Elasticity (margins: eyex) with Generalized Linear Model?

[Jose Alcides Santos]

Dear,

I am estimating temporal changes between cohorts in the association (in terms of elasticity) between the occupational status of the parents as measured by the ISEI and the income of the children. I'm using a Generalized Linear Model (GLM) model because it allows using zero income cases.
According to a post of Nick Cox: “Alternatively -glm- with log link gives all the advantages of log transformation without any of the difficulties of dealing with zeros. The assumption is that means are always positive, not the raw data”.
I have used the margins command with the eyex option for elasticity.
[CODE]

svy, subpop(id66): glm income  c.isei##i.cohort  i.state [where he lived at 15 years]  i.sex  i.collor, family(poisson) link(log)

margins cohort, eyex(isei)

This model with the logarithm link takes the exponential of the predictor instead of making the logarithmic transformation of the dependent variable (see comment in Hardin and Hilbe's book on the model algorithm).
The results seem to make sense. Measures of elasticity usually range between zero and one.
[CODE]

  ey/ex w.r.t. : isei
Expression   : Predicted mean income, predict()
Model VCE    : Linearized
Average marginal effects                          Number of obs   =      25772  ------------------------------------------------------------------------------
             |            Delta-method
             |      ey/ex   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      isei   |
     cohort  |
         48  |   .8300628    .052158    15.91   0.000     .7278351    .9322906
         58  |   .8401456   .0765742    10.97   0.000     .6900629    .9902282
         68  |   .8403638   .0455364    18.45   0.000     .7511141    .9296136
         78  |   .6801567   .0286101    23.77   0.000      .624082    .7362315
------------------------------------------------------------------------------

However, I am asking whether some distortion could be introduced due to the fact that the original model takes the exponential of the independent variable.

[Clyde Schechter]

No, you are fine. The -margins- command you wrote is using -predict()- for the "y" in "eyex" and -predict- after -glm- calculates mu, the expected value of income. So you will be getting the rate of relative change in income per unit of relative change in isei --which is exactly the elasticity of income with respect to isei.

[Jose Alcides Santos]

Dear Clyde Schechter
Thanks for the clarification.
I had some doubt. Your comment clarifies the argument of James Hardin & Joseph Hilbe: “The log link exponentiates the linear predictor, or x β, rather than log-transforming the response to linearize the relationship between the response and predictors. This procedure, implicit within the algorithm, allows easy interpretation of estimates and of fitted values”.
As you put: “-predict- after -glm- calculates mu, the expected value of income”.