing Between Fixed and Random Effects

The generally accepted way of choosing between fixed and random effects is running a Hausman test.

Statistically, fixed effects are always a reasonable thing to do with panel data (they always give consistent results) but they may not be the most efficient model to run. Random effects will give you better P-values as they are a more efficient estimator, so you should run random effects if it is statistcally justifiable to do so.

The Hausman test checks a more efficient model against a less efficient but consistent model to make sure that the more efficient model also gives consistent results.

To run a Hausman test comparing fixed with random effects in Stata, you need to first estimate the fixed effects model, save the coefficients so that you can compare them with the results of the next model, estimate the random effects model, and then do the comparison.

. xtreg dependentvar independentvar1 independentvar2 independentvar3 ... , fe

. estimates store fixed

. xtreg dependentvar independentvar1 independentvar2 independentvar3 ... , re

. estimates store random

. hausman fixed random

The hausman test tests the null hypothesis that the coefficients estimated by the efficient random effects estimator are the same as the ones estimated by the consistent fixed effects estimator. If they are (insignificant P-value, Prob>chi2 larger than .05) then it is safe to use random effects. If you get a significant P-value, however, you should use fixed effects.

In the panel data where the value of dependent variable repeats for different cross sections, would it affect or cause any technical difficulties? 

Hi, 

when you deal with this set of data, I am sure variables do show cointegration, thus i suggest looking for fully modified OLS, and error correction approaches.

Wishes

chandana