Negative marginal effects

[bmee inthakesone]

Dear Statalist;
I am a new member of this site and also stata program. I used probit model for my paper. is there any problem when my marginal effects turn to negative? How can I interpret is?
Your contribution is highly aprriciated

Best
Bounmy

[Carlo Lazzaro]

bmee:
welcome to the list.
Your chance of getting helpful repiies is conditional on posting what you typed and what Stata gave you back (as per FAQ). Thanks.

[Julian Pritsch]

First, I agree with Carlo: Please post the code and output so we can see what you did.

Now turning to your problem:
I read somewhere that marginal effects can turn negative. If I recall it correctly, the argument was that marginal effects are technically linear predictors and therefore can turn below zero. But I would be interested in a more technical discussion about this matter, too.

I am sure that the well known experts on non-linear modeling within this forum will jump in on this matter.

[Joshua D Merfeld]

I'm afraid I don't see what the problem here is. If by marginal effects we are talking about the change in the (conditional) probability of y based on a one-unit change in x, then there is no reason they can't be negative.

[Julian Pritsch]

Joshua D Merfeld:This is an interesting point. In my opinion the difficulties lie in the word "probabilities". Since probabilities have a range from 0 to 1, it is difficult to understand why predicted probabilities might be below zero. The important phrase in your statement is that marginal effects in logit/probit models describe the

... change in the (conditional) probability..

Therefore, the change (or difference?) in probabilities might indeed be negative.

Am I correct so far?

But why is that so seldom the case? In my experience the case of negative predicted probabilities (marginal effects) in non-linear models is rather rare.

[Maarten Buis]

You need to make a distinction between marginal effect and marginal prediction: the first is a change in predicted probability (in case of a logistic regression) and the latter is the predicted probability itself (also in case of logistic regression). The default for margins is marginal prediction and these are forced to remain between 0 and 1 if you estimated a logistic regression. If you specify the dydx() option you will get (average) marginal effects and these can be negative without problem.

As to why negative marginal effects happen so rarely, I just disagree that these happen rarely. It can happen that in your particular sub-field there are just many more positive effect than negative effects, or that there is a tradition in your sub-field to code variables in such a way that the effects are (likely to be) positive.

[Julian Pritsch]

Thanks Maarten,
this explains a lot.

So for example:

use http://www.stata-press.com/data/r13/nhanes2

logistic highbp i.sex c.age c.bmi

Logistic regression                               Number of obs   =      10351
                                                  LR chi2(3)      =    2410.54
                                                  Prob > chi2     =     0.0000
Log likelihood = -5845.4948                       Pseudo R2       =     0.1709

------------------------------------------------------------------------------
      highbp | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         sex |
     Female  |   .6172715   .0278625   -10.69   0.000     .5650079    .6743695
         age |   1.047058   .0014632    32.91   0.000     1.044194     1.04993
         bmi |   1.147006    .005827    27.00   0.000     1.135642    1.158484
       _cons |   .0029059   .0004428   -38.33   0.000     .0021556    .0039174
------------------------------------------------------------------------------

. margins sex

Predictive margins                                Number of obs   =      10351
Model VCE    : OIM

Expression   : Pr(highbp), predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         sex |
       Male  |   .4699816   .0062053    75.74   0.000     .4578195    .4821436
     Female  |   .3771019   .0059595    63.28   0.000     .3654215    .3887823
------------------------------------------------------------------------------

. margins, dydx(sex)

Average marginal effects                          Number of obs   =      10351
Model VCE    : OIM

Expression   : Pr(highbp), predict()
dy/dx w.r.t. : 2.sex

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         sex |
     Female  |  -.0928796   .0086035   -10.80   0.000    -.1097421   -.0760172
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

Here after margins sex the output shows the two positive values for males

(.4699816 )

and for female

(.3771019  )

which are "marginal predictions". The calculations after margins, dydx(sex) yield a negative value. This is a way of testing if the difference of the "marginal predictions" between male and female

di 0.3771019 - 0.4699816
-.0928797

is significant. The marginal effect ( in this case the change from the dummy variable from 1 (male) to 2 (female)) is therefore negative. But it could have been positive, if the dummy variable would have been coded the other way around.

See ouput:

tab sex, nolab

    1=male, |
   2=female |      Freq.     Percent        Cum.
------------+-----------------------------------
          1 |      4,915       47.48       47.48
          2 |      5,436       52.52      100.00
------------+-----------------------------------
      Total |     10,351      100.00


recode sex (1=2) (2=1), gen(sex_r)
(10351 differences between sex and sex_r)

tab sex_r

  RECODE of |
        sex |
   (1=male, |
  2=female) |      Freq.     Percent        Cum.
------------+-----------------------------------
          1 |      5,436       52.52       52.52
          2 |      4,915       47.48      100.00
------------+-----------------------------------
      Total |     10,351      100.00



logistic highbp i.sex_r c.age c.bmi

Logistic regression                               Number of obs   =      10351
                                                  LR chi2(3)      =    2410.54
                                                  Prob > chi2     =     0.0000
Log likelihood = -5845.4948                       Pseudo R2       =     0.1709

------------------------------------------------------------------------------
      highbp | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     2.sex_r |   1.620033   .0731254    10.69   0.000     1.482867    1.769887
         age |   1.047058   .0014632    32.91   0.000     1.044194     1.04993
         bmi |   1.147006    .005827    27.00   0.000     1.135642    1.158484
       _cons |   .0017937   .0002805   -40.44   0.000     .0013203     .002437
------------------------------------------------------------------------------




margins sex_r

Predictive margins                                Number of obs   =      10351
Model VCE    : OIM

Expression   : Pr(highbp), predict()

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       sex_r |
          1  |   .3771019   .0059595    63.28   0.000     .3654215    .3887823
          2  |   .4699816   .0062053    75.74   0.000     .4578195    .4821436
------------------------------------------------------------------------------

margins, dydx(sex_r)

Average marginal effects                          Number of obs   =      10351
Model VCE    : OIM

Expression   : Pr(highbp), predict()
dy/dx w.r.t. : 2.sex_r

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     2.sex_r |   .0928796   .0086035    10.80   0.000     .0760172    .1097421
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

This time the marginal effect (margins, dydx(sex_r)) is positive.

[Clyde Schechter]

Yes, of course, if you reverse the coding of a dichotomous predictor variable you will change the sign of the corresponding marginal effect. This is, I think, what Maarten was referring to when he said "or that there is a tradition in your sub-field to code variables in such a way that the effects are (likely to be) positive." One always has a choice which we to code such predictors, and coding them in a way that is expected to result in a positive marginal effect is a frequent preference in some settings.

[Joshua D Merfeld]

And of course if you use odds ratios as the output then the coefficients will never be negative, although with a logit (0,1 dependent variable) any log odds less than one will result in negative marginal effects (as you see in your example with female above). I would also disagree with the statement that negative marginal effects are rare; they happen all the time unless, as Clyde and Maarten commented, you code your variables specifically to avoid negative effects (like, as in the example Clyde gave, simply reversing the coding of your dichotomous sex variable). So, if you specify the -dydx- option, then marginal effects can absolutely be negative, but if you omit the -dydx- option, then the default, marginal probabilities, will never be negative.

[bmee inthakesone]

Thank you all for your kind contribution

[Anneke Boesche]

Thanks for all the great answers! But can you also explain how to interpret negative marginal effects in a probit function? For instance, I have the following result of -0.16537284 of price and I would like to interpret the price dummy on the dependent variable of the "willingness to pay higher prices for the protection of the environment". What would this negative marginal coefficient tell me?

[Emad Shehata]

Interpretation of probit model marginal effect

one unit change in x variable, will increases or decrease the probability of y variable by #

[Emad Shehata]

Interpretation of probit model elasticity

one unit percentage 1% change in x variable, will increases or decrease the probability of y variable by #%

[Anneke Boesche]

thanks a lot! does this also count for a linear probability model?

[Mischa Kleis]

Dear colleagues,

looking for an answer to my econometric issue, I decided to join Statalist.

My issue: oprobit with dependent variable "SWB" coded 0-10 => all positive (negative) coefficients turn to negative (positive) marginal effects at mean/APE. Why could that happen?
In the margins output it is written "Pr(swb==0)". According to Greene "Assuming that β is positive (for this x), Prob(y = 0|x) must decline. Alternatively, from the previous expression, it is obvious that the derivative of Prob(y = 0 | x) has the opposite sign from β."
Is that the crucial point?

How can I interpret my results (coefficients and marginal effects)? I don't want to just refer to Pr(swb==0) and talk about the whole range of possible values from 0 to 10.

In the following you can see what happens to the predictors "AGE" and "AGE2".

Code for the oprobit command:

oprobit swb age age_sq ib0.female_rec ib1.corigin_rec log_hh_oec
> d_equ_inc ib1.lf_emp_rec hours_worked_act hours_worked_act_sq ib
> 1.educ_rec ib2.marital_status ib0.no_children_hh_rec ib1.health_
> subj_rec ib1.worries_finances_rec if syear==2005, vce(cluster pi
> d)

             |               Robust
         swb |      Coef.   Std. Err.      z    P>|z|     [95% Con
> f. Interval]
-------------+----------------------------------------------------
> ------------
         age |  -.0837142   .0143802    -5.82   0.000    -.1118989
>    -.0555294
      age_sq |   .0982188   .0174849     5.62   0.000      .063949
>     .1324885

Code for the APE effects:

margins, dydx(*) post

Average marginal effects                          Number of obs  
> =       5519
Model VCE    : Robust

Expression   : Pr(swb==0), predict()
dy/dx w.r.t. : age age_sq 1.female_rec 2.corigin_rec
3.corigin_rec 4.corigin_rec 5.corigin_rec
6.corigin_rec 7.corigin_rec 8.corigin_rec
log_hh_oecd_equ_inc 2.lf_emp_rec 3.lf_emp_rec
4.lf_emp_rec 5.lf_emp_rec hours_worked_act
hours_worked_act_sq 2.educ_rec 3.educ_rec
4.educ_rec 5.educ_rec 1.marital_status
3.marital_status 4.marital_status
5.marital_status 1.no_children_hh_rec
2.no_children_hh_rec 3.no_children_hh_rec
4.no_children_hh_rec 2.health_subj_rec
3.health_subj_rec 4.health_subj_rec
5.health_subj_rec 2.worries_finances_rec
3.worries_finances_rec
> ------------
Delta-method
dy/dx   Std. Err.      z    P>z     [95% Con
> f. Interval]

> ------------
age    .0008576   .0002034     4.22   0.000      .000459
>     .0012562
age_sq   -.0010062   .0002425    -4.15   0.000    -.0014815
>    -.0005309

The same happens to marginal effects at the mean - and for all predictors.
Thanks a lot for your support!

Kind regards,
Mischa