How to calculate marginal effects in McFadden's choice model?

Question

I am estimating a multinomial choice model in stata with the command asclogit. One of the case specific variable is a dummy variable. I want to get the marginal effects of the dummy variable on choice probabilities. I have two ways of getting this. The first method is to use command

estat mfx, varlist(policy)

provided by stata's asclogit model. Another method is I predict the choice probabilities under two scenarios. One is with the dummy variable value being set to 0. Another is with the dummy variable set to 1. I compute the change in choice probabilies for all alternatives.

The problem is that the results given by two methods are not the same. I am not sure how to interpret this.

Answer 1

By default, estat mfx calculates the marginal effects at the mean values of the covariates (MEM). This special MEM uses an overall average for each case-specific covariate, and the choice-specific averages for the covariates that characterize the alternatives. The manual and help-files could be a lot clearer on this point.

Your finite difference of predictions way sounds like an average marginal effect (AME), and not a MEM. Since the expected value function is non-linear, the AME and the MEM will usually be fairly different.

You can use the code below to illustrate my claim and replicate what estat mfx is doing under the hood:

cls
webuse choice, clear
qui asclogit choice dealer, case(id) alternatives(car) casevars(income i.sex)

/* discrete MEMs */
estat mfx, varlist(sex) k(1)
table car, c(mean dealer mean income) row format(%16.0g)
estat mfx, varlist(sex) k(1) at(income = 42.096611022949 American:dealer = 18.976270675659 Japan:dealer = 7.5423727035522 Europe:dealer =  3.4610168933868)

/* By Hand */    
bys car: egen double mean_dealer = mean(dealer)
replace income = 42.096611022949
replace dealer = 18.976270675659 if car=="American":nation 
replace dealer = 7.5423727035522 if car=="Japan":nation 
replace dealer = 3.4610168933868 if car=="Europe":nation 
replace sex = 0
predict double p0, k(1)
replace sex = 1
predict double p1, k(1)
gen double diff = p1 - p0
table car, c(mean diff) format(%9.6f)

Here's the output:

. webuse choice, clear

. qui asclogit choice dealer, case(id) alternatives(car) casevars(income i.sex)

. 
. /* discrete MEMs */
. estat mfx, varlist(sex) k(1)

Pr(choice = American|1 selected) = .62729282
-------------------------------------------------------------------------------
variable     |   dp/dx   Std. Err.    z     P>|z|  [    95% C.I.    ]       X
-------------+-----------------------------------------------------------------
casevars     |                                                                 
       1.sex |  .040054    .06418    0.62   0.533  -.085735   .165844         0
-------------------------------------------------------------------------------
dp/dx for factor levels is the discrete change from the base level

Pr(choice = Japan|1 selected) =  .2929863
-------------------------------------------------------------------------------
variable     |   dp/dx   Std. Err.    z     P>|z|  [    95% C.I.    ]       X
-------------+-----------------------------------------------------------------
casevars     |                                                                 
       1.sex | -.110364   .058198   -1.90   0.058   -.22443   .003702         0
-------------------------------------------------------------------------------
dp/dx for factor levels is the discrete change from the base level

Pr(choice = Europe|1 selected) = .07972088
-------------------------------------------------------------------------------
variable     |   dp/dx   Std. Err.    z     P>|z|  [    95% C.I.    ]       X
-------------+-----------------------------------------------------------------
casevars     |                                                                 
       1.sex |   .07031   .038236    1.84   0.066  -.004632   .145252         0
-------------------------------------------------------------------------------
dp/dx for factor levels is the discrete change from the base level

    variable       base 
    sex            0

. table car, c(mean dealer mean income) row format(%16.0g)

----------------------------------------------
nationali |
ty of car |     mean(dealer)      mean(income)
----------+-----------------------------------
 American |  18.976270675659   42.096611022949
    Japan |  7.5423727035522   42.096611022949
   Europe |  3.4610168933868   42.096611022949
          | 
    Total |  9.9932203292847   42.096611022949
----------------------------------------------

. estat mfx, varlist(sex) k(1) at(income = 42.096611022949 American:dealer = 18.976270675659 Japan:dealer = 7.542372703552
> 2 Europe:dealer =  3.4610168933868)

Pr(choice = American|1 selected) = .62729282
-------------------------------------------------------------------------------
variable     |   dp/dx   Std. Err.    z     P>|z|  [    95% C.I.    ]       X
-------------+-----------------------------------------------------------------
casevars     |                                                                 
       1.sex |  .040054    .06418    0.62   0.533  -.085735   .165844         0
-------------------------------------------------------------------------------
dp/dx for factor levels is the discrete change from the base level

Pr(choice = Japan|1 selected) =  .2929863
-------------------------------------------------------------------------------
variable     |   dp/dx   Std. Err.    z     P>|z|  [    95% C.I.    ]       X
-------------+-----------------------------------------------------------------
casevars     |                                                                 
       1.sex | -.110364   .058198   -1.90   0.058   -.22443   .003702         0
-------------------------------------------------------------------------------
dp/dx for factor levels is the discrete change from the base level

Pr(choice = Europe|1 selected) = .07972088
-------------------------------------------------------------------------------
variable     |   dp/dx   Std. Err.    z     P>|z|  [    95% C.I.    ]       X
-------------+-----------------------------------------------------------------
casevars     |                                                                 
       1.sex |   .07031   .038236    1.84   0.066  -.004632   .145252         0
-------------------------------------------------------------------------------
dp/dx for factor levels is the discrete change from the base level

    variable       base 
    sex            0

. 
. /* By Hand */    
. bys car: egen double mean_dealer = mean(dealer)

. replace income = 42.096611022949
(885 real changes made)

. replace dealer = 18.976270675659 if car=="American":nation 
variable dealer was byte now float
(295 real changes made)

. replace dealer = 7.5423727035522 if car=="Japan":nation 
(295 real changes made)

. replace dealer = 3.4610168933868 if car=="Europe":nation 
(295 real changes made)

. replace sex = 0
(648 real changes made)

. predict double p0, k(1)
(option pr assumed; Pr(car))

. replace sex = 1
(885 real changes made)

. predict double p1, k(1)
(option pr assumed; Pr(car))

. gen double diff = p1 - p0

. table car, c(mean diff) format(%9.6f)

----------------------
nationali |
ty of car | mean(diff)
----------+-----------
 American |   0.040054
    Japan |  -0.110364
   Europe |   0.070310
----------------------