Interpretation of coefficients in gravity model using PPML (dummies)

[Julia Veje]

Hello,

I was wondering if anybody could help me how to interpret the coefficients of my gravity equation. I have used the ppml for estimation with time and country fixed effects and specifically i am trying to estimate the effect of the single market. MY model:
ppml rtrade log_distance loggdp_imp loggdp_eks colony comlang_off contig EEA aar_* ex_dum_* , cluster(dist)

aar_* is time fixed effects ex_dum_* is country fixed effects.
rtrade is the trade volume in mio. US$ in levels!
log_distance, loggdp_eks loggdp_imp are distance, GDP for exporter and GDP for importer country, all three are ing log.
EEA is my FTA dummy, with the value 1 if both countries are a member for the single market.
colony, comlang_off and contig are dummies (former colony, common official language and common border)

result after running my model:

rtrade = -0,38*log_distance + 0,85*loggdp_imp + 0,38*loggdp_eks +1,14*colony - 0,40*comlang_off - 0,5*contig + 0,6*EEA

So i know that interpreting a dummy you need to say: (e^beta-1)*100 = xx % meaning fx. beta for EEA variable is 0,6, so (e^0,6-1)*100 = 82 %, so being in the EEA increases trade by 82 %.
BUT my question is what it the coefficient is negative!? fx. comlang_off variable the coefficient is -0,4, implying a negative effect, BUT interpretation: (e^-0,4-1)*100 = 25 %, so is the effect positive or negative??

I'm so lost and hope somebody can help as i am defending my thesis and have become totally confused...

Best Julia

[Joao Santos Silva]

Dear Julia Veje,

If I am not mistaken, (e^-0,4-1)*100 = -0.33%, so the effect is negative. More generally, the coefficient and the effect always have the same sign because e^0-1=0.

Best wishes,

Joao

[Ainhoa Oses]

Dear Joao,

I have a similar doubt as above. I have a gravity model on bilateral migration flows, though more general on interpretation of the coefficients (not just on dummies). I estimate my equation through -xtpqml- as I include country-pair fixed effects. The code is as below, i.e. migration flows depend on the lagged-log of migration stocks in the previous period, and a number of other variables that are logarithmic and lagged to the previous period.

xtpqml migrafl llmigra llmigrasq llgdpc_o llgdpc_d llold_pop_o llold_pop_d llwork_pop_o llwork_pop_d llyoung_pop_o llyoung_pop_d  if migra>migra[_n-1],fe i(countrypair)  

I'm not sure how to interpret the coefficients. For example, when focusing on GDP per capita at origin (llgdpc_o, which is in log terms and lagged to the previous period), I get the expected sign. Is the interpretation exactly the same as in OLS? From your paper Log of Gravity, I understand this is the case. But I've seen in some posts that the exponent of the coefficient is taken for interpretation (for example, using the exponent here would imply a positive effect of GDP of origin, which is not in line with what I expect; I'm also unsure of your calculation above in #2 when calculating the e^-0,4-1, there I get a positive value as well).

migrafl = 0.61*llmigra - 0.04* llmigrasquare - 0.26*llgdpc_o + 0.44*llgdpc_d + 0.19*llold_pop_o - 0.13*llold_pop_d + 0.49*llwork_pop_o +1.81*llwork_pop_d + 0.43*llyoung_pop_o - 0.96*llyoung_pop_d

Many thanks for your excellent assistance and key contributions in this forum.

[Joao Santos Silva]

Dear OA Stata,

The coefficients on logged regressors are elasticities and there is no need to transform those. For regressors not in logs, the semi-elasticity is given by 100*(exp(beta) - 1)%. This is negative for negative beta and positive for positive beta; is is also approximately equal to 100*(beta)% for beta close to zero.

About #2, note that it should be (e^(-0.4)-1)*100 = -0.33%.

Best wishes,

Joao

[Ainhoa Oses]

Dear Joao,

That's perfect and really clear. Thanks a million for your help, I really appreciate it as it's being key for my work.

Best wishes,

Ainhoa

[Maryam Zia]

Dear Professor Joao,

I wanted to ask how do I interpret my PPML coefficient if my regressor is an index, taking values between 0 and 1.

Regression is Investment = Beta(GovernanceScore)

For example if my coefficient is 1.78(GovernanceScore), I calculate coefficient as e^(1.78-1) and I am confused how do I interpret it . Do I say if the score goes up by 1standard deviation, the investment goes up by 4.68 millions.

Best,
Maryam Zia

[Joao Santos Silva]

Dear Maryam Zia,

I think it is better if you use your index in the scale of 0 to 100. In that case the coefficient is 0.0178 and you can say that if the index increases by one point, investment increases by 1.8%.

Does it make sense?

Best wishes,

Joao

[Maryam Zia]

Dear Professor Joao,

Thank You. I wanted to ask if I keep my index from 0 to 1, then could I say if index goes up by 0.01 point, the investment will go up by 1.8% . Similarly can I say that the investment will be exp(1.78)= 5.92 times higher in countries with highest possible score(1) than in countries with lowest possible score (0).

Are these two explanations correct ?

Best,
Maryam

[Maryam Zia]

Sorry I think it would be if the index goes up by 1 percentage point, the investment will go up by 1.8% . Please let me know if my explanations are correct.


Thank You

[Joao Santos Silva]

I still think it is clearer to work in the 0 to 100 scale, but it is up to you. The first part of what you say sounds correct, but if the index goes from 0 to 1, I would say that the expected investment goes up by 100(exp(1-78) - 1)% = 493%.

Joao

[Maryam Zia]

Thank You. Could you please explain how can I get (exp(beta)-1)100 as my slope. I tried taking derivatives but couldn't end up with the latter expression. I am trying to understand the logic behind it.

Best,
Maryam

[Joao Santos Silva]

To see the effect of increasing x by 1 unit, do:

[exp(a+(x+1)b) - exp(a+b) ]/exp(a+xb) = exp(b) - 1

Best wishes,

Joao
PS: Apologies for the typo in #10; it should read: 100(exp(1.78) - 1)% = 493%

[Monicucha Paganini]

Dear Prof Silva,

may I ask you how would you interpret the constant in the gravity model?
what is the difference between the gravity and glm.
I am assessing a trade model and the impact of FDI restrictiveness.

thanks

[Joao Santos Silva]

Dear Monicucha Paganini,

The constant is generally the base level for the dummy variables.

the gravity equation for trade is a theoretical model describing trade flows; glm are generalized linear models. The gravity equation can be estimated using Poisson regression (PPML), which is a member of the glm family.

Best wishes,

Joao

[muhammad akhtaruzzaman]

Hello All,
I am learning about the gravity model and how the data structure look for a gravity model analysis. My question is:
Can I construct a dataset where ONE source country against all partners or destination countries? For example, I am interested in identifying the determinants of outward FDI of New Zealand. In this case, ONE source country (NZ) against all partner countries or destination countries to examine the determinants of NZ's outward FDI using Gravity model for say, 10-15 years data? Or, I need to paring each country at a time as source country?
Look forward to hearing from you.
Regards
Muhammad