Marginal effects in probit model - age and age2

[Annalena Juenger]

Hey everyone,

I am doing probit regressions with Stata 14.1 and I want to get average marginal effects for the variables age and age squared. I tried two different ways:

1)

probit y c.age c.age2 x3 x4 ...
margins, dydx(*)

2)

probit y c.age##c.age x3 x4 ...
margins, dydx(*)

In 1) i get two different results for age and age2, whereas I only have one result for age (in total) in 2).

Why is there only one result in 2) and which notification would be appropriate to identfy an effect that is u-shaped for age?

Thanks a for any suggestions! :-)

[Stephen Jenkins]

(1) is not legitimate. (2) is the way to go

. sysuse auto
(1978 Automobile Data)

. probit foreign c.mpg##c.mpg

Iteration 0:   log likelihood =  -45.03321  
Iteration 1:   log likelihood = -39.273692  
Iteration 2:   log likelihood = -39.258883  
Iteration 3:   log likelihood =  -39.25888  

Probit regression                               Number of obs     =         74
                                                LR chi2(2)        =      11.55
                                                Prob > chi2       =     0.0031
Log likelihood =  -39.25888                     Pseudo R2         =     0.1282

------------------------------------------------------------------------------
     foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |   .0934728   .1933909     0.48   0.629    -.2855663    .4725119
             |
 c.mpg#c.mpg |    .000054   .0039903     0.01   0.989    -.0077668    .0078748
             |
       _cons |  -2.606064   2.261608    -1.15   0.249    -7.038733    1.826606
------------------------------------------------------------------------------


. margins

Predictive margins                              Number of obs     =         74
Model VCE    : OIM

Expression   : Pr(foreign), predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .2972403   .0486994     6.10   0.000     .2017913    .3926893
------------------------------------------------------------------------------


. margins, at(mpg==21)

Adjusted predictions                            Number of obs     =         74
Model VCE    : OIM

Expression   : Pr(foreign), predict()
at           : mpg             =          21

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |    .267857   .0630984     4.25   0.000     .1441864    .3915276
------------------------------------------------------------------------------

[Annalena Juenger]

Thanks for your reply!

If I get one marginal effect for age and age2, how can I interpret this?

My dependant variable is participation in further training. I get a negative average marginal effect for age and age2 (let's say -0.22), that is significant.

Can I interpret like this: If age increases by one year, the probability of taking part in further training decreases c. p. on average by 0.22 percentage points.

How is the quadratic term involved in this??

[Phil Bromiley]

The easiest way to do this is by using factor variable notation to do the estimate and then predictive margins as Stephen explained.
margins, at(mpg=(21 22)) will give you the predicted value for mgp at 21 and 22. With a non-linear model (like profit) and a non-linear rhs variable (like x and x-squared) the impact of a one unit change will vary with the level of the change so you might want to do predictive margins for the entire range of your x variable.

[Stephen Jenkins]

How is the quadratic term involved in this??

-margins- understands factor variable notation, and so the prediction involved uses the correct formula -- based on all the terms in the model including main effect and its square. Phil's suggestion to look at the whole range can be accomplished by the following. Use the help for -margins- and -marginsplot- too.

. sysuse auto
(1978 Automobile Data)

. probit foreign c.mpg##c.mpg

Iteration 0:   log likelihood =  -45.03321  
Iteration 1:   log likelihood = -39.273692  
Iteration 2:   log likelihood = -39.258883  
Iteration 3:   log likelihood =  -39.25888  

Probit regression                               Number of obs     =         74
                                                LR chi2(2)        =      11.55
                                                Prob > chi2       =     0.0031
Log likelihood =  -39.25888                     Pseudo R2         =     0.1282

------------------------------------------------------------------------------
     foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |   .0934728   .1933909     0.48   0.629    -.2855663    .4725119
             |
 c.mpg#c.mpg |    .000054   .0039903     0.01   0.989    -.0077668    .0078748
             |
       _cons |  -2.606064   2.261608    -1.15   0.249    -7.038733    1.826606
------------------------------------------------------------------------------

. su mpg

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
         mpg |         74     21.2973    5.785503         12         41


. margins, at(mpg=(12(1)41))

Adjusted predictions                            Number of obs     =         74
Model VCE    : OIM

Expression   : Pr(foreign), predict()

1._at        : mpg             =          12

2._at        : mpg             =          13

3._at        : mpg             =          14

4._at        : mpg             =          15

5._at        : mpg             =          16

6._at        : mpg             =          17

7._at        : mpg             =          18

8._at        : mpg             =          19

9._at        : mpg             =          20

10._at       : mpg             =          21

11._at       : mpg             =          22

12._at       : mpg             =          23

13._at       : mpg             =          24

14._at       : mpg             =          25

15._at       : mpg             =          26

16._at       : mpg             =          27

17._at       : mpg             =          28

18._at       : mpg             =          29

19._at       : mpg             =          30

20._at       : mpg             =          31

21._at       : mpg             =          32

22._at       : mpg             =          33

23._at       : mpg             =          34

24._at       : mpg             =          35

25._at       : mpg             =          36

26._at       : mpg             =          37

27._at       : mpg             =          38

28._at       : mpg             =          39

29._at       : mpg             =          40

30._at       : mpg             =          41

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .0698902   .0746504     0.94   0.349     -.076422    .2162023
          2  |   .0835187   .0721787     1.16   0.247     -.057949    .2249864
          3  |   .0990725   .0685299     1.45   0.148    -.0352436    .2333887
          4  |    .116667   .0641047     1.82   0.069     -.008976    .2423099
          5  |   .1363935   .0595545     2.29   0.022     .0196688    .2531182
          6  |   .1583141   .0557889     2.84   0.005     .0489699    .2676582
          7  |   .1824554    .053839     3.39   0.001     .0769329    .2879779
          8  |   .2088048   .0544779     3.83   0.000     .1020302    .3155794
          9  |    .237306    .057782     4.11   0.000     .1240553    .3505567
         10  |    .267857   .0630984     4.25   0.000     .1441864    .3915276
         11  |   .3003093   .0694473     4.32   0.000      .164195    .4364236
         12  |   .3344687   .0759274     4.41   0.000     .1856538    .4832836
         13  |   .3700977   .0819112     4.52   0.000     .2095547    .5306407
         14  |   .4069203   .0871062     4.67   0.000     .2361954    .5776452
         15  |   .4446279   .0915651     4.86   0.000     .2651636    .6240922
         16  |   .4828868   .0956674     5.05   0.000     .2953822    .6703914
         17  |   .5213472   .1000482     5.21   0.000     .3252562    .7174381
         18  |   .5596522   .1054493     5.31   0.000     .3529755     .766329
         19  |   .5974484   .1124974     5.31   0.000     .3769575    .8179394
         20  |   .6343952   .1214906     5.22   0.000     .3962781    .8725123
         21  |    .670174   .1322935     5.07   0.000     .4108834    .9294645
         22  |   .7044969   .1443857     4.88   0.000     .4215061    .9874876
         23  |   .7371133   .1570025     4.69   0.000      .429394    1.044833
         24  |   .7678156   .1692897     4.54   0.000     .4360139    1.099617
         25  |   .7964425   .1804229     4.41   0.000     .4428201    1.150065
         26  |   .8228804   .1896863     4.34   0.000     .4511021    1.194659
         27  |   .8470642   .1965189     4.31   0.000     .4618943    1.232234
         28  |   .8689745   .2005363     4.33   0.000     .4759305    1.262018
         29  |   .8886344   .2015373     4.41   0.000     .4936284     1.28364
         30  |   .9061053   .1994962     4.54   0.000     .5150999    1.297111
------------------------------------------------------------------------------


. marginsplot

  Variables that uniquely identify margins: mpg

Graph not shown here, but you'll see a non-linear relationship

[Annalena Juenger]

Thanks a lot to both of you!

I did as you suggested.. the predictive margins and the marginsplot for the entire range of the age variable made the inversed u-shape clear.