Interpret semi-elasticity with eydx option, Stata

[Nabila Bouraima]

Sorry for posting this again. I have initially posted it in the wrong forum. Below is my request.
After a probit regression, I estimated semi-elasticities using eydx option with margins. I get a value of -0.03. My outcome variable (Y) is dummy (0, 1) and the independent variable (X) is log-transformed. How can I interpret the semi-elasticity?

Is it correct if I say a 1 % increase in X leads to a 3% decrease in Y, giving that exp(0.03)=1,030?
Or should I say a 1% increase in X leads to 0.03 % decrease in Y?

[Dimitriy V. Masterov]

The definition of an eydx semielasticity S is that a 100*S % change in the outcome is associated with a one-unit increase in the independent variable.

This means the interpretation here is

-0.03*100 = -3% decrease in Pr(y=1) for a 1 unit change in LOG X, which is an e = 2.7182818 factor increase in X.

So, when X almost triples, the probability of success declines slightly, on average. Note that this is in percent, not percentage points.

I find percent change in probabilities difficult to understand and explain to others. I recommend that (1) you don't log X and (2) calculate a dyex semielasticity instead of eydx. Divided by 100, this gives you the additive change in Pr(y=1) on a [0,1] scale associated with a +1% increase in x.

For example,

. sysuse auto, clear
(1978 automobile data)

. probit foreign c.price, nolog

Probit regression                                       Number of obs =     74
                                                        LR chi2(1)    =   0.18
                                                        Prob > chi2   = 0.6727
Log likelihood = -44.943972                             Pseudo R2     = 0.0020

------------------------------------------------------------------------------
     foreign | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       price |   .0000222   .0000522     0.42   0.671    -.0000802    .0001245
       _cons |  -.6701415   .3605534    -1.86   0.063    -1.376813    .0365303
------------------------------------------------------------------------------

. margins, dyex(price)

Average marginal effects                                    Number of obs = 74
Model VCE: OIM

Expression: Pr(foreign), predict()
dy/ex wrt:  price

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/ex   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       price |    .047956   .1139996     0.42   0.674    -.1754792    .2713912
------------------------------------------------------------------------------

This says that a 1% increase in price is associated with .047956/100 = .00047956 increase in Pr(foreign) on a [0,1] scale.

The baseline is that ~1/3 of cars are foreign, so that is roughly a 0.16% increase (i.e., considerably less than 1%). Margins, eyex() gives you something very close to that:

. margins, eyex(price)

Average marginal effects                                    Number of obs = 74
Model VCE: OIM

Expression: Pr(foreign), predict()
ey/ex wrt:  price

------------------------------------------------------------------------------
             |            Delta-method
             |      ey/ex   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       price |   .1563539   .3619077     0.43   0.666    -.5529722      .86568
------------------------------------------------------------------------------

[Nabila Bouraima]

Many thanks for your answer. This is well-detailed. One last question regarding the last part of your answer. Does it mean that I could use eyex instead of dyex. My understanding was that eyex is used when both dependent and independent variables are continuous.

[Dimitriy V. Masterov]

I think you should use

margins, dyex(log_x)

The expected value here is a probability, which is continuous (though bounded by zero and one), so you should be good.

I added the eyex elasticity example at the end to elucidate what dyex is calculating.

[Nabila Bouraima]

Well noted. Thank a lot.