How to compute manually margins (eyex and dydx) after a multinomial regression

[Justin Ndoutoumou]

Hello,
Can someone help me with computing the margins eyex and dydx manually, similar to what Stata does with the commands margins, eyex() and margins, dydx()?

I am working for an institution using a platform where Stata is not available, and we do not have the option to install packages in other statistical software's.
Therefore, I need to understand how Stata computes these margins so that I can replicate the process manually in other available applications on the platform.

However, I am not obtaining the same results when I compute them manually compared to when I use the commands margins, eyex() and margins, dydx() on my Stata.

Please find my data and scripts below.

Many thanks for your help!








. ssc install dataex, replace
checking dataex consistency and verifying not already installed...
all files already exist and are up to date.

. dataex

----------------------- copy starting from the next line -----------------------

* Example generated by -dataex-. To install: ssc install dataex
clear
input double(income ipc) str3 prem_emission float(testclass dicho)
              5185 .08737096837606123 "nc"  1 0
              6900  .2751666645772861 "nc"  1 0
              7150 .44855189232301124 "nc"  1 0
              5700  .5503496211136313 "nc"  1 0
              6180  .6150057483911646 "nc"  1 0
              7160  12.38556978124845 "nc"  3 0
              7080 1.3336560869358243 "nc"  3 0
              6650 1.4300558710753521 "nc"  3 0
              7940  4.936040486312466 "nc"  3 0
             10590  2.285560283724503 "nc"  3 0
              4810  .6379019006512485 "nc"  1 0
              5910 .10172782199131047 "nc"  1 0
              8610  4.725078450252779 "nc"  3 0
              6680  .8463431101088507 "nc"  2 0
              6860 1.0357571481480847 "nc"  2 0
              7630  1.854448838311463 "nc"  3 0
              3040  91.72693891176144 "nc"  3 0
              4340  90.03352982721125 "nc"  3 0
           34254.6 .05051347388196038 "nc"  1 0
           39848.4 .15006706827114039 "nc"  1 0
             42502  .8108797955127851 "nc"  2 0
           36233.9  .7376828109330235 "nc"  1 0
             36897  .9346414377752771 "nc"  2 0
           40701.2 16.456397743328935 "nc"  3 0
           34643.8 2.3534043016277897 "nc"  3 0
             41303  .6323109752132469 "nc"  1 0
           45267.6  6.465596085877682 "nc"  3 0
             50249 42.903106538453635 "nc"  3 0
           36119.9 .43771418545513907 "nc"  1 0
           36890.6 .08594027310274638 "nc"  1 0
           47784.2  9.053379009537554 "nc"  3 0
           37337.6  .7511179583907089 "nc"  1 0
           40425.8   .928748844024827 "nc"  2 0
           39739.2 2.3423877640346658 "nc"  3 0
28216.474203154732   89.9736677709193 "nc"  3 0
 28843.50696322484  91.76434419463205 "nc"  3 0
             61600  .2085018513959576 "13"  1 1
             69000  .2014138892114737 "18"  1 1
             74000 1.2771290982254766 "45"  2 1
                 0 1.0009551098376313 "33"  2 1
             64500 1.4021279178918629 "28"  3 1
             66000 10.198002759116184 "149" 3 1
             81000 2.3965958959220846 "23"  3 1
             73000 .30595178199915696 "12"  1 1
             76500  4.094888449590512 "12"  3 1
             96000  37.07914298207258 "3"   3 1
             62800 .17349063150589866 "11"  1 1
             72500  .4684846023188834 "28"  1 1
             85000  9.244558200225383 "35"  3 1
             79000 1.2183171332468152 "34"  2 1
             77500 1.5383212568171136 "50"  3 1
             80000  3.268682223143715 "108" 3 1
             86604 4.6343267360552405 "5"   3 1
            116570 11.645024425579592 "6"   3 1
end

------------------ copy up to and including the previous line ------------------

Listed 54 out of 54 observations

.
.
.
. // Multilogit and margins commands
. mlogit testclass income dicho, base(1) nolog

Multinomial logistic regression Number of obs = 54
LR chi2(4) = 2.58
Prob > chi2 = 0.6304
Log likelihood = -52.547628 Pseudo R2 = 0.0240

------------------------------------------------------------------------------
testclass | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
1 | (base outcome)
-------------+----------------------------------------------------------------
2 |
income | -8.88e-06 .0000214 -0.42 0.678 -.0000507 .000033
dicho | .9468991 1.366389 0.69 0.488 -1.731175 3.624973
_cons | -.8557658 .6622474 -1.29 0.196 -2.153747 .4422152
-------------+----------------------------------------------------------------
3 |
income | .0000173 .0000173 1.00 0.316 -.0000165 .0000512
dicho | -.3893059 1.10068 -0.35 0.724 -2.546598 1.767986
_cons | -.2064702 .5378888 -0.38 0.701 -1.260713 .8477725
------------------------------------------------------------------------------

. margins, dydx(*)

Average marginal effects Number of obs = 54
Model VCE : OIM

dy/dx w.r.t. : income dicho
1._predict : Pr(testclass==1), predict(pr outcome(1))
2._predict : Pr(testclass==2), predict(pr outcome(2))
3._predict : Pr(testclass==3), predict(pr outcome(3))

------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
income |
_predict |
1 | -2.48e-06 3.42e-06 -0.72 0.469 -9.19e-06 4.23e-06
2 | -2.32e-06 2.38e-06 -0.98 0.329 -6.98e-06 2.34e-06
3 | 4.80e-06 3.60e-06 1.33 0.183 -2.26e-06 .0000119
-------------+----------------------------------------------------------------
dicho |
_predict |
1 | .0157943 .2235704 0.07 0.944 -.4223956 .4539842
2 | .1445371 .1537585 0.94 0.347 -.1568241 .4458982
3 | -.1603314 .2379593 -0.67 0.500 -.6267231 .3060602
------------------------------------------------------------------------------

. margins, eyex(*)

Average marginal effects Number of obs = 54
Model VCE : OIM

ey/ex w.r.t. : income dicho
1._predict : Pr(testclass==1), predict(pr outcome(1))
2._predict : Pr(testclass==2), predict(pr outcome(2))
3._predict : Pr(testclass==3), predict(pr outcome(3))

------------------------------------------------------------------------------
| Delta-method
| ey/ex Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
income |
_predict |
1 | -.3409894 .462768 -0.74 0.461 -1.247998 .5660192
2 | -.6924964 .7032758 -0.98 0.325 -2.070892 .6858989
3 | .3455409 .2587482 1.34 0.182 -.1615963 .852678
-------------+----------------------------------------------------------------
dicho |
_predict |
1 | .0194882 .2456872 0.08 0.937 -.4620498 .5010262
2 | .3351212 .3353336 1.00 0.318 -.3221205 .9923629
3 | -.1102804 .1668992 -0.66 0.509 -.4373968 .2168359
------------------------------------------------------------------------------

.
.
. // Computing eyex margins manually
. quietly{

. sum elasticity_*

Variable | Obs Mean Std. Dev. Min Max
-------------+---------------------------------------------------------
elasticit~e1 | 54 0 0 0 0
elasticit~o1 | 54 -5.90e-07 8.46e-07 -2.09e-06 0
elasticit~e2 | 54 0 0 0 0
elasticit~o2 | 54 .0424143 .0634159 0 .2297057
elasticit~e3 | 54 0 0 0 0
-------------+---------------------------------------------------------
elasticit~o3 | 54 -.0673789 .0968364 -.213939 0

.
end of do-file

[Justin Ndoutoumou]

I fixed some errors on the previous post. I apologize.

. ssc install dataex, replace
checking dataex consistency and verifying not already installed...
all files already exist and are up to date.

. dataex

----------------------- copy starting from the next line -----------------------

* Example generated by -dataex-. To install: ssc install dataex
clear
input double(income ipc) str3 prem_emission float(testclass dicho)
              5185 .08737096837606123 "nc"  1 0
              6900  .2751666645772861 "nc"  1 0
              7150 .44855189232301124 "nc"  1 0
              5700  .5503496211136313 "nc"  1 0
              6180  .6150057483911646 "nc"  1 0
              7160  12.38556978124845 "nc"  3 0
              7080 1.3336560869358243 "nc"  3 0
              6650 1.4300558710753521 "nc"  3 0
              7940  4.936040486312466 "nc"  3 0
             10590  2.285560283724503 "nc"  3 0
              4810  .6379019006512485 "nc"  1 0
              5910 .10172782199131047 "nc"  1 0
              8610  4.725078450252779 "nc"  3 0
              6680  .8463431101088507 "nc"  2 0
              6860 1.0357571481480847 "nc"  2 0
              7630  1.854448838311463 "nc"  3 0
              3040  91.72693891176144 "nc"  3 0
              4340  90.03352982721125 "nc"  3 0
           34254.6 .05051347388196038 "nc"  1 0
           39848.4 .15006706827114039 "nc"  1 0
             42502  .8108797955127851 "nc"  2 0
           36233.9  .7376828109330235 "nc"  1 0
             36897  .9346414377752771 "nc"  2 0
           40701.2 16.456397743328935 "nc"  3 0
           34643.8 2.3534043016277897 "nc"  3 0
             41303  .6323109752132469 "nc"  1 0
           45267.6  6.465596085877682 "nc"  3 0
             50249 42.903106538453635 "nc"  3 0
           36119.9 .43771418545513907 "nc"  1 0
           36890.6 .08594027310274638 "nc"  1 0
           47784.2  9.053379009537554 "nc"  3 0
           37337.6  .7511179583907089 "nc"  1 0
           40425.8   .928748844024827 "nc"  2 0
           39739.2 2.3423877640346658 "nc"  3 0
28216.474203154732   89.9736677709193 "nc"  3 0
 28843.50696322484  91.76434419463205 "nc"  3 0
             61600  .2085018513959576 "13"  1 1
             69000  .2014138892114737 "18"  1 1
             74000 1.2771290982254766 "45"  2 1
                 0 1.0009551098376313 "33"  2 1
             64500 1.4021279178918629 "28"  3 1
             66000 10.198002759116184 "149" 3 1
             81000 2.3965958959220846 "23"  3 1
             73000 .30595178199915696 "12"  1 1
             76500  4.094888449590512 "12"  3 1
             96000  37.07914298207258 "3"   3 1
             62800 .17349063150589866 "11"  1 1
             72500  .4684846023188834 "28"  1 1
             85000  9.244558200225383 "35"  3 1
             79000 1.2183171332468152 "34"  2 1
             77500 1.5383212568171136 "50"  3 1
             80000  3.268682223143715 "108" 3 1
             86604 4.6343267360552405 "5"   3 1
            116570 11.645024425579592 "6"   3 1
end

------------------ copy up to and including the previous line ------------------

Listed 54 out of 54 observations

.
.
.
. // Multilogit and margins commands
. mlogit testclass income dicho, base(1) nolog

Multinomial logistic regression Number of obs = 54
LR chi2(4) = 2.58
Prob > chi2 = 0.6304
Log likelihood = -52.547628 Pseudo R2 = 0.0240

------------------------------------------------------------------------------
testclass | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
1 | (base outcome)
-------------+----------------------------------------------------------------
2 |
income | -8.88e-06 .0000214 -0.42 0.678 -.0000507 .000033
dicho | .9468991 1.366389 0.69 0.488 -1.731175 3.624973
_cons | -.8557658 .6622474 -1.29 0.196 -2.153747 .4422152
-------------+----------------------------------------------------------------
3 |
income | .0000173 .0000173 1.00 0.316 -.0000165 .0000512
dicho | -.3893059 1.10068 -0.35 0.724 -2.546598 1.767986
_cons | -.2064702 .5378888 -0.38 0.701 -1.260713 .8477725
------------------------------------------------------------------------------

. margins, dydx(*)

Average marginal effects Number of obs = 54
Model VCE : OIM

dy/dx w.r.t. : income dicho
1._predict : Pr(testclass==1), predict(pr outcome(1))
2._predict : Pr(testclass==2), predict(pr outcome(2))
3._predict : Pr(testclass==3), predict(pr outcome(3))

------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
income |
_predict |
1 | -2.48e-06 3.42e-06 -0.72 0.469 -9.19e-06 4.23e-06
2 | -2.32e-06 2.38e-06 -0.98 0.329 -6.98e-06 2.34e-06
3 | 4.80e-06 3.60e-06 1.33 0.183 -2.26e-06 .0000119
-------------+----------------------------------------------------------------
dicho |
_predict |
1 | .0157943 .2235704 0.07 0.944 -.4223956 .4539842
2 | .1445371 .1537585 0.94 0.347 -.1568241 .4458982
3 | -.1603314 .2379593 -0.67 0.500 -.6267231 .3060602
------------------------------------------------------------------------------

. margins, eyex(*)

Average marginal effects Number of obs = 54
Model VCE : OIM

ey/ex w.r.t. : income dicho
1._predict : Pr(testclass==1), predict(pr outcome(1))
2._predict : Pr(testclass==2), predict(pr outcome(2))
3._predict : Pr(testclass==3), predict(pr outcome(3))

------------------------------------------------------------------------------
| Delta-method
| ey/ex Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
income |
_predict |
1 | -.3409894 .462768 -0.74 0.461 -1.247998 .5660192
2 | -.6924964 .7032758 -0.98 0.325 -2.070892 .6858989
3 | .3455409 .2587482 1.34 0.182 -.1615963 .852678
-------------+----------------------------------------------------------------
dicho |
_predict |
1 | .0194882 .2456872 0.08 0.937 -.4620498 .5010262
2 | .3351212 .3353336 1.00 0.318 -.3221205 .9923629
3 | -.1102804 .1668992 -0.66 0.509 -.4373968 .2168359
------------------------------------------------------------------------------

.
.
. // Computing eyex margins manually
. mat list e(b)
. matrix a = e(b)
. mat a1 = a[1, "1:"]
. mat a2 = a[1, "2:"]
. mat coeff = (a1 \ a2)'
. mat list coeff

coeff[3,2]
y1 y1
1:o.income 0 -8.881e-06
1:o.dicho 0 .94689907
1:o._cons 0 -.85576583

.
. predict double prob_1 prob_2 prob_3, pr
. gen p=.
. forval i = 1/3 {
2. replace p=prob_`i'
3. gen elasticity_income`i' = coeff[`i', 1] *income* (1 - p)
4. gen elasticity_dicho`i' = coeff[`i', 2] *dicho* (1 - p)
5. }

. sum elasticity_*

Variable | Obs Mean Std. Dev. Min Max
-------------+---------------------------------------------------------
elasticit~e1 | 54 0 0 0 0
elasticit~o1 | 54 -2.14e-06 3.06e-06 -7.28e-06 0
elasticit~e2 | 54 0 0 0 0
elasticit~o2 | 54 .2630275 .3772945 0 .8805537
elasticit~e3 | 54 0 0 0 0
-------------+---------------------------------------------------------
elasticit~o3 | 54 -.1267801 .1881919 -.6775556 0

.
end of do-file

[Andrew Musau]

See https://www.stata.com/support/faqs/s...fects-methods/. The elasticity is related to derivative in the following way: https://en.wikipedia.org/wiki/Elasticity_of_a_function.

[Justin Ndoutoumou]

Many thanks, Andrew. I am taking a look at this material and will give a feedback.