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I am using a probit model, and margins says that my marginal effect is greater than 1. Can that be correct?

Title   Marginal effects of probabilities greater than 1
Authors May Boggess, StataCorp
Kristin MacDonald, StataCorp
Date April 2004; updated July 2009

The marginal effect of an independent variable is the derivative (that is, the slope) of the prediction function, which, by default, is the probability of success following probit. By default, margins evaluates this derivative for each observation and reports the average of the marginal effects. We can specify the point at which we want the marginal effect to be evaluated by using the at() option.

The important thing to remember is the slope of a function can be greater than one, even if the values of the function are all between 0 and 1.

Here are some examples:

 . sysuse auto, clear
 (1978 Automobile Data)
 
 . probit foreign mpg

 Iteration 0:   log likelihood =  -45.03321
 Iteration 1:   log likelihood = -39.264978
 Iteration 2:   log likelihood = -39.258972
 Iteration 3:   log likelihood = -39.258972

 Probit regression                                 Number of obs   =         74
                                                   LR chi2(1)      =      11.55
                                                   Prob > chi2     =     0.0007
 Log likelihood = -39.258972                       Pseudo R2       =     0.1282
 
 ------------------------------------------------------------------------------
      foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
          mpg |   .0960601   .0301523     3.19   0.001     .0369627    .1551575
        _cons |  -2.635268   .6841462    -3.85   0.000     -3.97617   -1.294366
 ------------------------------------------------------------------------------

 . margins, dydx(mpg) at(mpg=25)

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

 Expression   : Pr(foreign), predict()
 dy/dx w.r.t. : mpg
 at           : mpg             =          25
 
 ------------------------------------------------------------------------------
              |            Delta-method
              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
          mpg |   .0372895   .0125636     2.97   0.003     .0126653    .0619137
 ------------------------------------------------------------------------------
 
 . predict pmpg, p
 
 . sort mpg
 
 . twoway scatter pmpg mpg
graph

The graph is not steep at mpg=25, and that shows the result we see from margins, which is that the marginal effect is small. By comparison

 . sysuse auto, clear
 (1978 Automobile Data)

 . probit foreign gear_ratio

 Iteration 0:   log likelihood =  -45.03321
 Iteration 1:   log likelihood = -22.664339
 Iteration 2:   log likelihood = -21.653347
 Iteration 3:   log likelihood = -21.641904
 Iteration 4:   log likelihood = -21.641897
 Iteration 5:   log likelihood = -21.641897
 
 Probit regression                                 Number of obs   =         74
                                                   LR chi2(1)      =      46.78
                                                   Prob > chi2     =     0.0000
 Log likelihood = -21.641897                       Pseudo R2       =     0.5194
 
 ------------------------------------------------------------------------------
      foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
   gear_ratio |    3.45954   .7132767     4.85   0.000     2.061543    4.857537
        _cons |  -11.44249    2.30258    -4.97   0.000    -15.95546   -6.929517
 ------------------------------------------------------------------------------

 . margins, dydx(gear_ratio) at(gear_ratio=3.3)

 Conditional marginal effects                      Number of obs   =         74
 Model VCE    : OIM
 
 Expression   : Pr(foreign), predict()
 dy/dx w.r.t. : gear_ratio
 at           : gear_ratio      =         3.3
 
 ------------------------------------------------------------------------------
              |            Delta-method
              |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
   gear_ratio |    1.37969   .2867635     4.81   0.000     .8176435    1.941736
 ------------------------------------------------------------------------------
graph

Here we see the graph is quite steep at gear_ratio=3.3, so the marginal effect is large.

Many people expect the marginal effect to be less than one because we learn in calculus class that the derivative is the approximate change in y for a one-unit change in x. Because y is between 0 and 1, the change in y obviously cannot be greater than 1!

The issue comes from the word approximately. Remember the derivative at a point is the slope of the tangent line of the curve at that point.

Let’s draw the tangent line, at the point gear_ratio=3.3, on the graph produced by our last example.

graph

Now we see that the change in the y value on the line, between gear_ratio=3 and gear_ratio=4, is greater than 1 (because at gear_ratio=4, the line has y value greater than 1).

The approximation of a curve by a tangent line is good close to the point where the tangent is drawn, but if the slope of the curve is changing quickly, this approximation is not very good further away from the point.

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