How to interpret the marginal effect of a dummy regressors in a logit model

Question

I have a problem interpreting the marginal effect of a dummy variable in a logit model. (I am using Stata to estimate the logit regression)

I've run a simple logit say this:

logit w i.x i.y i.z 

where all variables are dummies. Then to compute the margins I ran this:

 margins, dydx(_all) at(x=1 y=1)

I wanted to know the margins where x=1 and y=1.

How should I interpret the margin for x or y?

Answer 1

Stata is smart enough to ignore the at() assignment for x when you calculate the AME for x (since otherwise you would get a zero). In the end, you have asked Stata to calculate this average of finite differences:

$$AME_x =\sum_{i=1}^N \left[ \hat p(x=1,y=1,z=z_i)-\hat p(x=0,y=1,z=z_i) \right],$$

where $\hat p(.)$ is the predicted probability from the logit model. Stata used differences here rather than derivatives since all your regressors are binary/categorical.

This is probably not a very sensible AME, but perhaps you have your reasons for doing it this way. I am calling this an AME, but it is actually a hybrid of AME and MER (marginal effect at representative values).

Here's a toy example showing the margins calculation by hand:

. sysuse auto, clear
(1978 Automobile Data)

. gen high_mpg = mpg>20

. gen high_rep = rep78>3

. gen heavy    = weight>3000

. 
. /* AME usig margins */
. logit foreign i.(high_mpg heavy high_rep), nolog

Logistic regression                             Number of obs     =         74
                                                LR chi2(3)        =      37.57
                                                Prob > chi2       =     0.0000
Log likelihood = -26.246142                     Pseudo R2         =     0.4172

------------------------------------------------------------------------------
     foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  1.high_mpg |  -1.118024   1.307539    -0.86   0.393    -3.680754    1.444706
     1.heavy |  -3.673601   1.417986    -2.59   0.010    -6.452802   -.8944001
  1.high_rep |   2.245017   .7705583     2.91   0.004     .7347502    3.755283
       _cons |  -.2405401   1.332215    -0.18   0.857    -2.851634    2.370554
------------------------------------------------------------------------------

. margins, dydx(high_mpg) at(high_mpg = 1 heavy = 1)

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

Expression   : Pr(foreign), predict()
dy/dx w.r.t. : 1.high_mpg
at           : high_mpg        =           1
               heavy           =           1

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  1.high_mpg |   -.053257   .0519245    -1.03   0.305     -.155027    .0485131
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

. 
. /* Calculate the same average marginal effect in-sample for high_mpg as above */
. /* (a) ME = phat(high_mpg=1, heavy=1, high_rep at own value) */
. /*        - phat(high_mpg=0, heavy=1, high_rep at own value) */
. gen double high_mpg_me =                                                                                                                        
>                    ///
>                                           [   exp(_b[_cons]+_b[1.high_mpg]+_b[1.heavy]+_b[1.high_rep]*high_rep)/   ///
>                                (1+exp(_b[_cons]+_b[1.high_mpg]+_b[1.heavy]+_b[1.high_rep]*high_rep))]  ///
>                                          -[   exp(_b[_cons]               +_b[1.heavy]+_b[1.high_rep]*high_rep)/   ///
>                                (1+exp(_b[_cons]               +_b[1.heavy]+_b[1.high_rep]*high_rep))]

. 
. /* (b) Calculate the average marginal effect (AME) */
. sum high_mpg_me, meanonly

. display "High MPG AME = " %9.6f r(mean)
High MPG AME = -0.053257

According to this model, when all cars are assumed to be heavy, but have their actual in-sample values of high repair record as they are observed. the probability of the car being foreign falls by 5.3 percentage points when it is high MPG (relative to low MPG).

---

Stata Code:

cls
sysuse auto, clear
gen high_mpg = mpg>20
gen high_rep = rep78>3
gen heavy    = weight>3000

/* AME usig margins */
logit foreign i.(high_mpg heavy high_rep), nolog
margins, dydx(high_mpg) at(high_mpg = 1 heavy = 1)

/* Calculate the same average marginal effect in-sample for high_mpg as above */
/* (a) ME = phat(high_mpg=1, heavy=1, high_rep at own value) */
/*        - phat(high_mpg=0, heavy=1, high_rep at own value) */
gen double high_mpg_me =                                                                       ///
                      [   exp(_b[_cons]+_b[1.high_mpg]+_b[1.heavy]+_b[1.high_rep]*high_rep)/   ///
                       (1+exp(_b[_cons]+_b[1.high_mpg]+_b[1.heavy]+_b[1.high_rep]*high_rep))]  ///
                     -[   exp(_b[_cons]               +_b[1.heavy]+_b[1.high_rep]*high_rep)/   ///
                       (1+exp(_b[_cons]               +_b[1.heavy]+_b[1.high_rep]*high_rep))]

/* (b) Calculate the average marginal effect (AME) */
sum high_mpg_me, meanonly
di "High MPG AME = " %9.6f r(mean)