Perfect multicollinearity when estimating a gravity model

Question

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.sstatic.net/TQDA3.png)

I appreciate any help!

Kind regards, Pedro!

Answer 1

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.

Answer 2

They will simply be sucked up into one variable. To "fix" this, you could simply drop all of Brazil's characteristics and leave a constant in: $\ln(EXP_D) = c + \beta_1 \ln(GDP_D) + \beta_2 \ln(POP_D) + \ldots$

Edit

Just understood the question from the comments below.

If you have only a cross-sectional data, you cannot distinguish the effects of GDP, population, etc. of the origin country (Brazil) on its exports.

One way you could do this is to run a panel regression... then you'll have variation in GDP, population, exports, etc.

In order to do this, you need to assume that the relationship remains the same throughout your sample period. If, for example, you have a change in government, then you might need to control for that, etc.