Constant elasticity model - poisson or OLS?

[Ariel Karlinsky]

Suppose you want to estimate the elasticity of y with regards to x, assuming it is constant (for ease of interpretation).
Up until now, I would have logged both variables, and then ran a standard OLS as:

reg ln_y ln_x

However, searching the web and statalist I came upon the highly cited paper "Log of Gravity" (direct here: http://www.mitpressjournals.org/doi/.../rest.88.4.641, further information here: http://personal.lse.ac.uk/tenreyro/LGW.html). which, in a nutshell, proposes to estimate constant elasticity model by:

poisson y ln_x

Though the above metnioned paper is not new and highly cited, I have to admit that was the first time I saw such a suggestion, and I'm wondering how generally acceptable it is.
So my question is simple - how do you estimate a constant elasticity model?

[Marcos Almeida]

I suspect - margins - would be among the most frequently used resources, so handy it is when it comes to estimate elasticities.

Here, some information on the way it is estimated under - margins - command: https://www.stata.com/support/faqs/s...using-margins/

[Carlo Lazzaro]

Ariel:
there is a popular entry in Stata blog on the topic you're interested in (https://blog.stata.com/2011/08/22/us...tell-a-friend/).
Personally, I've no experience with that use of Poisson and, being a pretty old fogey, my preferences go out to the double-logged OLS.

[Said Jafar]

Dear Ariel,

Firstly, i think it would be better if we call the estimator as Poisson Pseudo (Quasi) Maximum Likelihood as it is not really similar to poisson.

The authors provide detailed discussion in the third section of that paper about their choice. I can say this method is fully accepted in international economics literature. Several people brought up baseless claims about the performance of PPML and all have been successfully refuted. Fally (2015) also provide good support for PPML. There is a special page devoted to this paper where you can find more info.



Fally, T. (2015). Structural gravity and fixed effects. Journal of International Economics, 97(1), 76-85

CC:Joao Santos Silva

[Joao Santos Silva]

Dear Ariel,

First of all, thank you for the publicity to "The Log of Gravity" which is now officially a "classic"

Of course you know my answer to your question but, to be fair, there are many cases in which it does not really make a difference how you estimate the model. For example, my experience is that for wage equations the results tend to be very similar.

However, there are cases where it makes a difference (e.g., in estimating intergenerational mobility), and in some cases the difference is huge. As Dias Rafaj noted, PPML is currently the method of choice to estimate gravity equations because that is perhaps the case where the differences are more important. Anyway, Poisson regression is very easy to implement, is very robust, can be used with IV, and does not suffer from the incidental parameters problem. So, there are really not any good reasons not to use it, but I may be biased here.

Best wishes,

Joao

[Ariel Karlinsky]

Thank you Joao (and others) for the response. My particular interest here is in estimating the elasticity of of household expenditures on particular goods (such as Houthakker 1957) using expenditure surveys.

What drove me to search for an alternative to standard OLS of ln(y) on ln(x) is that I have many observations with 0 expenditures (as the products are very detailed, such as "expenditure on pickles"), and I was looking for ways to estimate the elasticity (and came upon the usual ln(y+1), dropping observations with 0's, etc.). I unfortunately have not found papers estimating expenditure elasticities using Poisson (or PPML) so I was wondering whether or not it was common practice, either by the general scientific community or by the economics discipline

[Joao Santos Silva]

Dear Ariel,

If you have zeros in the data, there there is a strong argument for using Poisson regression; as we show in "the Log of Gravity", using ln(y+1) is a really bad solution. It looks as if you will be the pioneer introducing Poisson regression in that context; what a great opportunity :-)

Best wishes,

Joao