Perfect multicollinearity when estimating a gravity model everyone!

I am estimating a gravity model in order to analyze the impacts of tighter environmental regulations on international trade. More specifically, I am analyzing Brazil's trade flow.

My (linearized) model is as follows:

$ \ln(EXP) = \ln(GDP_O) +  \ln(GDP_D) +  \ln(POP_O) + \ln(POP_D) + \dots $

In other words, the exports from Brazil (origin) to a country D (destination) depend on both countries GDPs,Populations, Total Area and  another couple of variables.

My problem is: because Brazil is the origin for the exports to all other countries, GDP_O, POP_O and any other variables representing Brazilian data will be equal for all observations and as such there will be perfect multicollinearity.

How do I circumvent this? Every gravity model uses both the exporter and importer variables in order to explain the bilateral trade flow.

EDIT: See for instance the estimation carried out here:  https://drive.google.com/file/d/1DczfoFI_mkI8Tm4H0WcaVZTz9ZXLGb2K/view?usp=drivesdk

EDIT2: Specifically, here:

[Regression table] (https://i.stack.imgur.com/TQDA3.png)

I appreciate any help!

Kind regards,
Pedro!
My problem is: because Brazil is the origin for the exports to all other countries, GDP_O, POP_O and any other variables representing Brazilian data will be equal for all observations and as such there will be perfect multicollinearity.


This isn't perfect multicollinearity. It's a lack of time/space variation.  Multicollinearity occurs between features, not observations.  You would of course not have space variation in any given year, for the reason you note - but in a comment you said you ran a panel regression, so you should have that time variation.


  I want to regress exports as a function of, among other things, Brazil's GDP. As such, I want to see the effect of GDP on Brazil's exports and not the effect of GDP on exports to each country 


Then, why are you using a gravity model for trade flows? Doing so spreads the effect size across a bunch of variables you seem to be uninterested in.  Total exports is just a number that relates to GDP through the accounting identity.  Since that identity is additive, a log-log form for your regression does not make sense.