Binary Response Model - Margins command & elasticity (dydx, eydx)

[Robert Niewoehner]

Hello everyone,

Hope you all are safe and well during these turbulent times. I have a quick question. I'm trying to make heads or tails of the output from the margins command.

Here are two other links that I found helpful:
1. https://stats.stackexchange.com/ques.../243463#243463
2. https://www.statalist.org/forums/for...centage-points

Here is my output from a binary response model (dependent variable takes values 0 or 1), thus the marginal effect is in a percentage point change. My explanatory variable is logged, hence leaving the 'dx' alone in the calculation.

Question: How do I reconcile the two effects? It doesn't makes sense to call for an elasticity (eydx) when y is already in proportional change format, but I'm not sure how to reconcile. Thanks for any and all help!

. margins, dydx(log_Fam)

Average marginal effects                        Number of obs     =     41,672
Model VCE    : Robust

Expression   : Pr(PickupNewPx), predict()
dy/dx w.r.t. : log_Fam

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     log_Fam |   .0112132   .0021374     5.25   0.000      .007024    .0154024
------------------------------------------------------------------------------

. margins, eydx(log_Fam)

Average marginal effects                        Number of obs     =     41,672
Model VCE    : Robust

Expression   : Pr(PickupNewPx), predict()
ey/dx w.r.t. : log_Fam

------------------------------------------------------------------------------
             |            Delta-method
             |      ey/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     log_Fam |   .0193496   .0037827     5.12   0.000     .0119356    .0267636
------------------------------------------------------------------------------

[Robert Niewoehner]

Also, I should note that the above marginal effects should be multiplied by 100% to get actual percentage point changes in y with respect to changes in x. Any help someone can provide reconciling these two is most helpful! Thank you!

[Robert Niewoehner]

Actually, I did end up finding my own answer to the question by simply working out the math given the definitions of elasticity on page 46 and on page 50 (under "Notation") for the margins help file (https://www.stata.com/manuals14/rmargins.pdf)

Definition:
elasticity and semielasticity. The elasticity of y with respect to x is d(lny)/d(lnx) = (x/y)×(dy/dx), which is approximately equal to the proportional change in y for a proportional change in x. The semielasticity of y with respect to x is either 1) dy/d(lnx) = x×(dy/dx) or 2) d(lny)/dx = (1/y) × (dy/dx), which is approximately 1) the change in y for a proportional change in x or 2) the proportional change in y for a change in x.

In summary, for eydx for a probit model, we are measuring the change in ln(probability) given a change in X, which makes very little sense. Hence, I conclude that if you already have a probability model (probit/logit/etc.), don't use the ey option with margins, instead use dy to get a change in probability.

[Phil Bromiley]

I often find it easier simply to use the margin statement to give predicted probabilities at different values of the X variables. Then you can easily talk about the change predicted probability for change in x.

[Robert Niewoehner]

Phil Bromiley : I agree, this feature is particularly useful, especially if you have interaction terms in the model. Being able to see how a coefficient of interest changes at different interaction levels is very useful. Thanks for responding!