Why do country-state pair fixed effects within gravity framework drastically change coefficients and significance?

[Michael Pont]

I am investigating the effects of migration on trade. My dataset contains approximately 130 countries and the 50 US states from 2013-2017, imports and exports. I am using the standard gravity framework with fixed effects. I want to show the development of varying fixed effects as seen in “Gravity for Dummies and Dummies for Gravity Equations” by Baldwin and Taglioni(2006). I have run multiple regressions and was curious about the reasoning behind the results. IVS and DVs are logged so coefficients are interpreted as elasticities. Standard errors are clustered by country-state pairs.

Stata command (with country, state, and year fixed effects) is the following:

reghdfe log_trade log_gdp log_gsp log_state_population log_country_population log_distance contiguity log_migration_stock absorb(i.year i.country i.state) vce(cluster i.country_state)

Stata command (with country-state pair fixed effects) is the following:

reghdfe log_trade log_gdp log_gsp log_state_population log_country_population log_migration_stock absorb(i.year i.country_state) vce(cluster i.country_state)

When I include country, state, and year fixed effects the coefficient on migration stock is 0.10*** and 0.19*** for exports and imports respectively. This falls in line with the literature.
However, when I include country-state pair fixed effects and year fixed effects the coefficient on migration stock becomes -.02* and .01 for exports and imports respectively (distance and contiguity dropped since time-invariant).

My main question is why the large difference in elasticities and significance?

From what I have gathered, fixed effects will subsume all unobserved time-invariant characteristics between the country and state in this case. I have also read from http://www.princeton.edu/~otorres/Panel101.pdf: “fixed-effects will not work well with data for which within-cluster variation is minimal or for slow changing variables over time”. The literature also suggests country pair fixed effects for panel data with gravity equations. This led me to examine the variation of the migration stock across the 5 years among country-state pairs. I calculated the coefficient of variation for each country-state pair and then took the average for the whole column. This was about 0.49 (49%) for imports and exports and was much higher than all other country and state variables (e.g. gdp, gsp, population) Trade was around 0.45 to 0.53.

Could it just be that there is no effect of migration? I am not trying to chase significance but I would like to know if country-state pairs is the way to go.
I am not really sure how to interpret all this. Would really appreciate the input of the community.
Thanks.

[Carlo Lazzaro]

MIchael:
welcome to this forum.
You may want to take a look at both http://personal.lse.ac.uk/tenreyro/lgw.html and Joao Santos Silva's posts on this forum related to gravity model.

[Phil Bromiley]

To add to Carlo's helpful response, when you change the basis of the fixed effects you're running a very different model. You're controlling for different factors. You're allowing different intercepts for each panel. This can change the results dramatically.

I'd wonder about how you code the state variable. I would normally think this would look like your country-state effect. It wouldn't make sense to have a panel that is a state that is not separate for each state in each country. You don't have states that span countries. Generally, if you include state, you won't need to include country - state stays constant within a country and so takes care of the country effect. So, when you get a big difference, I suspect you've done something odd with the coding of the state variable.

[Tom Zylkin]

Hi Michael,
To add to Carlo and Phil's replies (which already make some very good points), there are some standard references by Glick and Rose (2002) and Baier and Bergstrand (2007) that go into detail about the role played by pair fixed effects in this context.

I do share some of Phil's confusion about the meaning of "country-state" fixed effects. I assume what you mean is that you have two specifications: one specification where US states are the exporters and non-US countries are the importers and one specification where it is the opposite. If I have that wrong, then the references I just gave above may not apply.

Btw, you may also want to include exporter-year and and importer-year fixed effects to be consistent with Baldwin and Taglioni, 2006. It is tough to interpret your results without these fixed effects since we know there are important biases that these fixed effects help us control for.

Regards,
Tom