Interpreting Semi-Elasticities From "margins"

[Michael Boutros]

I am estimating a multinomial logit model using mlogit. My dependent variable, y, has three outcomes (1, 2, and 3), and my independent variable, x, is a continuous float between 0 and 15. My code:

. quiet mlogit y x
. margins, dyex(x)

Average marginal effects                        Number of obs     =      8,185
Model VCE    : OIM

dy/ex w.r.t. : x
1._predict   : Pr(y==1), predict(pr outcome(1))
2._predict   : Pr(y==2), predict(pr outcome(2))
3._predict   : Pr(y==3), predict(pr outcome(3))

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/ex   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x          |
    _predict |
          1  |  -.0049955   .0030267    -1.65   0.099    -.0109277    .0009367
          2  |  -.0136631   .0032885    -4.15   0.000    -.0201085   -.0072177
          3  |   .0186586   .0035522     5.25   0.000     .0116965    .0256207
------------------------------------------------------------------------------

I read this as:

A 100% increase in "x" increases "Pr(y==1)" by 0.49955 percentage points.

Is this the correct interpretation of the reported semi-elasticities?

Thanks!

[Michael Boutros]

Any takers? I'm almost certain that I'm interpreting the estimates correctly, but wouldn't mind reassurance from more seasoned State users.

[Dimitriy V. Masterov]

I don't think that is quite right. Here's the code example that shows what Stata is doing under the hood with margins, dyex:

. webuse sysdsn1, clear
(Health insurance data)

. qui mlogit insure age i.(male nonwhite site), nolog

. margins, dyex(age) predict(outcome(1)) 

Average marginal effects                        Number of obs     =        615
Model VCE    : OIM

Expression   : Pr(insure==Indemnity), predict(outcome(1))
dy/ex w.r.t. : age

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/ex   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |    .118398   .0622018     1.90   0.057    -.0035153    .2403113
------------------------------------------------------------------------------

. predict double phat0, outcome(1)
(option pr assumed; predicted probability)
(1 missing value generated)

. replace age = age*1.01
(643 real changes made)

. predict double phat1, outcome(1)
(option pr assumed; predicted probability)
(1 missing value generated)

. gen double dp = (phat1 - phat0)*100
(1 missing value generated)

. sum dp if e(sample)

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
          dp |        615     .118397    .0381523   .0428492   .2267397

In other words, a 1% increase in x decreases Pr(y=1) by 100*0.0049955=0.49955 percentage points.

Mathematically, this comes about because the semi-elasticity is (dp/dx)*x = 100*dp/(100*dx/x). In my toy example, 100*(1.01*age - age)/age = 1% change in age, so you also need to multiply the change in probability in the numerator by 100.